\label{HI_theorem}
Suppose $p,q$ satisfy (\ref{eq_range_q}). Then there exists $\de_0=\de_0(\mathit{data})\in(0,1)$ with $2-q(1-\delta_0)>0$
such that if $u$ is a very weak solution to (\ref{11}) with integrability deficit $\delta\in(\de_0,1)$ and $u\in C(0,T;L^2(\Omega_T,\mathbb{R}^N))$, we have
	\begin{align*}
		\begin{split}
int_{Q_{R} (z_0) }H(z,|\nabla u|)\,dz
int_{Q_{2R} (z_0) } (  H(z,|\nabla u|)^\delta+ H(z,|F|) ) \,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
		\end{split}
	\end{align*}
	 for every $Q_{2R}\subset\Om_T$ where $c\ge
     1$ depending on $\mathit{data},\|a\|_{L^\infty(\Om_T)},\| (H(z,|\nabla u|)+H(z,|F|)+1 )\|_{L^\delta(\Om_T)}$, and $R\in(0,R_0)$ for $R_0\in(0,1)$ depending on $\mathit{data}$, $\delta$, $\| H(z,|\nabla u|) \|_{L^\delta(\Om_T)}$, $\| H(z,|F|) \|_{L^\delta(\Om_T)}$, $\|u\|_{C(0,T;L^2(\Omega,\mathbb{R}^N))}$.

\begin{align}\label{def_E}
	E(\La)=\left\{z\in \RR^{n+1}: \Upsilon(z)\le\La\right\}.
\end{align}

\begin{definition}\label{def_weak}
A function $u\in L^1(0,T;W^{1,1}(\Omega,\RR^N))$ satisfying
\[
\iint_{\Omega_T} H(z,|\na u|)^\delta\,dz <\infty
\]
is a very weak solution to \eqref{11} with integrability deficit $\de \in (0,1)$, if
\[
    \iint_{\Omega_T}\left(-u\cdot \varphi_t+\mA(z,\na u)\cdot \na\varphi\right)\,dz
    =
    \iint_{\Om_T} \left( |F|^{p-2}F\cdot \na \varphi+a(z)|F|^{q-2}F\cdot \na\varphi  \right)\,dz
\]
for every $\varphi\in C_0^\infty(\Omega_T,\RR^N)$.
\end{definition}

\begin{align}\label{11}
	u_t-\dv\mA(z,\na u)=
    -\dv (|F|^{p-2}F+a(z)|F|^{q-2}F)
    \quad\text{in}\ \Omega_T,
\end{align}

\begin{theorem}\label{HI_theorem}
Suppose $p,q$ satisfy \eqref{eq_range_q}. Then there exists $\de_0=\de_0(\mathit{data})\in(0,1)$ with $2-q(1-\delta_0)>0$
such that if $u$ is a very weak solution to \eqref{11} with integrability deficit $\de\in(\de_0,1)$ and $u\in C(0,T;L^2(\Omega_T,\mathbb{R}^N))$, we have
	\begin{align*}
		\begin{split}
int_{Q_{R} (z_0) }H(z,|\na u|)\,dz
int_{Q_{2R} (z_0) } (  H(z,|\na u|)^\de+ H(z,|F|) ) \,dz+1\right)^{1+\frac{q(1-\de)}{2-q(1-\de)}}
		\end{split}
	\end{align*}
	 for every $Q_{2R}\subset\Om_T$ where $c\ge
     1$ depending on $\data,\|a\|_{L^\infty(\Om_T)},\| (H(z,|\na u|)+H(z,|F|)+1 )\|_{L^\delta(\Om_T)}$, and $R\in(0,R_0)$ for $R_0\in(0,1)$ depending on $\data$, $\delta$, $\| H(z,|\na u|) \|_{L^\delta(\Om_T)}$, $\| H(z,|F|) \|_{L^\delta(\Om_T)}$, $\|u\|_{C(0,T;L^2(\Omega,\mathbb{R}^N))}$.
\end{theorem}

In this paper, we study parabolic double-phase systems with the prototype equation
\[
u_t-\dv(|\nabla u|^{p-2}\nabla u + a(z)|\nabla u|^{q-2}\nabla u)=-\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),
\]
in $\Omega_T=\Omega\times(0,T)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $T>0$. Here $1<p<q<\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.

In the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\equiv 0$) in \cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed.
We refer to DiBenedetto’s book \cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \cite{MR1939688}.

This approach with intrinsic geometry has also been applied to other classes of nonlinear parabolic PDEs.
In the setting of porous medium equations, DiBenedetto, Gianazza and Vespri proved Hölder continuity and Harnack inequalities in \cite{MR2413134,MR2414742,MR2731161,MR2865434}, while Gianazza and Schwarzacher obtained gradient higher integrability in \cite{MR3928041,MR4019094}.
The same methodology has been adapted to equations that are nonlinear in both time and spatial derivative parts.
For doubly nonlinear equations, higher integrability, Hölder continuity and gradient Hölder continuity have been developed in a sequence of works \cite{bögelein2025schauderestimatesparabolicplaplace, MR4163123,  MR4287785, MR4603642, MR4500278,MR4373238, MR4797292} by Bögelein, Duzaar, Liao, Moring, Schätzler, Scheven and their collaborators.

Regarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \cite{MR3360738,MR3294408}.
Moreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.

The exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \begin{align}\label{11}
	u_t-\dv\mathcal{A}(z,\nabla u)=
    -\dv (|F|^{p-2}F+a(z)|F|^{q-2}F)
    \quad\text{in}\ \Omega_T,
\end{align}
where $\mathcal{A}(z,\nabla u):\Omega_T\times \mathbb{R}^{Nn}\longrightarrow \mathbb{R}^{Nn}$ with $N\ge1$ is a Carath\'eodory vector field satisfying the structure assumptions
\begin{align}\label{12}
		\mathcal{A}(z,\xi)\cdot \xi\ge L_1(|\xi|^p+a(z)|\xi|^q)
      \quad\text{and}\quad
		|\mathcal{A}(z,\xi)|\le L_2(|\xi|^{p-1}+a(z)|\xi|^{p-1})
\end{align}
for a.e. $z\in \Omega_T$ and every $\xi\in \mathbb{R}^{Nn}$ with constants $0<L_1\le L_2<\infty$. We assume that the coefficient function $a:\Omega_T\longrightarrow\mathbb{R}^+$ is non-negative and satisfies $a\in C^{\alpha,\alpha/2}(\Omega_T)$ for some $\alpha \in (0,1]$. Here $a\in C^{\alpha,\alpha/2}(\Omega_T)$ means that $a\in L^{\infty}(\Omega_T)$ and there exists a constant $[a]_{\alpha,\alpha/2;\Omega_T}=[a]_\alpha>0$, such that
\begin{align}\label{eq_holder_reg}
		|a(x,t)-a(y,s)|\le [a]_{\alpha,\alpha/2;\Omega_T}\max\{ |x-y|^\alpha,|t-s|^\frac{\alpha}{2} \}
\end{align}
for every $(x,y)\in\Omega$ and $(t,s)\in (0,T)$.
For the source term $F$, we assume that $F:\Om_T\longrightarrow\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\in L^1(\Om_T)$. Throughout the paper, we denote $H(z,s):\Omega_T\times \mathbb{R}^{+ }\longrightarrow \mathbb{R}^+$ as
\begin{align*}
	H(z,s)=s^p+a(z)s^q.
\end{align*}
Regarding the exponents, we assume that
\begin{align}\label{eq_range_q}
    q < p+\frac{2\alpha}{n+2} \quad \text{and} \quad p>2.
\end{align}

\begin{definition}\label{def_weak}
A function $u\in L^1(0,T;W^{1,1}(\Omega,\mathbb{R}^N))$ satisfying
\[
\iint_{\Omega_T} H(z,|\nabla u|)^\delta\,dz <\infty
\]
is a very weak solution to (\ref{11}) with integrability deficit $\delta \in (0,1)$, if
\[
    \iint_{\Omega_T}\left(-u\cdot \varphi_t+\mathcal{A}(z,\nabla u)\cdot \nabla\varphi\right)\,dz
    =
    \iint_{\Om_T} \left( |F|^{p-2}F\cdot \nabla \varphi+a(z)|F|^{q-2}F\cdot \nabla\varphi  \right)\,dz
\]
for every $\varphi\in C_0^\infty(\Omega_T,\mathbb{R}^N)$.
\end{definition}

Our main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\ref{11}) is actually a weak solution, i.e., satisfies Definition~\ref{def_weak} with $\delta = 1$. The constants depend on
\[
\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\alpha)
\]
and here, as in the rest of the paper, we denote $c=c(\mathit{data})$ for a positive constant depending at most on the constants above.

In this paper, we study parabolic double-phase systems with the prototype equation
\[
u_t-\dv(|\nabla u|^{p-2}\nabla u + a(z)|\nabla u|^{q-2}\nabla u)=-\dv(|F|^{p-2}F+a(z)|F|^{q-2}F),
\]
in $\Omega_T=\Omega\times(0,T)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $T>0$. Here $1<p<q<\infty$ and $a$ is a nonnegative function that is Hölder continuous in both space and time directions. Before stating our main theorem, we discuss the context of our methods.

In the regularity theory of parabolic PDEs, the classical techniques of DeGiorgi and Moser developed for elliptic equations have been extended to the linear parabolic setting ($p=2$ and $a\equiv 0$) in \cite{MR241822,MR159139}. Although this extension was already nontrivial in the linear case, additional difficulties arise when dealing with nonlinear parabolic equations. These challenges demand analytical tools that can handle nonlinear structures. A major breakthrough in this direction was the introduction of intrinsic geometries for nonlinear parabolic PDEs, first established by DiBenedetto in the study of parabolic $p$-Laplace equations ($a\equiv0$). Intrinsic geometries compensate for the lack of homogeneity in energy estimates and provide scaling invariant structures essential for nonlinear analysis. Based on this framework, various regularity properties have been developed.
We refer to DiBenedetto’s book \cite{MR1230384} for the proofs of Hölder and gradient Hölder continuity. Furthermore, Harnack inequalities were derived by DiBenedetto, Gianazza and Vespri in \cite{MR2413134,MR2414742,MR2731161,MR2865434}, while gradient higher integrability was proved by Kinnunen and Lewis in \cite{KL_HI}. In addition, Calderón–Zygmund type estimates were obtained by Acerbi and Mingione in \cite{MR2286632} and Schauder estimates were established by Mingione and Kuusi in \cite{MR2968162,MR3273649,MR3187676,MR3191979} and Misawa in \cite{MR1939688}.

Regarding the equation considered in this paper, the elliptic double-phase model was studied by Esposito, Leonetti, and Mingione in \cite{MR2076158} within the framework of nonstandard growth conditions introduced by Marcellini in \cite{MR969900,MR1094446}. The corresponding weak solution space was analyzed by Fonseca, Malý, and Mingione in \cite{MR2058167}. As the name double-phase indicates, the growth condition distinguishes two phases depending on whether the coefficient $a$ vanishes or not at a given point. The regularity theory for elliptic double-phase models was developed after the phase analysis introduced by Colombo and Mingione in \cite{MR3360738,MR3294408}.
Moreover, they established higher integrability, Hölder continuity, and gradient Hölder continuity. They also proved Calderón–Zygmund type estimates in \cite{MR3447716}, while delicate borderline cases were later shown by De Filippis and Mingione in \cite{MR3985927}. Furthermore, the Harnack inequality was established by Baroni, Colombo, and Mingione in \cite{MR3348922}. In the spirit of nonlinear parabolic equations, our main result will be proved by combining intrinsic geometry with this phase analysis framework.

The exact definition of the solutions that we study is the following. We consider very weak solutions $u=u(z)=u(x,t)$ to the system \begin{align}\label{11}
 u_t-\dv\mathcal{A}(z,\nabla u)=
 -\dv (|F|^{p-2}F+a(z)|F|^{q-2}F)
 \quad\text{in}\ \Omega_T,
\end{align}
where $\mathcal{A}(z,\nabla u):\Omega_T\times \mathbb{R}^{Nn}\longrightarrow \mathbb{R}^{Nn}$ with $N\ge1$ is a Carath\'eodory vector field satisfying the structure assumptions
\begin{align}\label{12}
 \mathcal{A}(z,\xi)\cdot \xi\ge L_1(|\xi|^p+a(z)|\xi|^q)
 \quad\text{and}\quad
 |\mathcal{A}(z,\xi)|\le L_2(|\xi|^{p-1}+a(z)|\xi|^{p-1})
\end{align}
for a.e. $z\in \Omega_T$ and every $\xi\in \mathbb{R}^{Nn}$ with constants $0<L_1\le L_2<\infty$. We assume that the coefficient function $a:\Omega_T\longrightarrow\mathbb{R}^+$ is non-negative and satisfies $a\in C^{\alpha,\alpha/2}(\Omega_T)$ for some $\alpha \in (0,1]$. Here $a\in C^{\alpha,\alpha/2}(\Omega_T)$ means that $a\in L^{\infty}(\Omega_T)$ and there exists a constant $[a]_{\alpha,\alpha/2;\Omega_T}=[a]_\alpha>0$, such that
\begin{align}\label{eq_holder_reg}
 |a(x,t)-a(y,s)|\le [a]_{\alpha,\alpha/2;\Omega_T}\max\{ |x-y|^\alpha,|t-s|^\frac{\alpha}{2} \}
\end{align}
for every $(x,y)\in\Omega$ and $(t,s)\in (0,T)$.
For the source term $F$, we assume that $F:\Om_T\longrightarrow\mathbb{R}^{Nn}$ is a measurable map such that $(|F|^p+a|F|^q)\in L^1(\Om_T)$. Throughout the paper, we denote $H(z,s):\Omega_T\times \mathbb{R}^{+ }\longrightarrow \mathbb{R}^+$ as
\begin{align*}
 H(z,s)=s^p+a(z)s^q.
\end{align*}
Regarding the exponents, we assume that
\begin{align}\label{eq_range_q}
 q < p+\frac{2\alpha}{n+2} \quad \text{and} \quad p>2.
\end{align}

\begin{definition}\label{def_weak}
A function $u\in L^1(0,T;W^{1,1}(\Omega,\mathbb{R}^N))$ satisfying
\[
\iint_{\Omega_T} H(z,|\nabla u|)^\delta\,dz <\infty
\]
is a very weak solution to (\ref{11}) with integrability deficit $\delta \in (0,1)$, if
\[
 \iint_{\Omega_T}\left(-u\cdot \varphi_t+\mathcal{A}(z,\nabla u)\cdot \nabla\varphi\right)\,dz
 =
 \iint_{\Om_T} \left( |F|^{p-2}F\cdot \nabla \varphi+a(z)|F|^{q-2}F\cdot \nabla\varphi \right)\,dz
\]
for every $\varphi\in C_0^\infty(\Omega_T,\mathbb{R}^N)$.
\end{definition}

Our main theorem states that if the integrability deficit is close enough to 1, a very weak solution to (\ref{11}) is actually a weak solution, i.e., satisfies Definition~\ref{def_weak} with $\delta = 1$. The constants depend on
\[
\mathit{data} = (n,N,p,q,L_1,L_2,[a]_\alpha)
\]
and here, as in the rest of the paper, we denote $c=c(\mathit{data})$ for a positive constant depending at most on the constants above.

Our main theorem states that if the integrability deficit is close enough to 1, a very weak solution to \eqref{11} is actually a weak solution, i.e., satisfies Definition~\ref{def_weak} with $\de = 1$. The constants depend on
\[
\data = (n,N,p,q,L_1,L_2,[a]_\al)
\]
and here, as in the rest of the paper, we denote $c=c(\data)$ for a positive constant depending at most on the constants above.

\item[(W10)]\label{viii} If $U_i=G_{i}$, then
 $\tfrac{a(z_i)}{2}\le a(z)\le 2a(z_i)$ for every $z\in 200 Q_{r_i}(z_i)$.
 \item[(W11)]\label{ix} For any $i\in\mathbb{N}$, the cardinality of $\mathcal{I}_i$, denoted by $|\mathcal{I}_i|$, is finite. Moreover, there exists a constant $c= c(n)$, such that $|\mathcal{I}_i|\le c$.
 \end{enumerate}
 Additionally, there exists a partition of unity $\{\om_i\}_{i\in\mathbb{N}}$ subordinate to the Whitney decomposition $\{2U_i\}_{i\in \mathbb{N}}$ with the following properties:
 \begin{enumerate}[resume*=theoremconditions]
 \item[(W12)]\label{x} $0\le \om_i\le 1$, $\om_i\in C_0^\infty(2U_i)$ for every $i\in\mathbb{N}$ and $\sum_{i\in\mathbb{N}}\om_i=1$ on $E(\La)^c$.
 \item[(W13)]\label{xi}There exists a constant $c=c(n)$ such that $\lVert\na \om_j\rVert_{\infty}\le cr_i^{-1}$,
 for every $i\in\mathbb{N}$ and $j\in \mathcal{I}_i$.
 \item[(W14)]\label{xii} There exists a constant $c=c(n)$ such that
 \[
 \lVert \pa_t\om_j\rVert_{\infty}\le cs_i^{-1}
 \]
 for every $i\in\mathbb{N}$ and $j\in \mathcal{I}_i$.
 \end{enumerate}
\end{proposition}
The proof of Proposition~\ref{prop_whitney} is otherwise the same as in \cite[Proposition~3.2]{KKS}, but there is difference in the proofs of \ref{UUi} and \ref{viii}. Thus, we provide only the proofs of \ref{UUi} and \ref{viii}.
\begin{proof}[Proof of \ref{UUi}]
 Since we have $Q_i\subset E(\La)^c$, there holds
 \[
int_{Q_i} \Upsilon^\mu\,dz=\frac{\la_i^{p-2}}{2|B_1|r_i^{n+2}} \iint_{\mathbb{R}^{n+1}} \Upsilon^{\mu}\,dz.
 \]
 Multiplying both sides by $\la_i^{-p\mu}r_i^{n+2}$ and then taking exponent $\tfrac{\alpha}{n+2}$, we get
 \[
 r_i^\alpha < K\la_i^{-\frac{2\alpha}{n+2}+\frac{\alpha p}{n+2} (1-\mu)}.
 \]
 Note that by the choice of $\mu$ in \eqref{def_mu} and $\mfs$ in \eqref{def_mfs}, it follows that
 \[
 -\frac{2\alpha}{n+2}+\frac{\alpha p}{n+2} (1-\mu)
<-\frac{2\alpha}{n+2}+\frac{\alpha q}{n+2} (1-\mu)<p-q-\mfs
 \]
 and therefore,
 \[
 r_i^\alpha < K \la_i^{-\mfs}\la_i^{p-q}.
 \]
 Moreover, since we have $\theta \mfc<\theta\Pi<\La<2\la_i^p$ and \eqref{def_mfc}, it follows that
 \[
 r_i^\alpha < K \la_i^{-\mfs}\la_i^{p-q}\le K \left( \frac{\theta \mfc}{2} \right)^{-\frac{\mfs}{p}}\la_i^{p-q}\le \frac{1}{800(1+[a]_\alpha)}\la_i^{p-q}.
 \]
\end{proof}
\begin{proof}[Proof of \ref{viii}]
 We claim that
 \[
 2[a]_\alpha (200r_i)^\alpha < a(z_i).
 \]
 Once the claim holds true, then we have
 \[
 2[a]_\alpha (200r_i)^\alpha < a(z_i)< \inf_{z\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\alpha (200 r_i)^\alpha.
 \]
 and therefore,
 \[
 [a]_\alpha (200 r_i)^\alpha< \inf_{z\in 200 Q_{r_i}(z_i)} a(z)
 \]
 Moreover, we get
 \begin{align*}
 \begin{split}
 \sup_{z\in 200 Q_{r_i}(z_i)} a(z)\le \inf_{z\in 200 Q_{r_i}(z_i)} a(z)+ [a]_\alpha (200 r_i)^\alpha< 2\inf_{z\in 200 Q_{r_i}(z_i)} a(z)
 \end{split}
 \end{align*}
 and thus, the proof is completed.
 To prove the claim, we argue as in the proof of \ref{UUi}. We have
 \[
 (a(z_i)\la_i^{q})^\mu < \frac{\La\la_i^{-2}}{2|B_1|r_i^{n+2}}\iint_{\mathbb{R}^{n+1}}\Upsilon^\mu\,dz<\frac{a(z_i)\la_i^{q-2}}{|B_1|r_i^{n+2}}\iint_{\mathbb{R}^{n+1}}\Upsilon^\mu\,dz,
 \]
 where to obtain the last inequality, we used the fact that $\la_i^p<a(z_i)\la_i^q$. Multiplying both sides by $(a(z_i)\la_i^{q})^{-\mu} r_i^{n+2}$ and then taking exponent $\tfrac{\alpha}{n+2}$, we have
 \[
 r_i^\alpha < \la_i^{-\frac{2\alpha}{n+2}+\frac{\alpha q}{n+2}(1-\mu)} \left(\frac{a(z_i)^{1-\mu}}{|B_1|} \iint_{\mathbb{R}^{n+1}} \Upsilon^\mu \,dz \right)^\frac{\alpha}{n+2}
 <K \la_i^{-\mfs} \la_i^{p-q}.
 \]
 Again, since $\theta\mfc<\theta\Pi<\La\le 2a(z_i)\lai^q \le 2\|a\|_\infty \lai^q $ and \eqref{def_mfc} holds, we get
 \[
 r_i^\alpha < K\left( \frac{\theta \mfc}{2\|a\|_\infty} \right)^{-\frac{\mfs}{q}}\lai^{p-q}<\frac{1}{800[a]_\alpha}\lai^{p-q}.
 \]

\subsection{The $p$-intrinsic case}
We begin by proving a parabolic Poincar\'e inequality in the $p$-intrinsic case.
First we estimate the last term in Lemma~\ref{lem_parabolic_poincare}.
\begin{lemma}\label{sec4:lem:1}
 Let $u$ be a very weak solution to \eqref{11}. Then, for $\theta\in((q-1)/p,\de]$ and $s\in[2\rho,4\rho]$,
 there exists a constant $c=c(\data)$ such that
 \begin{align*}
 \begin{split}
int_{Q_{s}^\pi(z_0)}(|\na u|^{p-1}+a(z)|\na u|^{q-1}+|F|^{p-1} + a(z) |F|^{q-1} )\,dz\\
int_{Q_{s}^\pi(z_0)} (|\na u| + |F| )^{p-1}\,dz\\
int_{Q_{s}^\pi(z_0)}a(z)^\frac{q-1}{q} ( |\na u|^{q-1} + |F|^{q-1} )\,dz\\
int_{Q_{s}^\pi(z_0)} ( |\na u| + |F| )^{\theta p}\,dz\right)^{\frac{p-1}{\theta p}},
 \end{split}
 \end{align*}
 whenever $Q_{16\rho}^{\pi}(z_0)\subset\Omega_T$ satisfies \eqref{eq_pcase}-\eqref{eq_p2}.
\end{lemma}

We remark that we have calculated the following in the proof of the previous lemma.
\begin{lemma}\label{coro_p_phase}
 Let $u$ be a very weak solution to \eqref{11}. Then for $\theta\in(\mu,\de]$ and $s\in[2\rho,4\rho]$, there exist constants $c=c(\data)$ and $R_0=R_0(\data, \delta ,\| u\|_{C_{\loc}(0,T;L_{\loc}^2(\Om,\mathbb{R}^N))} )\in (0,1)$ such that
 \begin{equation*}
 \begin{split}
int_{Q_{s}^\pi(z_0)} H \left( z, \frac{|u-u_{Q_{s}^\pi(z_0)}| }{s} \right)^\theta \,dz
int_{Q_{s}^\pi(z_0)}H(z,|\na u|)^\theta\,dz\\
int_{Q_{s}^\pi(z_0)} H(z,|F|)^\delta \,dz \right)^\frac{\theta}{\delta}
 \end{split}
 \end{equation*}
 whenever $Q_{16\rho}^{\pi}(z_0)\subset\Omega_T$ satisfies \eqref{eq_pcase}-\eqref{eq_p2} with $\rho\in(0,R_0)$.
\end{lemma}

\begin{lemma}\label{lem_reverse_Hölder}
Let $u$ be a very weak solution to \eqref{11}.
There exist constants $c=c(\data)$, $\delta_2=\delta_2(\data)\in(\delta_1,1)$ with $2-q(1-\delta_2)>0$, $\theta_0=\theta_0(\data)\in(0,\delta_2) $, $R_0=R_0(\data, \delta ,\| u\|_{C_{\loc}(0,T;L_{\loc}^2(\Om,\mathbb{R}^N))} )\in (0,1)$ such that
 \begin{align*}
 \begin{split}
 \pi^{\de p}
int_{Q_{2\rho}^\pi(z_0)} H(z,|F|)^\delta \,dz,
 \end{split}
 \end{align*}
 whenever $Q_{16\rho}^{\pi}(z_0)\subset\Omega_T$ satisfies \eqref{eq_pcase}-\eqref{eq_p2} with $\rho\in(0,R_0)$ and $\delta\in (\delta_2,1)$.
 Moreover, we have
 \begin{align}\label{eq_revH_Psi}
 \begin{split}
 \iint_{Q_{16\rho}^\pi(z_0)}H(z,|\na u|)^\delta\,dz
 &\le c\Pi^{\delta-\theta_0}\iint_{Q_{2\rho}^\pi(z_0)\cap \Psi(c^{-1}\Pi)}H(z,|\na u|)^{\theta_0}\,dz\\
 &\qquad +c \iint_{Q_{2\rho}^\pi(z_0)\cap \Phi(c^{-1}\Pi)}H(z,|F|)^\delta \,dz,
 \end{split}
 \end{align}
 where $\Psi(\cdot)$ and $\Phi(\cdot)$ are defined in \eqref{sec2:2}.
\end{lemma}

\begin{lemma}\label{lem_KKM_4_3_pq}
 Let $u$ be a very weak solution to \eqref{11}. Then there exists a constant $c=c(n,N,p,q,L_1,L_2)$ such that
 \begin{align*}
nt_{B_{2\rho}(x_0)}\frac{|u-u_{G_{2\rho}^\pi(z_0)}|^2}{(2\rho)^2\mh^{1-\de}}\,dx\le c\pi^2\Pi^{\de-1},
 \end{align*}
 whenever $G_{16\rho}^{\pi}(z_0)\subset\Omega_T$ satisfies \eqref{eq_pqcase}-\eqref{eq_pq4}.
\end{lemma}
\begin{proof}
 By Lemma~\ref{lem_caccioppoli}, there exists a constant $c=c(n,p,q,L_1,L_2)$ such that
 \begin{equation}\label{sec5:61}
 \begin{split}
nt_{B_{2\rho}(x_0)}\frac{|u-u_{G_{2\rho}^\pi(z_0)}|^2}{(2\rho)^2\mh^{1-\de}}\,dx\\
int_{G_{4\rho}^\pi(z_0)}\left(\frac{|u-u_{G_{4\rho}^\pi(z_0)}|^{\de p}}{(4\rho)^{\de p}}+a(z)^\delta\frac{|u-u_{G_{4\rho}^\pi(z_0)}|^{\de q}}{(4\rho)^{\de q}}\right)\,dz\\
int_{G_{4\rho}^\pi(z_0)} H(z,|F|)^\delta \,dz.
 \end{split}
 \end{equation}

For weak solutions to elliptic $p$-Laplace systems, gradient higher integrability results go back to Elcrat and Meyers in \cite{Meyers}. The core principle of the proof is already the same as in our paper. First a Caccioppoli inequality and a Sobolev-Gagliardo-Nirenberg type inequality are used to obtain a reverse Hölder inequality, after which higher integrability follows from an application of Gehring's lemma. Higher integrability result similar to Theorem~\ref{HI_theorem} for elliptic $p$-Laplace systems was proven for very weak solutions by Iwaniec and Sbordone in \cite{Iwaniec_Sbordone} and independently by Lewis in \cite{Lewis_elliptic}. We mention that a conjecture in \cite{Iwaniec_Sbordone} on minimal integrability deficit for elliptic $p$-Laplace problems was  recently answered by Colombo and Tione in \cite{colombo_jems}. There is a difficulty in showing the Caccioppoli inequality for very weak solutions, as a very weak solution is not an admissible test function. This problem was overcome in \cite{Lewis_elliptic} by using a type of Lipschitz truncation based on Whitney decomposition.

Theorem~\ref{HI_theorem} was proven for parabolic $p$-Laplace systems in \cite{KL_veryweak}. In the parabolic case, intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse Hölder inequality. Based on \cite{KL_veryweak}, gradient higher integrability of very weak solutions was extended to variable exponent problem and higher order problem of $p$-Laplace type in \cite{MR4074602,verena_thesis,bogelein_li,Li}. Furthermore, corresponding arguments have been implemented in the context of Calder\'on-Zygmund type estimates in \cite{MR3928653,MR3483172} and applied to establish existence of very weak solutions for elliptic $p$-Laplace problems \cite{MR3531368,MR3500290} and linear parabolic problems \cite{MR3985550}.
The parabolic Lipschitz truncation method has been considered also in relation to caloric approximations in \cite{MR3672391,MR4525732} and existence of solutions to fluid models in \cite{MR2668872}.

     For parabolic double-phase problems higher integrability for weak solutions was shown in \cite{KKM,KS}. In there, phase analysis is used to distinguish whether the energy estimates of double-phase systems are perturbed into that of $p$-Laplace systems or $(p,q)$-Laplace systems depending on the pointwise value of $a$. Based on the phase division, we construct a corresponding intrinsic geometry.
 Similar phase analysis was used in \cite{KKS} to obtain standard energy estimates for weak solutions. Although a Lipschitz truncation method was applied in \cite{KKS}, the argument is different for very weak solutions.
Other recent result for parabolic double-phase problems include partial regularity in the nondegenerate case in \cite{MR4774133,kim2025boundedsolutionsinterpolativegap,MR4926910,oh2025gradient,ok2025partialregularityparabolicsystems,sen2025lipschitzregularityparabolicdouble}.

Corresponding result to Theorem~\ref{HI_theorem} for very weak solutions to the elliptic case was proved in \cite{BBK}. Obtaining the higher integrability result for very weak solutions to parabolic double-phase problems presents several new challenges. In order to show that there exists $\de_0$ satisfying Theorem~\ref{HI_theorem}, the higher integrability gained in the argument has to be quantified and independent from $\delta$. In \cite{KKM} the phase analysis in the reverse Hölder inequality argument for weak solutions is done with respect to a constant $K$ depending on the integral of the gradient. Utilizing the same argument for very weak solutions would lead to estimates depending on $\delta$. To avoid with this problem, we consider cylinders with radii bounded by a constant $R_0$ appearing in Theorem~\ref{HI_theorem} and use a new phase analysis argument.

Also in the proof of the Caccioppoli inequality there are novel challenges. In \cite{KKS} the Lipschitz truncation method is used to prove a Caccioppoli inequality for weak solutions to parabolic double-phase problems. In the argument in \cite{KKS}, the term $\Lambda|E(\Lambda)^c|$ in (\ref{def_E}) vanishes as $\Lambda \to \infty$. For very weak solutions this does not happen and finer estimates are required. In particular, we have to compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders defined in Section~\ref{sec_setup}. In our context these cylinders have different geometries and to compare them we provide a new argument in Lemma~\ref{lem_theta_2}. Our proof of the Caccioppoli inequality is self-contained. To simplify some of the arguments of previous papers, we have included the term $\tfrac{|u-u_0|}{\rho}$ in the definition of $E(\Lambda)$.

Finally, we note that the range of $q$ in the parabolic double-phase system is not completely understood, however, the strict inequality for the upper bound of $q$ in (\ref{eq_range_q}) seems to be natural. For the elliptic case, the range of $q$ is well understood and its optimality is known. When we consider very weak solutions in the elliptic setting, the natural integrability of the gradient does not allow the borderline case, see \cite{BBK} for the details.

To prove Theorem~\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \emph{Caccioppoli inequality} and combines it with a \emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \emph{reverse H\"older inequality}; then the higher integrability conclusion follows by applying \emph{Gehring's lemma}.

For \emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\"older inequality.”

In the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \(\delta_0\) as in Theorem~\ref{HI_theorem}, “the higher integrability gained in the argument has to be quantified and independent from \(\delta\).” Since using the phase analysis from \cite{KKM} for very weak solutions “would lead to estimates depending on \(\delta\),” the approach is to “consider cylinders with radii bounded by a constant \(R_0\) appearing in Theorem~\ref{HI_theorem} and use a new phase analysis argument.”

There are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \cite{KKS}, the term “\(\Lambda|E(\Lambda)^c|\) … vanishes as \(\Lambda\to\infty\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries,” and this comparison is handled via “a new argument in Lemma~\ref{lem_theta_2}.” The authors also note a simplification: they “included the term \(\tfrac{|u-u_0|}{\rho}\) in the definition of \(E(\Lambda)\).”

To prove Theorem~\ref{HI_theorem}, the text describes the following proof strategy (in the spirit of Meyers/Elcrat for higher integrability): first one proves a \emph{Caccioppoli inequality} and combines it with a \emph{Sobolev--Gagliardo--Nirenberg type inequality} to obtain a \emph{reverse H\"older inequality}; then the higher integrability conclusion follows by applying \emph{Gehring's lemma}.

For \emph{very weak solutions}, there is an additional obstacle: “a very weak solution is not an admissible test function,” so establishing the Caccioppoli inequality requires a \emph{Lipschitz truncation} argument “based on Whitney decomposition.” In the parabolic setting “intrinsic geometry plays a role both in the Lipschitz truncation and in obtaining the reverse H\"older inequality.”

In the parabolic double-phase case, new issues arise in making the gain uniform in the integrability deficit: to ensure the existence of a \(\delta_0\) as in the main theorem, “the higher integrability gained in the argument has to be quantified and independent from \(\delta\).” Since using the phase analysis from \cite{KKM} for very weak solutions “would lead to estimates depending on \(\delta\),” the approach is to “consider cylinders with radii bounded by a constant \(R_0\) appearing in the main theorem and use a new phase analysis argument.”

There are also “novel challenges” in the Caccioppoli step for very weak solutions: unlike in \cite{KKS}, the term “\(\Lambda|E(\Lambda)^c|\) … vanishes as \(\Lambda\to\infty\)” does not vanish here, so “finer estimates are required.” In particular, one must “compare intrinsic cylinders in the Whitney decomposition with the underlying intrinsic cylinders,” which “have different geometries.” This comparison is handled via the following lemma.

\begin{lemma}\label{lem_theta_2}
	Suppose $i\in \Theta_2$ and $50U_i\cap \Q_2\ne\emptyset$. Then there exists $c=c(\data)$ such that
    \[
    \rho_2-\rho_1 \leq c\theta^{\frac
    {2-q}{p}}r_i.
    \]
\end{lemma}

The authors also note a simplification: they “included the term \(\tfrac{|u-u_0|}{\rho}\) in the definition of \(E(\Lambda)\).”

\label{HI_theorem}
Suppose $p,q$ satisfy
\begin{align}\label{eq_range_q}
    q < p+\frac{2\alpha}{n+2} \quad \text{and} \quad p>2.
\end{align}
Then there exists $\de_0=\de_0(\mathit{data})\in(0,1)$ with $2-q(1-\delta_0)>0$
such that if $u$ is a very weak solution to
\begin{align}\label{11}
\tu_t-\dv\mA(z,\na u)=
    -\dv (|F|^{p-2}F+a(z)|F|^{q-2}F)
    \quad\text{in}\ \Omega_T,
\end{align}
with integrability deficit $\delta\in(\de_0,1)$ and $u\in C(0,T;L^2(\Omega_T,\mathbb{R}^N))$, we have
	\begin{align*}
		\begin{split}
int_{Q_{R} (z_0) }H(z,|\nabla u|)\,dz
int_{Q_{2R} (z_0) } (  H(z,|\nabla u|)^\delta+ H(z,|F|) ) \,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
		\end{split}
	\end{align*}
	 for every $Q_{2R}\subset\Om_T$ where $c\ge
     1$ depending on $\mathit{data},\|a\|_{L^\infty(\Om_T)},\| (H(z,|\nabla u|)+H(z,|F|)+1 )\|_{L^\delta(\Om_T)}$, and $R\in(0,R_0)$ for $R_0\in(0,1)$ depending on $\mathit{data}$, $\delta$, $\| H(z,|\nabla u|) \|_{L^\delta(\Om_T)}$, $\| H(z,|F|) \|_{L^\delta(\Om_T)}$, $\|u\|_{C(0,T;L^2(\Omega,\mathbb{R}^N))}$.,

Existential–Universal
Inequality or Bound

Question:
Let \(\Omega_T=\Omega\times(0,T)\), and define \(H(z,s)=s^p+a(z)s^q\). Assume \(a\ge 0\) belongs to \(C^{\alpha,\alpha/2}(\Omega_T)\), \(\alpha\in(0,1]\), and that \(\mathcal A:\Omega_T\times\mathbb R^{Nn}\to\mathbb R^{Nn}\) is a Carath\'eodory vector field satisfying the double-phase structure conditions
\[
\mathcal A(z,\xi)\cdot \xi\ge L_1\big(|\xi|^p+a(z)|\xi|^q\big),
\qquad
|\mathcal A(z,\xi)|\le L_2\big(|\xi|^{p-1}+a(z)|\xi|^{q-1}\big)
\]
for a.e. \(z\in\Omega_T\) and all \(\xi\in\mathbb R^{Nn}\). Suppose also that
\[
q<p+\frac{2\alpha}{n+2},\qquad p>2.
\]
If \(u\) is a very weak solution of
\[
u_t-\operatorname{div}\mathcal A(z,\nabla u)=-\operatorname{div}\big(|F|^{p-2}F+a(z)|F|^{q-2}F\big)
\quad\text{in }\Omega_T,
\]
with integrability deficit \(\delta\in(\delta_0,1)\) and \(u\in C(0,T;L^2(\Omega,\mathbb R^N))\), and where \(Q_r(z_0)\) denotes a standard parabolic cylinder, which local higher-integrability estimate holds for every \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\)?

There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and \(R_0\in(0,1)\) depends on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\).

Choices:
B.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends only on the structural data and \(\|a\|_{L^\infty(\Omega_T)}\), and one may take \(R_0=1\).
C.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\), for some constants \(c\ge 1\) and \(R_0\in(0,1)\).
D.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta)>0\) for every \(\delta\in(\delta_0,1)\), and for every such \(\delta\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)^\delta\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and \(R_0\in(0,1)\) depends on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\).
E.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and \(R_0\in(0,1)\) depends on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\).

small-radius restriction and dependence of constants on global \(L^\delta\) data

explicit dependence of \(c\) and \(R_0\)

source term appears as \(H(z,|F|)\), not \(H(z,|F|)^\delta\)

precise Gehring exponent \(1+\frac{q(1-\delta)}{2-q(1-\delta)}\)

The stem does not explicitly reveal the correct option or uniquely signal A. It states the assumptions and asks for the valid conclusion, without giving away the dependence structure or exact estimate.

The item is essentially a theorem-statement recognition task: under the listed hypotheses, the student must pick the exact existence/local estimate conclusion. This is very close to a direct restatement rather than an application to a new situation.

Some reasoning is needed to distinguish subtle variants (radius restriction, norm dependence of constants, whether H(z,|F|) or H(z,|F|)^delta appears, and the Gehring-type exponent). However, the task is still mainly identification of the canonical theorem statement rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: over-uniformity in R, omission of dependency data, incorrect source-term integrability, and loss of the higher exponent. They are distinct and well aligned with theorem-misstatement errors.

A technically solid MCQ with strong, plausible distractors and no obvious answer leakage, but it mainly tests recall/recognition of a theorem statement rather than genuinely generative mathematical reasoning.


Question:
Let \(\Omega_T=\Omega\times(0,T)\), and define \(H(z,s)=s^p+a(z)s^q\). Assume \(a\ge 0\) belongs to \(C^{\alpha,\alpha/2}(\Omega_T)\), \(\alpha\in(0,1]\), and that \(\mathcal A:\Omega_T\times\mathbb R^{Nn}\to\mathbb R^{Nn}\) is a Carath\'eodory vector field satisfying the double-phase structure conditions
\[
\mathcal A(z,\xi)\cdot \xi\ge L_1\big(|\xi|^p+a(z)|\xi|^q\big),
\qquad
|\mathcal A(z,\xi)|\le L_2\big(|\xi|^{p-1}+a(z)|\xi|^{q-1}\big)
\]
for a.e. \(z\in\Omega_T\) and all \(\xi\in\mathbb R^{Nn}\). Suppose also that
\[
q<p+\frac{2\alpha}{n+2},\qquad p>2.
\]
If \(u\) is a very weak solution of
\[
u_t-\operatorname{div}\mathcal A(z,\nabla u)=-\operatorname{div}\big(|F|^{p-2}F+a(z)|F|^{q-2}F\big)
\quad\text{in }\Omega_T,
\]
with integrability deficit \(\delta\in(\delta_0,1)\) and \(u\in C(0,T;L^2(\Omega,\mathbb R^N))\), and where \(Q_r(z_0)\) denotes a standard parabolic cylinder, which local higher-integrability estimate holds for every \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\)?

Choices:
B.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends only on the structural data and \(\|a\|_{L^\infty(\Omega_T)}\), and one may take \(R_0=1\).
C.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\), for some constants \(c\ge 1\) and \(R_0\in(0,1)\).
D.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta)>0\) for every \(\delta\in(\delta_0,1)\), and for every such \(\delta\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)^\delta\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and \(R_0\in(0,1)\) depends on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\).
E.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,[a]_{\alpha,\alpha/2})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and for every \(\delta\in(\delta_0,1)\), if \(u\) is as above, then
\[
\fint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\fint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2}}
\]
for every \(Q_{2R}(z_0)\subset\Omega_T\), where \(c\ge 1\) depends on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and \(R_0\in(0,1)\) depends on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\).

The stem gives assumptions and asks for the resulting local estimate, but it does not explicitly state the conclusion or uniquely reveal the correct option. The answer must be identified from the choices.

This is essentially a theorem-recall item: the assumptions are listed and the task is to select the exact theorem conclusion. It is very close to a direct restatement rather than an independent problem.

There is some reasoning pressure because the options differ in subtle but important ways (dependence of constants, radius restriction, source-term exponent, Gehring exponent, weaker-vs-sharper statement). However, the item mainly tests recognition of the precise theorem statement rather than generative mathematical reasoning.

The distractors are strong: they are mathematically nearby variants involving common failure modes such as omitting small-scale restrictions, weakening dependence statements, altering the source-term integrability, or changing the precise exponent. They are distinct and plausible.

A technically well-constructed theorem-identification MCQ with strong distractors and little answer leakage, but it is largely tautological and tests precise recall/comparison more than genuine generative reasoning.


\begin{theorem}\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP \eqref{kdV} admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}
\end{theorem}

In mathematics and physics, a quantum graph is a network composed of vertices connected by edges, where each edge carries a differential or pseudo-differential equation. When every edge is endowed with a natural metric, the structure is referred to as a metric graph. A common illustration is a power network: the transmission lines act as edges connecting transformer stations (vertices). In such a case, the governing differential equations may describe the voltage along each line, with boundary conditions at the vertices ensuring that the total current flowing into and out of each vertex sums to zero.

The concept of quantum graphs traces back to the work of Linus Pauling in the 1930s, who used them to model free electrons in organic molecules. Since then, they have emerged in a wide range of contexts, including quantum chaos, waveguide analysis, photonic crystals, and Anderson localization—the suppression of wave propagation in disordered media. Quantum graphs also arise as limit models of systems composed of thin wires and serve as important theoretical tools in mesoscopic physics and nanotechnology. A simplified formulation of quantum graphs was later proposed by Freedman \textit{et al.} in \cite{Freedman}.

Beyond solving the differential equations in specific applications, the study of quantum graphs encompasses fundamental questions such as well-posedness, controllability, and identifiability. For example, controllability concerns determining the inputs required to steer the system toward a desired configuration, such as ensuring stable power distribution across a grid. Identifiability, on the other hand, focuses on optimal measurement placement—for instance, choosing pressure gauges in a water network to detect and locate potential leaks.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Other nonlinear dispersive systems on graphs have also been the subject of extensive study. For example, in the context of well-posedness theory, Cavalcante \cite{Cav} established local well-posedness for the Cauchy problem associated with the Korteweg–de Vries (KdV) equation on a metric star graph composed of one negative and two positive half-lines joined at a common vertex $\nu = 0$ (the so-called $\mathcal{Y}$-junction).

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Significant progress has also been achieved in the fields of control theory and inverse problems on networks. Ignat \textit{et al.} \cite{Ignat2011} examined inverse problems for the heat and Schrödinger equations on tree-like structures, while Baudouin and Yamamoto \cite{Baudouin} proposed a unified and simplified framework for coefficient identification problems. Furthermore, Ammari and Crépeau \cite{Ammari and Crepeau 2018}, Cerpa \textit{et al.} \cite{Cerpa, Cerpa1}, and Parada \textit{et al.} \cite{Parada2022a, Parada2022b} established stabilization and boundary controllability results for the KdV equation on star-shaped graphs, contributing substantially to the control-theoretic understanding of dispersive models on networks. Recently, in \cite{CapistranoParadaDaSilva25}, we studied the analytical structure of the entire functions associated with the KdV operator, addressing control problems involving mixed Dirichlet–Neumann boundary controls.

\subsection{Setting of problem and functional framework} In the context of graph structure, let us consider the following: given $N\geq 2$, consider a star-shaped network $\mathcal{T}$ with $N$ edges described by intervals $I_j=(0,l_j)$, $l_j>0$ for $j=1,...,N$. Denoting by $e_1,...,e_N$ the edges of $\mathcal{T}$ one has $\mathcal{T}=\bigcup_{j=1}^Ne_j$. Here, we study the local  well-posedness for a system composed of $N$ nonlinear KdV equations posed on $\mathcal{T}$ and coupled by the boundary, namely
		\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
where $u=(u_1,...,u_N)$ stands the state of the system, $p_j$ and $g_j$ are the controls inputs and $\alpha>\frac{N}{2}$. We represent this situation in the figure \eqref{fig:network}.
		\begin{figure}[H]
			\centering
			\begin{tikzpicture}
				\draw(0, 0) circle (1mm);
				\draw[line width=0.8pt](0.02, 0.09) -- (0.5, 1.95);
				\draw[fill=black](0.53, 2.04) circle (1mm);
				\draw[line width=0.8pt](0.07, 0.07) -- (1.33, 1.5);
				\draw[fill=black](1.39, 1.57) circle (1mm);
				\draw[line width=0.8pt](0.1, 0.04) -- (1.85, 0.78);
				\draw[fill=black](1.93, 0.815) circle (1mm);
				\draw[line width=0.8pt](0.08, -0.04) -- (1.75, -1);
				\draw[fill=black](1.825, -1.045) circle (1mm);
				\draw[line width=0.8pt](0.05, -0.09) -- (1.02, -1.73);
				\draw[fill=black](1.07, -1.8) circle (1mm);
				\draw[fill=black](1.45, 0.2) circle (0.3mm);
				\draw[fill=black](1.45, 0.05) circle (0.3mm);
				\draw[fill=black](1.45, -0.1) circle (0.3mm);
				\draw node (no1) at (-0.3,0) {$0$};
				\draw node (no1) at (0.19, 1.7) {$e_1$};
				\draw node (no1) at (0.86, 1.35) {$e_2$};
				\draw node (no1) at (1.35, 0.8) {$e_3$};
				\draw node (no1) at (1.7, -0.6) {$e_{_{N-1}}$};
				\draw node (no1) at (1.06, -1.25) {$e_{_N}$};
				\draw(3.95,-0.5) circle (1mm);
				\draw node (no1) at (6.4,-0.5) {{\footnotesize \hspace{10mm} internal controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
				\draw[fill=black](3.95,-1) circle (1mm);
				\draw node (no1) at (6.4,-1) {{\footnotesize \hspace{13mm} external controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
			\end{tikzpicture}
			\label{fig:network}
			\caption{Network with $N$ edges}
	\end{figure}

Let us introduce some notation used throughout this work. From now on, we have the following:
	 \begin{itemize}
	 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
	and initial data is $u^0=(u^0_1,...,u^0_N)$.
	 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by  $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
	Moreover,  $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
	 \end{itemize}
	\begin{itemize}
		\item[iii.] Define also the following spaces:
		$$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
		and
		$$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
		where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
		$$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
		\item[iv.] Consider the following characterization:
		$$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
		as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
	L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in 	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Let us introduce some notation used throughout this work. From now on, we have the following:
 \begin{itemize}
 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
 and initial data is $u^0=(u^0_1,...,u^0_N)$.
 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
 Moreover, $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
 \end{itemize}
 \begin{itemize}
 \item[iii.] Define also the following spaces:
 $$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
 and
 $$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
 where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
 $$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
 \item[iv.] Consider the following characterization:
 $$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
 as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
 C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
 L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\begin{lemma}\label{lemma for v}
 Let $(u^0,g_0, g,p)\in \mathbb{X}_{3,T}$, $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ and $v$ the solution of \eqref{v_j}. Then for $\beta=0,1,2,3$ we have:
 \begin{enumerate}
 \item[i.] The function
 \begin{align*}
 \begin{array}{rcl}
 [0,T]&\rightarrow&\mathcal{D}'(0,l_j)\\
 s&\mapsto&\partial_x^\beta v_j(s,\cdot)
 \end{array}
 \end{align*}
 is integrable in $[0,T]$.
 \item[ii.]For any $t \in [0,T]$ we have $\displaystyle\int_0^tv_j(s,\cdot)ds \in H^4(0,l_j)$ and for $\beta=0,1,2,3$ we have
 \begin{align*}
 \partial_x^\beta\int_0^tv_j(s,\cdot)ds=\int_0^t\partial_x^\beta v_j(s,\cdot)ds.
 \end{align*}
 \end{enumerate}
\end{lemma}
The previous lemma can give us the regularity for $u$.
\begin{proposition}\label{W.P._s=3}
Let $(u^0,g_0,g,p) \in \mathbb{H}^3(\mathcal{T})\times H^{\frac{3-1}{3}}(0,T)\times [H^\frac{3}{3}(0,T)]^N\times [H^{\frac{3+1}{3}}(0,T)]^N$ satisfying the linear compatibility conditions for $s=3$ and $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ be. For $j=1,...,N$ the function $t\mapsto u_j(t,\cdot)$ given by \eqref{u_j} satisfies the following items:
\begin{enumerate}
 \item[i.] $u_j \in C^1([0,T],L^2(0,l_j))\cap C([0,T];H^3(0,l_j))$;
 \item[ii.] $u\in \mathbb{B}_{3,T}=C([0,T];\mathbb{H}^3(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^4(\mathcal{T}))$ is the unique classical solution to the problem \eqref{LkdV}; and
 \begin{equation*}
 \begin{split}
 \|u\|_{\mathbb{B}_{3,T}}\leq &C\left( \|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\
 &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right)
\end{split}
 \end{equation*}
 for some constant $C>0$.
 \item[iii.] $u \in H^1(0,T;\mathbb{H}^1(\mathcal{T}))$ and
 \begin{equation*}
 \begin{split}
 \|u\|_{H^1(0,T;\mathbb{H}^1(\mathcal{T}))}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right),
 \end{split}
 \end{equation*}
 for some constant $C>0$.
 \item [iv.] $\partial_x^ku_j\in L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)$ and
 \begin{align*}
 \|\partial_x^ku_j\|_{L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right).
 \end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
Consider $j \in\{1,...,N\}$.
\vspace{0.2 cm}
\noindent\textbf{i.} Since $v_j\in C([0,T],L^2(0,l_j))$ it is immediate that by \eqref{u_j} that $u_j \in C^{1}([0,T],L^2(0,l_j))$.

\section{Local well-posedness of the nonlinear problem}\label{sec4}
Now, we are in a position to prove the first main result of the article, namely, the local well-posedness of the following problem
\begin{align}\label{NLKdV}
 \left\{
 \begin{array}{ll}
 \partial_t u_j+\partial_x [(a_j+1)u_j]+\partial_x^3 u_j+u_j\partial_x u_j=f_j,&t \in (0,T),\ x \in (0,l_j),\\
 u_j(t,0)=u_1(t,0),&t\in (0,T),\\
 \displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,T)\\
 u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\\
 u_j(0,x)=u_j^0(x),&x \in (0,l_j),
 \end{array}
 \right.
\end{align}
where $j=1,...,N$ and $a$ is a given function. Remember that: for $s\geq 0$ and $T>0$ the space
\begin{align*}
\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}_{s,T}^*;\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\}
\end{align*}
is a Banach space with the norm
\begin{align*}
\|v\|_{\mathcal{X}_{s,T}}=\|v\|_{\mathbb{B}_{s,T}^*}+\sum_{k=0}^2\|\partial_x^kv\|_{\prod_{j=1}^NL_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T))}.
\end{align*}
Note that \eqref{kdV} is a particular case of \eqref{NLKdV} with $a=0$.
\begin{proof}[Proof of Theorem \ref{main}] Fix $\theta \in (0,T)$ and define $\Gamma_a:\mathcal{X}_{s,\theta}\rightarrow \mathcal{X}_{s,\theta}$ putting, for each $u \in \mathcal{X}_{s,\theta}$, $\Gamma_au$ as being the solution $v$ of
\begin{align*}
 \left\{
 \begin{array}{ll}
 \partial_t v_j+\partial_x [(a_j+1)v_j]+\partial_x^3 v_j=-u_j\partial_x u_j+f_j,&t \in (0,\theta),\ x \in (0,l_j),\ j=1,...,N,\\
 v_j(t,0)=v_1(t,0),&t\in (0,\theta),\ j=1,...,N,\\
 \displaystyle\sum_{j=1}^N\partial_x^2 v_j(t,0)=-\alpha v_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,\theta,)\\
 v_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x v_j(t,l_j)=g_j(t),&t\in (0,\theta),\ j=1,...,N,\\
 v_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N.
 \end{array}
 \right.
\end{align*}
From Proposition \ref{W.P.LKdV-a} we have
\begin{align*}
& \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0-\frac{N}{3}(u_1+a_1)^2(\cdot,0)\|_{H^\frac{s-1}{3}(0,\theta)}+\|g\|_{[H^\frac{s}{3}(0,\theta)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,\theta)]^N}+\|u\partial_xu\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}+\|f\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))} \right).
\end{align*}
Using Lemma \ref{lemma 2.4} we obtain
\begin{align*}
 \|\partial_xu v\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\leq C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathbb{B}_{s,\theta}^*}\|u\|_{\mathbb{B}_{s,\theta}^*}.
\end{align*}
Furthermore, Lemma \ref{lemma 2.1} gives us
\begin{align*}
\left\|\frac{N}{3}(u_1+a_1)^2(\cdot,0)\right\|_{H^\frac{s-1}{3}(0,\theta)}&\leq C\theta^\alpha\frac{N}{3}\left\|(u_1+a_1)(\cdot,0)\right\|_{H^\frac{s+1}{3}(0,\theta)}^2\\
&\leq C\theta^\alpha\frac{N}{3}\|u+a\|_{\mathcal{X}_{s,\theta}}^2\\
&\leq \frac{4}{3}CN\theta^\alpha \left(\|u\|_{\mathcal{X}_{s,\theta}}^2+\|a\|_{\mathcal{X}_{s,\theta}}^2\right),
\end{align*}
where $\alpha>0$ is a constant. Thus,
\begin{align*}
 \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq& C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\\
&\left.+C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha \|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha\|a\|_{\mathcal{X}_{s,\theta}}^2 \right).
\end{align*}
Defining
\begin{align*}
r=&4C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\right),
\end{align*}
it follows that, for $u \in B=\{v \in \mathcal{X}_{s,\theta};\ \|v\|_{\mathcal{X}_{s,\theta}}\leq r\}$,
\begin{align}\label{eq 3.2}
\|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq \frac{r}{4}+C^2(\theta^\frac{1}{2}+\theta^\frac{1}{3})r^2+\frac{4}{3}C^2N\theta^\alpha r^2+\frac{4}{3}C^2N\theta^\alpha\|a\|_{\mathcal{X}_{s,T}}^2.
\end{align}

The proof of the previous theorem relies on several analytical tools combined with a fixed-point argument. In addition, a new notion of $s$-compatibility adapted to the graph structure is introduced (see Section \ref{sec2}), inspired by the single-edge KdV framework in \cite{BSZ 2003}. This theorem provides a sharp local well-posedness result, along with regularity properties, for the associated functions. Furthermore, we establish the existence of global solutions at the $H^s$-norm, which enables applications to control problems on star graphs and resolves several open questions that will be addressed in a forthcoming paper. The second main result of this work can be stated as follows.

\begin{theorem}\label{global_(0,3)}
Let $T>0$ and $\lambda\geq 1$ be. Consider $(u^0,g_0,g,p)\in \mathbb{X}_{s,T}^\lambda$, where $\mathbb{X}_{s,T}^\lambda$ is the set of $s$-compatible data
$$u^0 \in \mathbb{H}^{s}(\mathcal{T}), g_0 \in H^{\frac{s+\lambda-1}{3}}(0,T),\ g \in [H^\frac{s}{3}(0,T)]^N\quad \text{and}\quad p\in [H^\frac{s+\lambda+1}{3}(0,T)]^N.$$
Thus, the problem \eqref{kdV} has a unique solution $u\in \mathcal{X}_{s,T}$ which satisfies
\begin{align*}
	\|u\|_{\mathcal{X}_{s,T}}&\leq C\left(\|(u^0,g_0,g,p)\|_{0,\lambda}\right)\left(
	\begin{array}{c}
		\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s+\lambda-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}+\|p\|_{[H^\frac{s+\lambda+1}{3}(0,T)]^N}
	\end{array}
	\right)
\end{align*}
where $C>0$ is a positive constant.
\end{theorem}
Together, these two theorems provide, for the first time, a comprehensive framework for the well-posedness of general graph structures with boundary controls. Moreover, they extend the classical result for a single KdV equation \cite{BSZ 2003} to star-shaped graphs composed of $N$ equations.
\vspace{0.2cm}

The paper is organized as follows. In Section \ref{sec2}, we introduce a general notion of $s$-compatibility for star graphs. Building on this framework, Section \ref{sec3} addresses the well-posedness of the linear system associated with \eqref{kdV}. After that, Section \ref{sec4} is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem \ref{main}. The second main result, concerning global well-posedness, in the $H^s$-regime, for the system \eqref{kdV} is established in Section \ref{sec6}.

To prove Theorem~\ref{main}, the introduction states that it "relies on several analytical tools combined with a fixed-point argument." It also mentions that "a new notion of $s$-compatibility adapted to the graph structure is introduced" (Section~\ref{sec2}), "inspired by the single-edge KdV framework in \cite{BSZ 2003}." The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat "the well-posedness of the linear system associated with \eqref{kdV}" (Section~\ref{sec3}); and then Section~\ref{sec4} "is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem~\ref{main}."

To prove the main theorem, the introduction states that it “relies on several analytical tools combined with a fixed-point argument.” It also mentions that “a new notion of $s$-compatibility adapted to the graph structure is introduced” (Section~\ref{sec2}), “inspired by the single-edge KdV framework in \cite{BSZ 2003}.” The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat “the well-posedness of the linear system associated with
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
” (Section~\ref{sec3}); and then Section~\ref{sec4} “is devoted to proving the local well-posedness result for the system above, giving the proof of the main theorem.”

\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}

Question:
Let \(T>0\), \(l_j>0\) for \(j=1,\dots,N\), and \(s\in[0,3]\). Let \(\mathcal T\) be the star-shaped network with edges \((0,l_j)\), and write \(\mathbb L^2(\mathcal T)=\prod_{j=1}^N L^2(0,l_j)\) and \(\mathbb H^s(\mathcal T)=\prod_{j=1}^N H^s(0,l_j)\). Assume
\[
f\in W^{\frac{s}{3},1}(0,T;\mathbb L^2(\mathcal T))\cap L^{\frac{6}{6-s}}(0,T;\mathbb H^{\frac{s}{3}}(\mathcal T)),
\]
and
\[
(u^0,g_0,g,p)\in \mathbb H^s(\mathcal T)\times H^{\frac{s-1}{3}}(0,T)\times [H^{\frac{s}{3}}(0,T)]^N\times [H^{\frac{s+1}{3}}(0,T)]^N
\]
satisfy the \(s\)-compatibility conditions. Consider, for \(j=1,\dots,N\), the initial-boundary value problem
\[
\begin{cases}
\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,& t\in(0,T),\ x\in(0,l_j),\\
u_j(t,0)=u_1(t,0),& t\in(0,T),\\
\displaystyle\sum_{j=1}^N \partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),& t\in(0,T),\\
u_j(t,l_j)=p_j(t),\quad \partial_x u_j(t,l_j)=g_j(t),& t\in(0,T),\\
u_j(0,x)=u_j^0(x),& x\in(0,l_j).
\end{cases}
\]
Also set
\[
\mathbb B_{s,\tau}=C([0,\tau],\mathbb H^s(\mathcal T))\cap L^2(0,\tau;\mathbb H^{s+1}(\mathcal T)),
\]
\[
\mathbb B^*_{s,\tau}=\mathbb B_{s,\tau}\cap H^{\frac{s}{3}}(0,\tau;\mathbb H^1(\mathcal T)),
\]
and
\[
\mathcal X_{s,\tau}=\left\{v\in \mathbb B^*_{s,\tau}:\ \partial_x^k v_j\in L_x^{\infty}(0,l_j;H^{\frac{s+1-k}{3}}(0,\tau)),\ k=0,1,2,\ j=1,\dots,N\right\}.
\]
Which statement about existence and uniqueness is valid for this problem?

There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).

Choices:
B. There exists \(T^*\in(0,T)\), depending only on \(T,s,N\) and the lengths \(l_j\), such that for every admissible \((u^0,g_0,g,p,f)\) satisfying the \(s\)-compatibility conditions, the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).
C. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits at most one solution in \(\mathcal X_{s,T^*}\).
D. For every \(T>0\), the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T}\).
E. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathbb B^*_{s,T^*}\).

The stem does not explicitly reveal the correct conclusion. It lists hypotheses and asks which well-posedness statement follows; the key features of the answer (local time interval, uniqueness, and the exact solution space) are not directly stated in the stem.

This is essentially a theorem-recall item: the stem reproduces the full hypotheses and the options are slight variants of the theorem's conclusion. It is not a pure verbatim restatement because the choices alter quantifiers, uniqueness, and function spaces, but it remains very close to the source theorem.

Some reasoning is needed to distinguish local vs global existence, uniqueness vs mere existence, and the precise solution space. However, the task mainly tests recognition of the exact theorem statement rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: global-in-time instead of local, existence without uniqueness, incorrect dependence of the lifespan, and a weaker/incorrect solution space. These reflect common ways a theorem statement can be misstated.

A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it is still close to theorem restatement and only moderately tests genuine generative reasoning.


\begin{thm}\label{thm:1.1}
Given $\phi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\phi$ on $\Omega$ and an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ such that $\Phi-u_\phi\in H^1_0(\Omega)$.
\end{thm}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\mathbb{R}^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\varphi\in C(\partial\Omega)$, the \textbf{\boldmath Dirichlet problem\unboldmath} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\overline{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\varphi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \textbf{\boldmath classical solution\unboldmath} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this paper we consider a generalised solution of the Dirichlet problem which is described as follows.

\begin{equation}
  \tag*{$D(\varphi)$}
  \label{eq:D}
  \left\{
    \begin{aligned}
      &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
      &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
    \end{aligned}
  \right.
\end{equation}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\RR^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\phi\in C(\partial\Omega)$, the \emphdef{Dirichlet problem} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \emphdef{classical solution} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\cite[Corollary~4.3]{AtES24}.
\begin{thm}[Extension result]\label{thm:2.1}
Let $\phi\in C(\partial\Omega)$. Then there exist an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ and $M \ge 0$ such that
\begin{equation}\label{eq:2.1}
 \abs*{\int_\Omega\Phi\Delta w\dx} \le M\norm{w}_{H^1(\Omega)}
\end{equation}
for all $w\in C^\infty_\cpt(\Omega)$. This means that $\Delta\Phi\in H^{-1}(\Omega)$, the dual space of $H^1_0(\Omega)$.
\end{thm}
Note that for $u\in H^1(\Omega)$ one has
\[
 \abs*{\int_\Omega u\Delta w\dx} = \abs*{-\int_\Omega\nabla u\nabla w\dx}\le\norm{u}_{H^1(\Omega)}\norm{w}_{H^1(\Omega)}
\]
for all $w\in C^\infty_\cpt(\Omega)$ and therefore $\Delta u\in H^{-1}(\Omega)$.

Let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying~\eqref{eq:2.1}.
By the Riesz--Fréchet representation theorem there exists a unique $v\in H^1_0(\Omega)$ such that
\[
 \int_\Omega \Phi\Delta w\dx = -\int_\Omega\nabla v\nabla w\dx = \int_\Omega v\Delta w\dx
\]
for all $w\in C^\infty_\cpt(\Omega)$. This means that $u:=\Phi-v$ satisfies $\Delta u=0$ weakly. Hence $u\in C^\infty(\Omega)$ by Weyl's theorem and $\Delta u=0$ in the classical sense;
that is, $u$ is a harmonic function.
We call $u$ the \emphdef{variational solution} of~\ref{eq:D} and set $u_\phi := u$.
To do so, we have to show that the function does not depend on the extension $\Phi$ of $\phi$. This follows from the next lemma.

Next we talk about the Dirichlet principle.
We start by mentioning that the variational solution $u_\phi$ is obtained by a minimising process.
In fact, let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying the estimate~\eqref{eq:2.1}, i.e.\ $\Delta\Phi\in H^{-1}(\Omega)$.
The solution $v\in H^1_0(\Omega)$ of $\Delta v=\Delta\Phi$ is the unique function $v\in H^1_0(\Omega)$ such that
\begin{equation}\label{eq:2.5}
 \frac{1}{2}\int_\Omega\abs{\nabla v}^2\dx + \langle\Delta\Phi,v\rangle = \min\Bigl\{\frac{1}{2}\int_\Omega\abs{\nabla w}^2\dx + \langle\Delta\Phi,w\rangle \setcolon w\in H^1_0(\Omega)\Bigr\}
\end{equation}
(see e.g.~\cite[Theorem~4.24]{AU23}).
Now we can characterise when $\int_\Omega\abs{\nabla u_\phi}^2\dx<\infty$; i.e.\ when $u_\phi$ has finite energy.
Since $u_\phi\in C^\infty(\Omega)\cap C_\bdd(\Omega)$, this is equivalent to saying that $u_\phi\in H^1(\Omega)$.

\begin{thm}\label{thm:2.7}
Let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item\label{en:t2.7-i}
The variational solution $u_\phi$ of~\ref{eq:D} is an element of $H^1(\Omega)$.
\item\label{en:t2.7-ii}
There exists $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ with $\restrict{\Phi}{\partial\Omega}=\phi$.
\end{romanenum}
In that case
\[
 \int_\Omega\abs{\nabla u_\phi}^2\dx = \min\Bigl\{\int_\Omega\abs{\nabla w}^2\dx \setcolon w\in H^1(\Omega),\ \Phi-w\in H^1_0(\Omega)\Bigr\}.
\]
\end{thm}
\begin{proof}
\begin{parenum}
\item[\ref{en:t2.7-ii}$\Rightarrow$\ref{en:t2.7-i}.]
Since $\Phi\in H^1(\Omega)$, $u_\phi=\Phi-v\in H^1(\Omega)$, where $v\in H^1_0(\Omega)$ is such that $\Delta v=\Delta\Phi$.
Moreover, by~\eqref{eq:2.5} and considering $W:=\{w\in H^1(\Omega)\setcolon \Phi-w\in H^1_0(\Omega)\}$,
\begin{align*}
 \frac{1}{2}[u_\phi,u_\phi] -\frac{1}{2}[\Phi,\Phi] &= \frac{1}{2}[v,v]-[\Phi,v] \\
 &= \min\Bigl\{\frac{1}{2}[\Phi-w,\Phi-w] - [\Phi,\Phi-w] \setcolon w\in W\Bigr\} \\
 &= \min\Bigl\{\frac{1}{2}[w,w] - \frac{1}{2}[\Phi,\Phi] \setcolon w\in W\Bigr\} \\
 &= \frac{1}{2} \min\Bigl\{[w,w] \setcolon w\in W\Bigr\}-\frac{1}{2}[\Phi,\Phi].
\end{align*}
Thus $[u_\phi,u_\phi] = \min\{[w,w]\setcolon w\in W\}$ and this property characterises $u_\phi$.
\item[\ref{en:t2.7-i}$\Rightarrow$\ref{en:t2.7-ii}.]
Let $\Phi\in C(\clos{\Omega})$ be such that $\restrict{\Phi}{\partial\Omega}=\phi$ and $\Delta\Phi\in H^{-1}(\Omega)$.
Let $v\in H^1_0(\Omega)$ be such that $\Delta v=\Delta\Phi$.
Then $u_\phi=\Phi-v$.
Since $u_\phi\in H^1(\Omega)$ by hypothesis, it follows that $\Phi\in H^1(\Omega)$.\qedhere
\end{parenum}
\end{proof}

If $\Omega$ is more regular, the following holds.
\begin{thm}\label{thm:2.9}
Assume that $\Omega$ has Lipschitz boundary and let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item $u_\phi\in H^1(\Omega)$
\item $\phi\in H^{1/2}(\partial\Omega)$
\end{romanenum}
\end{thm}
Here, since $\Omega$ has Lipschitz boundary, $H^{1/2}(\partial\Omega)=\{\operatorname{tr}\Phi \setcolon \Phi\in H^1(\Omega)\}$, where $\operatorname{tr}\colon H^1(\Omega)\to L^2(\partial\Omega,\mathcal{H}^{d-1})$ is the trace operator and $\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.
If $\phi\in C(\partial\Omega)\cap H^{1/2}(\partial\Omega)$, there even exists a $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $\restrict{\Phi}{\partial\Omega}=\phi$.
We refer to~\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\ref{thm:2.9}.
A more general result on very rough domains is~\cite[Theorem~5.8]{AtES24}.

Theorem~\ref{thm:5.1} is contained in Landkof~\cite[p.\,243, lines 1--5]{La72}.
We will give a very different proof based on the following result. We assume that $d\ge 3$.
\begin{thm}\label{thm:5.2}
Let $z_0\in \partial\Omega$ be a singular point of $\Omega$.
Then there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$, $z_0\in\partial\Omega_0$ and $z_0$ is a singular point of $\Omega_0$.
\end{thm}
We postpone the proof of Theorem~\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.
More precisely, given the potential $V\colon D\to(0,\infty)$ from Lemma~\ref{lem:lebesgue-potential-basics}, we had $\Omega=\{x\in D\setcolon \frac{1}{2}<V(x)<2\}$. We can choose e.g.~$\Omega_0=\{x\in D\setcolon \frac{1}{c}<V(r,z)<c\}$ for any $c\in (1,2)$.
Now we can easily prove Theorem~\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\cite[Theorem~6.74]{AU23}.
\begin{prop}\label{prop:5.4}
Let $z_0\in \partial\Omega$. Then $z_0$ is regular if there exists a \emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\ a function $b\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $b(x)>0$ for all $x\in\clos{\Omega}\setminus\{z_0\}$, $b(z_0)=0$ and $-\Delta b\ge 0$ weakly on $\Omega$.
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:5.1}]
Let $z_0\in \partial\Omega$ be a singular point for $\Omega$.
By Theorem~\ref{thm:5.2} there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$ and $z_0\in\partial\Omega_0$ is a singular point with respect to $\Omega_0$.
Let $0\le\phi\in C(\partial\Omega)$ be such that $\phi(z_0)=0$ and $\phi(w)>0$ at a regular point $w\in \partial\Omega$.
Then there exists $0\le\Phi_0\in C^\infty_\cpt(\RR^d)$ such that $\phi_0:=\restrict{\Phi_0}{\partial\Omega}$ satisfies $\phi_0\le\phi$ and $\phi_0(w)>0$.
Let $u_\phi$ and $u_{\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\Omega$.
Then $u_\phi\ge u_{\phi_0}\ge 0$ on $\Omega$ by the maximum principle, and $u_{\phi_0}\ne 0$.
Moreover, $u_{\phi_0}\in H^1(\Omega)$ by Theorem~\ref{thm:2.7}.

We call $u_\varphi$ the \textbf{\boldmath variational solution\unboldmath} of~\ref{eq:D}.
It attains the given boundary value $\varphi$ in the Sobolev sense.
The existence of such a variational solution is difficult to prove and was an open problem for general $\varphi\in C(\partial\Omega)$ until recently.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.
We then establish diverse properties of the variational solution defined as above.
We show that it is well-defined; i.e.\ $u_\varphi$ does not depend on the extension $\Phi$ of $\varphi$ and it coincides with the classical solution of~\ref{eq:D} whenever a classical solution exists.
Moreover, we show that the weak maximum principle is valid for variational solutions.
In Section~\ref{sec:3} we will see that $u_\varphi$ coincides with the Perron solution.
This notion was defined by Perron in 1923~\cite{Per1923} and has been a subject of intense research.
One subject is the way in which $u_\varphi(x)$ attains the value $\varphi(z)$ as $x\to z$.
So Theorem~\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.

In the second part of the paper, from Section~\ref{sec:4} on, we investigate the behaviour of $u_\varphi(x)$ at a fixed singular point $z_0\in\partial\Omega$, i.e.\ a point where $\lim_{x\to z_0} u_\varphi(x)=\varphi(z_0)$ is violated for some $\varphi\in C(\partial\Omega)$.

In a single page paper Lebesgue~\cite{Leb1913} considered a potential in $\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.
He chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.
We call this region \textbf{\boldmath Lebesgue's domain\unboldmath}.

Lebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \textbf{\boldmath Lebesgue cusp\unboldmath}.
Taking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.
This is of great interest since in $\mathbb{R}^2$ each simply connected domain is \textbf{\boldmath Dirichlet regular\unboldmath}; i.e.\ $u_\varphi$ is a classical solution for all $\varphi\in C(\partial\Omega)$.
Our point is to consider the original Lebesgue domain, whose boundary has two connected components.
It demonstrates two striking features of the generalised solution $u_\varphi$.

\begin{enumerate}[1.]
\item It is impossible to deduce from the regularity properties of $\varphi$ whether or not $u_\varphi$ is a classical solution.
The boundary of Lebesgue's domain is the disjoint union of two closed sets $\Gamma_2$ and $\Gamma_{\frac{1}{2}}$ (which are level sets of the considered potential).
If $\varphi$ is constant on each of these two closed sets with two different constants, $u_\varphi$ is not a classical solution.
To prove this, the variational definition of $u_\varphi$ will be most useful.

\item The irregular behaviour of $u_\varphi(x)$ as $x\to z_0$ is a non-local and unstable phenomenon.
For $\phi_0\equiv 0$ one has $u_{\phi_0}\equiv 0$, but if we perturb $\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\varphi$ is no longer a classical solution.
This can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\ref{sec:5}.
\end{enumerate}

As explained above, there seems to be no reasonable condition on the boundary function $\varphi$ to ensure that $u_\varphi$ is a classical solution of~\ref{eq:D} -- even though there are always many classical solutions.
The situation is different if we instead consider the question of when $u_\varphi$ has finite energy $\int_\Omega\abs{\nabla u_\varphi}^2$.
This is the case if and only if $\varphi$ has an extension $\Phi\in C(\overline{\Omega})\cap H^1(\Omega)$, and then $u_\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.
As an easy consequence, if $\varphi$ is Lipschitz continuous, then $u_\varphi$ has finite energy.
These results are proved in Section~\ref{sec:2} along with other properties of the variational solution.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a \emph{nontrivial extension result} that we shall take as given.

\label{thm:1.1}
Given $\varphi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\varphi$ on $\Omega$ and an extension $\Phi\in C(\overline{\Omega})$ of $\varphi$ such that $\Phi-u_\varphi\in H^1_0(\Omega)$.,

Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). Given a boundary function \(\varphi\in C(\partial\Omega)\), which existence-and-uniqueness statement holds? (Here \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\).)

There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that there exists at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) with \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u_\varphi\in H^1_0(\Omega)\).

Choices:
B. There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that for every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) one has \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. There exists at least one harmonic function \(u\) on \(\Omega\) and at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u\in H^1_0(\Omega)\).
D. There exists exactly one function \(u_\varphi\in C(\overline{\Omega})\) that is harmonic on \(\Omega\) and satisfies \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may choose \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\), there exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1(\Omega)\).

The stem gives only the setup and definitions; it does not explicitly state the correct conclusion or provide a direct hint toward choice A over the nearby variants.

The item is close to theorem-recognition: the correct option is essentially a precise theorem statement under the given hypotheses. However, it is not a pure restatement because the alternatives vary quantifiers, uniqueness, and regularity in meaningful ways.

To choose correctly, the solver must distinguish subtle logical differences: existence vs uniqueness, existential vs universal choice of extension, and classical boundary regularity vs weaker harmonicity statements. That creates substantial reasoning pressure.

The distractors are mathematically plausible and target common failure modes: overclaiming boundary continuity (B), dropping uniqueness (C), mishandling quantifiers (D), and imposing unnecessary H^1/weak-solution structure (E). They are distinct and well-designed.

A strong MCQ with subtle, high-quality distractors and little answer leakage. Its main weakness is that it still functions partly as theorem-statement recognition rather than fully generative problem solving.


Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). Given a boundary function \(\varphi\in C(\partial\Omega)\), which existence-and-uniqueness statement holds? (Here \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\).)

Choices:
B. There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that for every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) one has \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. There exists at least one harmonic function \(u\) on \(\Omega\) and at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u\in H^1_0(\Omega)\).
D. There exists exactly one function \(u_\varphi\in C(\overline{\Omega})\) that is harmonic on \(\Omega\) and satisfies \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may choose \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\), there exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1(\Omega)\).

The stem does not explicitly state the correct theorem or its key quantifier structure. It gives standard setup and notation, but the correct choice is not leaked; the test-taker must distinguish among subtly different existence/uniqueness formulations.

The item is close to a standard theorem statement about the weak Dirichlet problem, so it has some theorem-restatement flavor. However, it is not purely tautological because the options introduce meaningful variations in quantifiers, regularity, and uniqueness.

The question requires moderate reasoning: the solver must recognize why the exact weak formulation is correct and why stronger or weaker alternatives fail. Still, it mainly tests precise theorem recall/recognition rather than deeper generative inference.

The distractors are strong and mathematically plausible. They target common failure modes: over-strong universal quantification, loss of uniqueness, unjustified upgrade to classical boundary attainment, and confusion between H^1 and H^1_0.

A solid MCQ with no major answer leakage and high-quality distractors, but it remains somewhat close to theorem recognition rather than a strongly generative reasoning task.