\begin{lem}$\mathrm{(Lemma \textbf{ }2.6 \text{ of }\cite{Zhu})}$\label{separation}
		If $ f:M\to M$ is a non-degenerate endomorphism, or, more generally, if $ f:M\to M $ is a local homeomorphism, then $ f $ has uniform separation of preimages.
	\end{lem}

In modern dynamical systems and ergodic theory, both entropy and pressure are the fundamental quantities that measure the complexity of a system. These concepts are closely related to physics and thermodynamic mechanisms; see the monographs by Ruelle \cite{Ruelle} and Bowen \cite{Bowen book}. Roughly speaking, the topological pressure can be interpreted as the maximal free energy that a system can attain. Then, the Gibbs distribution, which can maximize free energy, corresponds to the equilibrium state in ergodic theory. If one neglects the potential energy, topological pressure reduces to topological entropy.

	Invariant measures play central roles in ergodic theory. Let $f$ be a continuous map on a compact metric space $X$. Given an $f$-invariant measure $\mu$, one can define the measure-theoretic entropy $h_{\mu}(f)$. The variational principle establishes a relationship between the measure-theoretic entropy and the topological entropy  $h(f)$ as
	$$
	h(f)=\sup\{h_{\mu}(f) \;|\; \mu\in \mathcal{M}_f\},
	$$
	where $ \mathcal{M}_f $ is the set of $ f $-invariant Borel probability measures.	The element in $ \mathcal{M}_f $ that makes the equality hold is called the \emph{measure of maximal entropy}.
	Given a continuous function $\phi$ on $X$, referred to as a potential, the variational principle for pressure then yields
	$$
	P(f,\phi)=\sup\{h_{\mu}(f)+\int \phi d\mu \;|\; \mu\in \mathcal{M}_f\}.
	$$
	The element in $ \mathcal{M}_f $ that reaches the supremum is called the \emph{equilibrium state} for $\phi$. It follows that when the potential function vanishes, the topological pressure reduces to the topological entropy, and the corresponding equilibrium state coincides with a measure of maximal entropy.

	The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic,  there are two classical ones established in the  1970s. The first one is using the transfer operator theory, see in \cite{Bowen book} and \cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\Lambda$ and then obtain a symbolic extension of $f_\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\Lambda$ is a topologically mixing compact locally maximal hyperbolic set
	for a diffeomorphism $f$, then $f_\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.

	In recent years, the study of dynamics of the differentiable systems beyond uniform hyperbolicity, such as non-uniformly hyperbolic systems and partially hyperbolic systems,  has attracted much attention and made a series of important advances. For partially hyperbolic diffeomorphisms, a center direction is allowed in addition to the hyperbolic directions. The presence of the center direction permits a very rich type of structure. Investigating the thermodynamic mechanisms, especially the existence and uniqueness of the equilibrium state, for partially hyperbolic systems require more exquisite techniques.

	Recently, Climenhaga and Thompson \cite{CT21} ingeniously developed the criterion of Bowen \cite{Bowen74} in settings beyond uniform hyperbolicity
	using weakened versions of specification and expansivity. They first considered the case of symbolic systems.
	Rather than assuming specification on the entire symbolic system, they identified a collection of ``good" words on which the specification property holds, while showing that the set of ``bad" words has topological entropy strictly smaller than that of the full system. Using this decomposition, they established the uniqueness of the measure of maximal entropy and the equilibrium state in \cite{CT12} and \cite{CT13}, respectively. Further, Climenhaga and Thompson weakened expansivity by introducing a weaker, non-uniform notion applicable to general compact metric spaces; they also formulate correspondingly weakened, non-uniform versions of specification, expansivity, and the Bowen property in \cite{CT14} and \cite{CT16}. For applications of the method mentioned above, one can see \cite{CFT18} and \cite{CFT19}. The general strategy is to first identify obstructions to expansivity and specification, and then to control their entropy or pressure. Pacifico, F. Yang, and J. Yang \cite{Yang} further refined the conditions in the Climenhaga-Thompson approach, allowing the framework to be applied in settings with fixed parameters.

	From \cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \cite{Mongez} substituted the condition on the central Lyapunov exponent used in \cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \cite{Yan21, HHW}.  Liu and Liao  \cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\"{o}lder continuous potentials with not very large oscillation.

	Partially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \cite{Wang, Zhu}.   Álvarez and Cantarino \cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and  Cabezas  \cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.

	Our main purpose in this paper is to generalize the results in \cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \ref{separation}), then we can deal with the preimages on a small domain. Denote  $U_a(\phi):=\{\psi\in C(X)\;|\; \lVert \phi-\psi \rVert <a\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \lVert \phi \rVert=\sup_{x\in X}\lvert \phi(x)\rvert $.

	\begin{mainthm}\label{Main}
		Let $ f:M\to M $ be a non-degenerate $ C^2 $  partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \phi:M\to \mathbb{R} $ be a H\"{o}lder continuous potential. Assume that  $ P^u(f,\phi)>P^s(f,\phi)$ and  the unstable foliation $\tilde{ \mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2} $
		there exists a $ C^1 $ neighborhood $ \mathcal{U}_f $ of $ f $ such that for any H\"{o}lder continuous potential $\psi\in U_{a}(\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\in \mathcal{U}_f $ there   has a unique equilibrium state for $(g,\psi)$.
	\end{mainthm}

The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic, there are two classical ones established in the 1970s. The first one is using the transfer operator theory, see in \cite{Bowen book} and \cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\Lambda$ and then obtain a symbolic extension of $f_\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\Lambda$ is a topologically mixing compact locally maximal hyperbolic set
 for a diffeomorphism $f$, then $f_\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.

From \cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \cite{Mongez} substituted the condition on the central Lyapunov exponent used in \cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \cite{Yan21, HHW}. Liu and Liao \cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\"{o}lder continuous potentials with not very large oscillation.

Partially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \cite{Wang, Zhu}. Álvarez and Cantarino \cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and Cabezas \cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.

Our main purpose in this paper is to generalize the results in \cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \ref{separation}), then we can deal with the preimages on a small domain. Denote $U_a(\phi):=\{\psi\in C(X)\;|\; \lVert \phi-\psi \rVert <a\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \lVert \phi \rVert=\sup_{x\in X}\lvert \phi(x)\rvert $.

\begin{mainthm}\label{Main}
 Let $ f:M\to M $ be a non-degenerate $ C^2 $ partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \phi:M\to \mathbb{R} $ be a H\"{o}lder continuous potential. Assume that $ P^u(f,\phi)>P^s(f,\phi)$ and the unstable foliation $\tilde{ \mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2} $
 there exists a $ C^1 $ neighborhood $ \mathcal{U}_f $ of $ f $ such that for any H\"{o}lder continuous potential $\psi\in U_{a}(\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\in \mathcal{U}_f $ there has a unique equilibrium state for $(g,\psi)$.
 \end{mainthm}

The following results in \cite{CT16} and \cite{Yang} respectively proved the uniqueness of equilibrium states for certain systems.
 \begin{thm} $(\mathrm{Theorem \text{ }5.6 \text{ of } \cite{CT16}})$\label{CT}
 Let $ X $ be a compact metric space, $ f:X\to X $ a homeomorphism and $ \phi:X\to \mathbb{R} $ a continuous potential function. Suppose that there are $ \varepsilon>0$, $\delta>0 $ with $ \varepsilon>40\delta $ such that $P^\perp_{\mathrm{exp}}(f,\phi,\varepsilon)<P(f,\phi) $ and there exists $ \mathcal{D}\subset X\times \mathbb{N} $ admits a decomposition $ (\mathcal{P},\mathcal{G},\mathcal{S}) $ with the following properties:
 \begin{flushleft}
 (1) For every $ N\in\mathbb{N} $, $ \mathcal{G}^N $ has tail $ (W) $-specification at scale $ \delta $;\\
 (2) $ \phi $ has the Bowen property at scale $ \varepsilon $ on $ \mathcal{G} $;\\
 (3) $P(\mathcal{P}\cup\mathcal{S}\cup\mathcal{D}^c,\phi,\delta,\varepsilon)<P(f,\phi) $.\\
 Then there exists a unique equilibrium state for $ (f,\phi) $.
 \end{flushleft}
 \end{thm}
 \begin{thm} $(\mathrm{Theorem \text{ }A} \text{ of }\cite{Yang})$\label{2.2}
 Let $ X $ be a compact metric space, $ f:X\to X $ a homeomorphism and $ \phi:X\to \mathbb{R} $ a continuous potential function. Suppose that there exist $ \varepsilon>0$, $\delta>0 $ with $ \varepsilon\geq 2000\delta $ such that $ P^\perp_{\mathrm{exp}}(f,\phi,\varepsilon)<P(f,\phi) $ and $ X\times \mathbb{N} $ admits a decomposition $ (\mathcal{P},\mathcal{G},\mathcal{S})$ with the following properties:
 \begin{flushleft}
 (1) $ \mathcal{G} $ has $ (W) $-specification at scale $ \delta $;\\
 (2) $ \phi $ has the Bowen property at scale $ \varepsilon $ on $ \mathcal{G} $;\\
 (3) $ P(\mathcal{P}\cup\mathcal{S},\phi,\delta,\varepsilon)<P(f,\phi) $.\\
 Then there exists a unique equilibrium state for $ (f,\phi) $.
 \end{flushleft}
 \end{thm}
 Theorem \ref{CT} is sometimes referred to as the Climenhaga-Thompson Criterion (abbrev. CT Criterion). As an application of the above criteria, Mongez and Pacifico \cite{Mongez} gave the following result for certain partially hyperbolic diffeomorphisms.
 \begin{thm}$(\mathrm{Theorem \text{ }A} \text{ of }\cite{Mongez})$ \label{Mon}
 Let $f: M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact manifold $M$ with $TM=E^u\oplus E^c\oplus E^s$ and $\phi:M\to \mathbb{R}$ a H\"{o}lder continuous potential. Assume that dim($E^c$)=1 and the unstable foliation $\mathcal{F}^u(f)$ is minimal. If $h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$, then there exists a $C^1$ neighborhood $\mathcal{U}$ of $f$ so that $(g,\phi)$ has a unique equilibrium state for every $C^{1+}$ diffeomorphism $g\in \mathcal{U}$.
 \end{thm}
 The main task of this paper is to obtain the counterpart of Theorem \ref{Mon} in the setting of partially hyperbolic endomorphisms under weaker conditions.

Next, we will use the results in \cite{Yun, Chung, Gelfert} that for any $ C^{1+\alpha} $ mapping, there exists a horseshoe whose topological pressure can approximate the sum of the metric entropy and the integral of the potential whenever the metric entropy is positive for an ergodic hyperbolic measure.

As we know that the horseshoe for $ C^{1+\alpha} $ mapping is structurally stable under small $ C^1 $ perturbation in \cite{Chung}, i.e. there is a $ C^1 $ neighborhood $ \mathcal{U}_1 $ of $ f $ such that there exists a hyperbolic horseshoe $ H_g $ and $
 \tau|_{(H_g)^g}: (H_g)^g \to (H_g)^g $ is topologically conjugate to $\tau|_{(H_\varepsilon)^f}: (H_\varepsilon)^f \to (H_\varepsilon)^f $ by $ h_g:(H_g)^g \to (H_\varepsilon)^f $ with $ \lVert h_g-id \rVert<\delta $
 for any $ g\in \mathcal{U}_1 $ where $(H_g)^g$, $(H_\varepsilon)^f$ are the inverse limit spaces corresponding to the horseshoes respectively and $ \lVert \phi^*\circ h_g-\phi^*\rVert<\varepsilon $ if $ \lVert h_g-id \rVert<\delta $. Then we have
 \begin{align*}
 P(\tau|_{(H_{\varepsilon})^f},\phi^*|_{(H_{\varepsilon})^f})=P(\tau|_{(H_g)^g}, \phi^*\circ h_g).
 \end{align*}
 and
 \begin{align*}
 P^u(f,\phi)-2\varepsilon<P(g|_{H_g}, \phi|_{H_g}).
 \end{align*}
 In order to relate the above inequality to the unstable pressure of $g$, we need to establish the existence of the equilibrium state for $g|_{H_g}$ as follows.
 \begin{claim}
 If $ H_g $ is a hyperbolic set for the endomorphism $ g $, then $ \tau|_{(H_g)^g} $ is expansive.
 \end{claim}
 In other words, we claim that there is a constant $ \eta>0 $ small enough such that for any $ \tilde{x},\tilde{y} \in (H_g)^g$ with $ \tilde{x}\neq \tilde{y}$ then $ \tilde{d}(\tau^k\tilde{x}, \tau^k\tilde{y})>\eta $ for some $ k\in \mathbb{Z} $. Otherwise $ \tilde{y}\in \tilde{W}^u_{\eta}(\tilde{x})\cap \tilde{W}^s_{\eta}(\tilde{x})$.
 Since $ \tilde{x}\in \tilde{W}^u_{\eta}(\tilde{x})\cap \tilde{W}^s_{\eta}(\tilde{x})$ and $ \Pi_{\tilde{W}^u_{loc}(\tilde{x})}: \tilde{W}^u_{loc}(\tilde{x}) \to W^u_{loc}(\tilde{x}) $ is bijective, then by the local product structure in Theorem IV.2.3 of \cite{Qian}, we have $\tilde{x}=\tilde{y} $. This completes the proof of this claim.

Now we assume that $f $ is a $ C^2 $ non-degenerate partially hyperbolic endomorphism on the compact Riemannian manifold with dim$(E^c)=1 $ and $ \phi $ is a H\"{o}lder continuous potential function on $ M $. Recall that let $\phi^*=\phi\circ \Pi$. Given an ergodic measure $ \mu\in \mathcal{M}^e_f $, we can consider the central Lyapunov exponents by Theorem I.2.1 and Proposition I.3.5 of \cite{Qian}. We use $\tilde{\mu} $ to denote the ergodic $ \tau $-invariant Borel probability measure on $ M^f $ such that $ \Pi \tilde{\mu}=\mu $. We denote $ \lambda^c(\tilde{\mu},\tau) $ (resp. $\lambda^c(\mu,f)$) be the central Lyapunov exponent of $ \tilde{\mu} $ (resp. $ \mu$) whenever $ \tilde{\mu}$ (resp. $\mu $) is ergodic, i.e. for $\tilde{\mu}$-$a.e.$ $ \tilde{x}=(x_n)_{n\in\mathbb{Z}}\in M^f$ (resp. $ \mu$-$a.e.$ $x_0\in M $), we have
 \begin{align*}
 \lambda^c(\mu,f)=\lambda^c(\tilde{\mu},\tau)
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lvert D_{x_0}f^m \xi\rvert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lVert D_{x_0}f^m |_{E^c(\tilde{x})}\rVert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lVert D_{x_{m-1}}f\circ \cdots \circ D_{x_1}f \circ D_{x_0}f |_{E^c(\tilde{x})}\rVert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}\log\lVert D_{x_k}f |_{E^c(\tau^k\tilde{x})}\rVert\\
 &=\int_{M^f} \log\lVert Df |_{E^c(\tilde{x})}\rVert d\tilde{\mu},
 \end{align*}
 where $ \tilde{x}\in M^f $ with $ \Pi(\tilde{x})=x_0 $, $ 0\neq\xi\in E^c(\tilde{x}) $, and define $\tilde{\varphi}^c(\tilde{x}):=\log\lVert Df |_{E^c(\tilde{x})}\rVert$.

\begin{lem}
 There exists $ r>0 $ such that
 $ \sup\{h_{\tilde{\mu}}(\tau)+\int\phi^* d\tilde{\mu}: \tilde{\mu}\in \mathcal{M}_\tau, \lambda^c(\tilde{\mu},\tau)\geq -r\}<P(\tau,\phi^*) $.
 \end{lem}
 \begin{pf}
 First, we claim that
 $ \sup\{h_{\tilde{\mu}}(\tau)+\int\phi^* d\tilde{\mu}: \tilde{\mu}\in \mathcal{M}_\tau, \lambda^c(\tilde{\mu},\tau)\geq 0\}<P(\tau,\phi^*) $. If not, we can assume there exists an invariant probability measure $ \tilde{\mu} $ with $ \lambda^c(\tilde{\mu},\tau)\geq 0 $, but $ \tilde{\mu} $ is an equilibrium state for $ (\tau,\phi^*) $. By the ergodic decomposition theorem, we have
 $ \lambda^c(\tilde{\mu},\tau)=\int \tilde{\varphi}^c d\tilde{\mu}=\int_{\mathcal{M}_\tau^e} (\int \tilde{\varphi}^c dm) d\tau(m) \geq 0$,
 and by Theorem 8.4 in \cite{Walters} we get
 $ h_{\tilde{\mu}}(\tau)=\int_{\mathcal{M}_\tau^e} h_m(\tau)d\tau(m) $.
 Then
 $$
 P(\tau,\phi^*)=h_{\tilde{\mu}}(\tau)+\int \phi^* d\tilde{\mu}
 =\int_{\mathcal{M}_\tau^e} (h_m(\tau)+\int \phi^* dm)d\tau(m),
 $$
 so each $ m\in \mathcal{M}_\tau^e $ must be the equilibrium state for $ (\tau,\phi^*) $. Therefore, there exists a measure $ m_0\in\mathcal{M}_\tau^e $ such that
 $ \lambda^c(m_0,\tau)=\int \tilde{\varphi}^c dm_0\geq 0 $
 and
 $ P(\tau,\phi^*)=h_{m_0}(\tau)+\int \phi^* dm_0 $,
 which contradicts with $ P^+(\tau,\phi^*)<P^-(\tau,\phi^*) $.

The proof of the main theorem in this paper is inspired by arguments established in \cite{CT21} and \cite{Mongez}. Compared with the case of diffeomorphisms, the main difficulty here comes from non-invertibility. In order to overcome this deficiency, we use the technique of inverse limit. For certain non-degenerate partially hyperbolic endomorphisms with one-dimensional center and any  H\"{o}lder continuous potential function satisfy specific conditions, we aim to construct a decomposition of orbit segments in the inverse limit spaces. An important condition in the main theorem is the gap between the unstable pressure and the stable pressure. For partially hyperbolic diffeomorphisms, the unstable entropy and unstable pressure for a continuous function were introduced in \cite{HHW, HWZ}, respectively. As a generalization, the concepts of unstable entropy and unstable pressure were given by Wang, Wu and Zhu in \cite{Wang} for partially hyperbolic endomorphisms. In \cite{Zhu}, Wu and Zhu introduced the stable entropy for non-degenerate $C^1$  partially hyperbolic endomorphisms and their stable pressure is given in \cite{Li}. Different from the condition given by Mongez and Pacifico in \cite{Mongez}, we change the condition ``$h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$'' to ``$P^u(f,\phi)>P^s(f,\phi)$'', where
	$P^u(f,\phi)=\sup\{h^u_{\mu}(f)+\int \phi d\mu\;|\; \mu\in \mathcal{M}_f^e\}$ and $P^s(f,\phi)=\sup\{h^s_{\mu}(f)+\int \phi d\mu\;|\; \mu\in \mathcal{M}_f^e\}$.
	Since for each ergodic $f$-invariant measure $\mu$, there is
	\begin{align*}
		h^u_\mu(f)+\min\phi \leq h^u_\mu(f)+\int \phi d\mu,
	\end{align*}
	we have $P^u(f,\phi)\geq h^u(f)+\min\phi$.
	Meanwhile, $h^s(f)+\max\phi \geq P^s(f,\phi)$. Therefore, the condition ``$P^u(f,\phi)>P^s(f,\phi)$'' is weaker than ``$h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$'' since the space we are considering is compact so $\sup\phi = \max\phi$ and $\inf\phi=\min\phi$.
	Under the condition $ P^u(f,\phi)>P^s(f,\phi) $ and assuming the existence of unstable foliation with some sense of minimality, we show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction. Moreover, the potential $ \phi $ satisfies the Bowen property on these good orbit segments. Additionally, the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system. As a consequence, there exists a unique equilibrium state on the inverse limit space, and its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism. Furthermore, we also demonstrate that the conditions of the theorem are inherently robust, which further implies the robustness of the equilibrium state.
	At the same time, H\"{o}lder continuous potential functions can also vary within a certain range,  thus allowing the uniqueness of the equilibrium state of the system under small perturbation can be obtained.

	This paper is organized as follows. In Section 2, we give some preliminaries, including the concepts of the decomposition of the orbit segments, the pressure of the obstructions to expansivity, partially hyperbolic endomorphisms, inverse limit space, (un)stable en        tropy and (un)stable pressure, and other necessary definitions. In Section 3, we demonstrate the robustness of the conditions in Main Theorem. In Section 4, we prove  Main Theorem using the improved Climenhaga-Thompson criterion. In Section 5, we give an example satisfying the conditions in our main theorem.

The proof is said to be “inspired by arguments established in \cite{CT21} and \cite{Mongez},” with the “main difficulty” (vs. diffeomorphisms) coming from “non-invertibility,” which is handled by “the technique of inverse limit.” Under the condition $P^u(f,\phi)>P^s(f,\phi)$ (introduced as a weakening of Mongez–Pacifico’s condition), the strategy is to “construct a decomposition of orbit segments in the inverse limit spaces.” Assuming “the existence of unstable foliation with some sense of minimality,” they “show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction,” and that the potential $\phi$ “satisfies the Bowen property on these good orbit segments.” They also show that “the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system.” Then, “as a consequence, there exists a unique equilibrium state on the inverse limit space,” and “its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism.” Finally, they “demonstrate that the conditions of the theorem are inherently robust,” implying “robustness of the equilibrium state,” and note that H\"older potentials “can also vary within a certain range,” giving uniqueness “under small perturbation.”

The results of the Main Theorem still hold for $ C^{1+\alpha} $ mappings. To cite some conclusions in \cite{Qian} conveniently, we therefore assume $ C^2 $ regularity here.,

Question:
Let $M$ be a compact Riemannian manifold, let $f:M\to M$ be a non-degenerate $C^{1+\alpha}$ partially hyperbolic endomorphism for some $\alpha>0$ with one-dimensional center bundle, and let $\phi:M\to\mathbb{R}$ be a H\"older continuous potential. Assume that the unstable pressure and stable pressure satisfy $P^u(f,\phi)>P^s(f,\phi)$, and that the unstable foliation $\tilde{\mathcal W}^u$ (defined on the inverse-limit space associated to $f$) is minimal, meaning every unstable leaf is dense. For $a>0$, define
$$U_a(\phi):=\{\psi\in C(M):\|\phi-\psi\|_\infty<a\}.$$
For any
$$0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2},$$
which statement holds for every such $f$, $\phi$, and $a$?

There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits a unique equilibrium state for the potential $\psi$.

Choices:
B. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits at least one equilibrium state for the potential $\psi$, and this equilibrium state is unique whenever $P^u(g,\psi)>P^s(g,\psi)$.
C. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that the system $g$ admits a unique equilibrium state for the potential $\phi$ for every $g\in\mathcal U$.
D. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits a unique equilibrium state for the potential $\psi$ whenever $0<a<P^u(f,\phi)-P^s(f,\phi)$.
E. For every $\psi\in U_a(\phi)$, there exists a $C^1$ neighborhood $\mathcal U_\psi$ of $f$ such that for every $g\in\mathcal U_\psi$, the system $g$ admits a unique equilibrium state for the potential $\psi$.

robust uniqueness tied to fixed pressure-gap/minimality hypotheses rather than rechecking perturbed pressures

dropped the uniformity over all nearby potentials $\psi\in U_a(\phi)$

half-gap bound on allowable oscillation radius

uniform neighborhood independent of the perturbed potential


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


Question:
Let \(\Omega_T=\Omega\times(0,T)\), let \(a\ge 0\) with \(a\in C^{\alpha,\alpha/2}(\Omega_T)\), and define \(H(z,s)=s^p+a(z)s^q\). Let \(\mathcal A:\Omega_T\times\mathbb R^{Nn}\to\mathbb R^{Nn}\) be a Carath\'eodory field satisfying the double-phase structure conditions \(\mathcal A(z,\xi)\cdot \xi\ge L_1(|\xi|^p+a(z)|\xi|^q)\) and \(|\mathcal A(z,\xi)|\le L_2(|\xi|^{p-1}+a(z)|\xi|^{q-1})\). Assume \(q<p+\frac{2\alpha}{n+2}\) and \(p>2\). If \(u\in C(0,T;L^2(\Omega,\mathbb R^N))\) is a very weak solution with integrability deficit \(\delta\) of
\[
u_t-\operatorname{div}\mathcal A(z,\nabla u)=-\operatorname{div}\big(|F|^{p-2}F+a(z)|F|^{q-2}F\big)\quad\text{in }\Omega_T,
\]
meaning in particular that \(H(z,|\nabla u|)^\delta\in L^1(\Omega_T)\), and if \(Q_r(z_0)\) denotes a standard parabolic cylinder of radius \(r\) centered at \(z_0\), which existence-and-uniform local estimate statement holds?

There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,\alpha,[a]_{\alpha,\alpha/2;\Omega_T})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and whenever \(\delta\in(\delta_0,1)\) and \(u\) is such a very weak solution, there exist constants \(c\ge 1\) and \(R_0\in(0,1)\) with \(c\) depending on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and with \(R_0\) depending on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\), such that for every parabolic cylinder \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\),
\[
\dashint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\dashint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}.
\]

Choices:
B.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,\alpha,[a]_{\alpha,\alpha/2;\Omega_T})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and whenever \(\delta\in(\delta_0,1)\) and \(u\) is such a very weak solution, there exist constants \(c\ge 1\) and \(R_0\in(0,1)\) depending only on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), such that for every parabolic cylinder \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,1)\),
\[
\dashint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\dashint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}.
\]
C.
Assume there exists \(\delta_0\in(0,1)\) such that \(2-q(1-\delta_0)>0\). Then whenever \(\delta\in(\delta_0,1)\) and \(u\) is such a very weak solution, there exist constants \(c\ge 1\) and \(R_0\in(0,1)\) such that for every parabolic cylinder \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\),
\[
\dashint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\dashint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}.
\]
D.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,\alpha,[a]_{\alpha,\alpha/2;\Omega_T})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and whenever \(\delta\in(\delta_0,1)\) and \(u\) is such a very weak solution, there exist constants \(c\ge 1\) and \(R_0\in(0,1)\) with the same dependence as in the theorem, such that for every parabolic cylinder \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\),
\[
\dashint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\dashint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)^\delta\big)\,dz+1\right)^{1+\frac{q(1-\delta)}{2-q(1-\delta)}}.
\]
E.
There exists \(\delta_0=\delta_0(n,N,p,q,L_1,L_2,\alpha,[a]_{\alpha,\alpha/2;\Omega_T})\in(0,1)\) such that \(2-q(1-\delta_0)>0\), and whenever \(\delta\in(\delta_0,1)\) and \(u\) is such a very weak solution, there exist constants \(c\ge 1\) and \(R_0\in(0,1)\) with \(c\) depending on the structural data, \(\|a\|_{L^\infty(\Omega_T)}\), and \(\|H(z,|\nabla u|)+H(z,|F|)+1\|_{L^\delta(\Omega_T)}\), and with \(R_0\) depending on the structural data, \(\delta\), \(\|H(z,|\nabla u|)\|_{L^\delta(\Omega_T)}\), \(\|H(z,|F|)\|_{L^\delta(\Omega_T)}\), and \(\|u\|_{C(0,T;L^2(\Omega,\mathbb R^N))}\), such that for every parabolic cylinder \(Q_{2R}(z_0)\subset\Omega_T\) with \(R\in(0,R_0)\),
\[
\dashint_{Q_R(z_0)} H(z,|\nabla u|)\,dz
\le c\left(\dashint_{Q_{2R}(z_0)}\big(H(z,|\nabla u|)^\delta+H(z,|F|)\big)\,dz+1\right).
\]

small-radius restriction R<R0 and non-uniform dependence of R0

explicit dependence of \delta_0, c, and R_0 on the data/norms

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

Gehring gain exponent 1+q(1-\delta)/(2-q(1-\delta))