\begin{teorema}\label{thm:main}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then $\mathcal{S}_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\mathcal{S}_\mu^k$ at $x$ coincides with $D_x$ for $\mathcal{H}^{k}$-a.e.\ $x \in \mathcal{S}_\mu^k$.
\end{teorema}

The notion of blow-up of a geometric or analytic object, i.e.\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale".
For this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.

For a Radon measure $\mu$ in $\mathbb{R}^n$, a natural definition of blow-up at a point $x \in \mathbb{R}^n$ is the one in the sense of Preiss, \cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\mathbb{R}^n)$) of the rescalings
\begin{equation}\label{eq:rescalings}
c_j (\tau_{x,r_j})_\# \mu,
\end{equation}
for some sequences $c_j>0$, $r_j \to 0$, where $\tau_{x,r}(y)\coloneqq \frac{y-x}{r}$ and ${}_\#$ is the push-forward of Radon measures.

We say that $\mu$ has unique blow-up at $x \in \operatorname{supp} \mu$ if there exists a Radon measure $\mu_x$ with either $|\mu_x|(B_1(0)) = 1$ or $\mu_x = 0$ and such that, for every sequence $r_j \to 0^+$, there are a subsequence, not relabeled, and a constant $c\neq 0$ for which
\begin{equation}\label{eq:def-unique_blowup}
\mu_{x,r_j} \coloneqq
\frac{1}{|\mu|(B_{r_j}(x))} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \mu_x,
\end{equation}
where $|\mu|$ is the total variation measure of $\mu$ and $\overset{\ast}{\rightharpoonup}$ denotes the weak*-convergence of Radon measures.
We will denote by $\mathcal{S}_\mu$ the set of points where $\mu$ has unique blow-up.

Mattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \cite[Theorem 3.2]{mattila_unique05} is that,
for $\mu$-a.e.\ $x \in \mathcal{S}_\mu$, the blow-up of $\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\mathcal{H}^k$ on a $k$-plane.
Although this describes the blow-ups of $\mu$ at $\mu$-a.e.\ point in $\mathcal{S}_\mu$, it can be interesting to study the behavior of $\mu$ in the remaining part of $\mathcal{S}_\mu$.

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

\item It is easily seen (Lemma \ref{lmm:0-hom}) that the unique blow-up $\mu_x$ of $\mu$ at a point in $x \in \mathcal{S}_\mu$ has an \emph{invariant linear subspace} $D_x$ of $\mathbb{R}^n$, namely $(\tau_{y,1})_\# \mu_x = \mu_x$ for $y \in D_x$. Therefore
\begin{equation}\label{eq:def_s_mu_k}
\mathcal{S}_\mu = \bigcup_{k=1}^n \mathcal{S}_\mu^k,
\qquad\text{where} \qquad
\mathcal{S}_\mu^k \coloneqq \left \{x \in \mathcal{S}_\mu \colon \dim D_x=k \right \}.
\end{equation}
Do these $\mathcal{S}_\mu^k$ enjoy finer rectifiability properties?
\end{itemize}

The main result of this note provides an answer to these questions.

For a Radon measure $\mu$ in $\mathbb{R}^n$, a natural definition of blow-up at a point $x \in \mathbb{R}^n$ is the one in the sense of Preiss, \cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\mathbb{R}^n)$) of the rescalings
\begin{equation}\label{eq:rescalings}
c_j (\tau_{x,r_j})_\# \mu,
\end{equation}
for some sequences $c_j>0$, $r_j \to 0$, where $\tau_{x,r}(y)\coloneqq \frac{y-x}{r}$ and ${}_\#$ is the push-forward of Radon measures.

We say that $\mu$ has unique blow-up at $x \in \operatorname{supp} \mu$ if there exists a Radon measure $\mu_x$ with either $|\mu_x|(B_1(0)) = 1$ or $\mu_x = 0$ and such that, for every sequence $r_j \to 0^+$, there are a subsequence, not relabeled, and a constant $c\neq 0$ for which
\begin{equation}\label{eq:def-unique_blowup}
\mu_{x,r_j} \coloneqq
\frac{1}{|\mu|(B_{r_j}(x))} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \mu_x,
\end{equation}
where $|\mu|$ is the total variation measure of $\mu$ and $\overset{\ast}{\rightharpoonup}$ denotes the weak*-convergence of Radon measures.
We will denote by $\mathcal{S}_\mu$ the set of points where $\mu$ has unique blow-up.

Mattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \cite[Theorem 3.2]{mattila_unique05} is that,
for $\mu$-a.e.\ $x \in \mathcal{S}_\mu$, the blow-up of $\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\mathcal{H}^k$ on a $k$-plane.
Although this describes the blow-ups of $\mu$ at $\mu$-a.e.\ point in $\mathcal{S}_\mu$, it can be interesting to study the behavior of $\mu$ in the remaining part of $\mathcal{S}_\mu$.

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

\item It is easily seen (Lemma \ref{lmm:0-hom}) that the unique blow-up $\mu_x$ of $\mu$ at a point in $x \in \mathcal{S}_\mu$ has an \emph{invariant linear subspace} $D_x$ of $\mathbb{R}^n$, namely $(\tau_{y,1})_\# \mu_x = \mu_x$ for $y \in D_x$. Therefore
\begin{equation}\label{eq:def_s_mu_k}
\mathcal{S}_\mu = \bigcup_{k=1}^n \mathcal{S}_\mu^k,
\qquad\text{where} \qquad
\mathcal{S}_\mu^k \coloneqq \left \{x \in \mathcal{S}_\mu \colon \dim D_x=k \right \}.
\end{equation}
Do these $\mathcal{S}_\mu^k$ enjoy finer rectifiability properties?
\end{itemize}

The main result of this note provides an answer to these questions.

\begin{lemma}\label{lmm:0-hom}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and let $\mu_x \neq 0$ be the unique blow-up of $\mu$ at $x \in \mathcal{S}_\mu$. Then there exists $\alpha=\alpha(x)\geq 0$ such that $\mu_x$ and $|\mu_x|$ are $(\alpha-n)$-homogeneous,\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely
\begin{equation}\label{eq:hom_blow-up}
(\tau_{0,\lambda})_\# \mu_x=\lambda^\alpha \mu_x,
\quad
(\tau_{0,\lambda})_\# |\mu_x|=\lambda^\alpha |\mu_x|
\qquad
\forall \lambda>0.
\end{equation}
For $\mu_x$, and more generally for every $(\alpha-n)$-homogeneous Radon measure, the set
\begin{equation}
C_x\coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,\lambda})_\# \mu_x=\lambda^\alpha (\tau_{y,1})_\# \mu_x \,\,\,\,\forall \lambda>0 \right \}
\end{equation}
coincides with the invariant subspace $D_x$ of $\mu_x$, that is
\begin{equation}
D_x \coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,1})_\# \mu_x= \mu_x \right \}.
\end{equation}
Moreover $D_x$ is a linear subspace of $\mathbb{R}^n$ and, provided $\mu_x \neq 0$, it holds $\dim D_x \leq \alpha$.
\end{lemma}

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

The main result of this note provides an answer to these questions.

The idea of the proof is that, for every $x \in \mathcal{S}_\mu^k$, there is a radius $r>0$ for which the points in $\mathcal{S}_\mu^k \cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \notin x+C$ with $r_j=|x-y_j| \to 0$ and such that the measures $c_j \mu_{y_j}$ and $\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$; on the other hand $\mu_{y_j,r_j}$, being translations of $\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$.

\begin{corollario}\label{cor:rectif_criterion}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Let us assume that, for $|\mu|$-a.e.\ $x \in \mathbb{R}^n$, there exists a Radon measure $\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \to 0^+$ there are a subsequence, not relabeled, and a constant $c\neq 0$ such that
\begin{equation}
\frac{1}{r_j^k} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \nu_x.
\end{equation}
Then $\mu$ is $k$-rectifiable: there exists a signed function $\theta \in L^1_{\mathrm{loc}}(\mathcal{S}_\mu^k,\mathcal{H}^k)$ such that \mbox{$\mu= \theta \mathcal{H}^k \llcorner \mathcal{S}_\mu^k$,} where $\mathcal{S}_\mu^k$ is defined in \eqref{eq:def_s_mu_k}.
\end{corollario}

\begin{corollario}\label{cor:almost_everywhere_rect}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then, for $|\mu|$-a.e.\ $x \in \mathcal{S}_\mu^k$, the unique blow-up of $\mu$ at $x$ is a multiple of $\mathcal{H}^k\llcorner D_x$.
\end{corollario}

\begin{lemma}\label{lmm:0-hom}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and let $\mu_x \neq 0$ be the unique blow-up of $\mu$ at $x \in \mathcal{S}_\mu$. Then there exists $\alpha=\alpha(x)\geq 0$ such that $\mu_x$ and $|\mu_x|$ are $(\alpha-n)$-homogeneous,\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely
\begin{equation}\label{eq:hom_blow-up}
(\tau_{0,\lambda})_\# \mu_x=\lambda^\alpha \mu_x,
\quad
(\tau_{0,\lambda})_\# |\mu_x|=\lambda^\alpha |\mu_x|
\qquad
\forall \lambda>0.
\end{equation}
For $\mu_x$, and more generally for every $(\alpha-n)$-homogeneous Radon measure, the set
\begin{equation}
C_x\coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,\lambda})_\# \mu_x=\lambda^\alpha (\tau_{y,1})_\# \mu_x \,\,\,\,\forall \lambda>0 \right \}
\end{equation}
coincides with the invariant subspace $D_x$ of $\mu_x$, that is
\begin{equation}
D_x \coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,1})_\# \mu_x= \mu_x \right \}.
\end{equation}
Moreover $D_x$ is a linear subspace of $\mathbb{R}^n$ and, provided $\mu_x \neq 0$, it holds $\dim D_x \leq \alpha$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm:main}]
The statement is trivial for $k=n$, thus we assume $k \in \{0,1,\dots,n-1\}$ be fixed throughout the proof.
Let us moreover fix $\gamma \in \left (0,\frac{1}{16}\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\{V_\ell\}_{\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\overline{C(0,V_\ell,\gamma)}$ for some $\ell \in \{1,\dots,m\}$. For every $\ell \in \{1,\dots,m\}$ and every $i \in \mathbb{N}$, we define the set
\begin{equation}
E_{\ell,i} \coloneqq \left \{ x \in \mathcal{S}^k \colon D_x \subset \overline{C(0,V_\ell,\gamma)},\,\,\, \alpha(x) \leq i,\,\,\, \eta(x)\geq \frac{1}{i},\,\,\, d(c\mu_{x},\omega)\geq \frac{1}{i} \,\,\,\forall |c| \in \left [\frac{1}{i},1 \right ],\, \forall\omega \in \operatorname{Hom}_{k+1}^i \right \},
\end{equation}
where $D_x$ and $\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\mu_x$ given by Lemma \ref{lmm:0-hom}, $\eta=\eta(x)$ is given by Lemma \ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\alpha-n)$-homogeneous measures described in Section \ref{sec:notations}.
Since $\mathcal{S}^k= \bigcup_{i \in \mathbb{N}}\bigcup_{\ell=1}^m E_{i,\ell}$, it is enough to prove that each $E_{i,\ell}$ can be covered by a countable union of Lipschitz graphs.

Let us fix $i \in \mathbb{N}$ and $\ell \in \{1,\dots,m\}$.
Then fix the value $\varepsilon$ given by Lemma \ref{lmm:too_close} for $\beta=i$, $\delta=\frac{1}{i}$, $M=i \cdot 2^\alpha$ and, for every $\rho>0$, define the set
\begin{equation}
E_{i,\ell,\rho} \coloneqq
\left \{x \in E_{i,\ell} \colon \inf_{|c| \in [\eta,1]}d(\mu_{x,r},c\mu_x)< \varepsilon \,\,\,\forall r \in (0,\rho) \right \}.
\end{equation}
If $x \in E_{i,\ell}$, then it belongs to $E_{i,\ell,\rho}$ for some $\rho>0$, because the existence of a sequence $r_j \to 0$ such that $d(\mu_{x,r_j},c\mu_x) \geq \varepsilon$ for every $j \in \mathbb{N}$ and every $|c| \in [\eta,1]$ would contradict $x \in \mathcal{S}$. Therefore
\begin{equation}
E_{i,\ell} = \bigcup_{\rho \in \mathbb{Q} \cap (0,+\infty)} E_{i,\ell,\rho}
\end{equation}
and it suffice to prove that, for a fixed $\rho>0$, the set $E_{i,\ell,\rho}$ can be covered with a countable union of Lipschitz graphs.
In order to do so, we claim that, for every $x \in E_{i,\ell,\rho}$, there exists $r>0$ such that
\begin{equation}\label{eq:outside_cones}
E_{i,\ell,\rho} \cap B_{r}(x) \setminus C(x,D_x,3\gamma)= \{x\}.
\end{equation}
Let us fix $x\in E_{i,\ell,\rho}$, let $\mu_x$ be $(\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \in \mathbb{N}$ there exists
\begin{equation}\label{eq:y_j_final}
y_j \in E_{i,\ell,\rho} \cap \overline{B_{\frac{r_j}{2}}(x)} \setminus\Big( B_{\frac{r_j}{4}}(x) \cup C(x,D_x,3\gamma) \Big),
\end{equation}
for some $r_j \leq \rho$ such that $r_j \downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \in \left [\frac{1}{i},1 \right ]$ such that
\begin{equation}\label{eq:distance_muj}
d( c_j\mu_{y_j}, \mu_{y_j,r_j})< \varepsilon
\qquad
\forall j \in \mathbb{N}.
\end{equation}
Up to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\mu_{x,r_j} \overset{\ast}{\rightharpoonup} c\mu_x$ for some $c \in \left [\frac{1}{i},1 \right ]$.
We now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \eqref{eq:distance_muj}.

The idea of the proof is that, for every $x \in \mathcal{S}_\mu^k$, there is a radius $r>0$ for which the points in $\mathcal{S}_\mu^k \cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \notin x+C$ with $r_j=|x-y_j| \to 0$ and such that the measures $c_j \mu_{y_j}$ and $\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$; on the other hand $\mu_{y_j,r_j}$, being translations of $\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$.

\medskip

A first outcome of Theorem \ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.

\begin{corollario}\label{cor:rectif_criterion}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Let us assume that, for $|\mu|$-a.e.\ $x \in \mathbb{R}^n$, there exists a Radon measure $\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \to 0^+$ there are a subsequence, not relabeled, and a constant $c\neq 0$ such that
\begin{equation}
\frac{1}{r_j^k}  (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \nu_x.
\end{equation}
Then $\mu$ is $k$-rectifiable: there exists a signed function $\theta \in L^1_{\mathrm{loc}}(\mathcal{S}_\mu^k,\mathcal{H}^k)$ such that \mbox{$\mu= \theta \mathcal{H}^k \llcorner \mathcal{S}_\mu^k$,} where $\mathcal{S}_\mu^k$ is defined in \eqref{eq:def_s_mu_k}.
\end{corollario}

The above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \cite[Theorem 11.8]{simonlectures83}, where one usually sets $\nu_x=\theta_0 \mathcal{H}^k \llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\tau_{x,r_j})_\# \mu \overset{\ast}{\rightharpoonup} \theta_0 \mathcal{H}^k \llcorner V_x$ as $r \to 0^+$.

We point out that Corollary \ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.

\medskip

A further consequence of Theorem \ref{thm:main} is the following extension of \cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\mathcal{S}_{\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\mu$ is more diffused than a $k$-dimensional measure is $|\mu|$-negligible.

\begin{corollario}\label{cor:almost_everywhere_rect}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then, for $|\mu|$-a.e.\ $x \in \mathcal{S}_\mu^k$, the unique blow-up of $\mu$ at $x$ is a multiple of $\mathcal{H}^k\llcorner D_x$.
\end{corollario}

We conclude this introduction with a comment on the definition of unique blow-up at a point.
As already said above, we say that $\mu_x$ is the unique blow-up of $\mu$ at $x \in \operatorname{supp}\mu$ if, for every sequence $r_j \to 0$, \emph{there exists} a subsequence for which \eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\{r_j\}_j$: according to \cite{mattila_unique05}, one says that  $\mu$ has unique blow-up at $x \in \mathbb{R}$ if there exists $\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\tau_{x,r_j})_\# \mu$, where $c_j>0$ and $r_j \to 0^+$, is of the form $c\nu$ for some $c>0$.

\cite[Lemma 2.5]{mattila_unique05} shows, provided $\mu$ is positive, that this condition actually implies $\mu_{x,r} \overset{\ast}{\rightharpoonup} c\mu_x$ as $r \to 0$ and, subsequently, our definition (they are actually equivalent).
However, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \rho_j$ such that
\begin{equation}
\lim_{j \to +\infty} \frac{1}{a_j} \sum_{i=j+1}^{+\infty} a_i = 0,
\qquad
\lim_{j \to +\infty} \frac{\rho_{j+1}}{\rho_j}=0.
\end{equation}
Then the Radon measure on $\mathbb{R}$ defined as
\begin{equation}
\mu \coloneqq \sum_{j \in \mathbb{N}} a_j\left ( \delta_{2\rho_j + \rho_j^2} - \delta_{2\rho_j - \rho_j^2} \right )
\end{equation}
has the null measure as unique blow-up at $x=0$ in the sense of \cite{mattila_unique05}, but $\mu_{0,\rho_j} =\frac{1}{|\mu|(B_{\rho_j}(0))} (\tau_{0,\rho_j})_\# \mu $ has no converging subsequences because $|\mu_{x,\rho_j}|\big(B_2(0)\big) \to +\infty$ as $j\to +\infty$.

\subsection*{Acknowledgments}
The author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems" at ESI (Wien), which the author wishes to thank.

The post-theorem introduction gives the following proof idea for Theorem~\ref{thm:main}.

1. **Cone containment near each point.** For every $x\in\mathcal{S}_\mu^k$, there exists $r>0$ such that the points in $\mathcal{S}_\mu^k\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.

2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\notin x+C$ with $r_j=|x-y_j|\to 0$ such that the measures $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$ are “as close as we want.”

3. **Two limiting descriptions of the same asymptotically close measures.**
   - On the one hand, $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$.
   - On the other hand, $\mu_{y_j,r_j}$ are translations of $\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$.

4. **Contradiction.** This “contradict[s] the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\mathcal{H}^k$-a.e. $x\in\mathcal{S}_\mu^k$.

The post-theorem introduction gives the following proof idea for the main theorem.

1. **Cone containment near each point.** For every $x\in\mathcal{S}_\mu^k$, there exists $r>0$ such that the points in $\mathcal{S}_\mu^k\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.

2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\notin x+C$ with $r_j=|x-y_j|\to 0$ such that the measures $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$ are “as close as we want.”

3. **Two limiting descriptions of the same asymptotically close measures.**
   - On the one hand, $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$.
   - On the other hand, $\mu_{y_j,r_j}$ are translations of $\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$.

4. **Contradiction.** This “contradict[s] the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\mathcal{H}^k$-a.e. $x\in\mathcal{S}_\mu^k$.

\label{thm:main}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then $\mathcal{S}_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\mathcal{S}_\mu^k$ at $x$ coincides with $D_x$ for $\mathcal{H}^{k}$-a.e.\ $x \in \mathcal{S}_\mu^k$.,

Question:
Let \(\mu\) be a signed Radon measure on \(\mathbb{R}^n\). For \(x\in \operatorname{supp}\mu\), say that \(\mu\) has a unique blow-up at \(x\) if there exists a Radon measure \(\mu_x\), with either \(|\mu_x|(B_1(0))=1\) or \(\mu_x=0\), such that for every sequence \(r_j\to 0^+\) there are a subsequence and a constant \(c\neq 0\) for which
\[
\mu_{x,r_j}:=\frac{1}{|\mu|(B_{r_j}(x))}(\tau_{x,r_j})_\#\mu \overset{*}{\rightharpoonup} c\mu_x,
\qquad \tau_{x,r}(y)=\frac{y-x}{r}.
\]
Let \(\mathcal S_\mu\) be the set of points where \(\mu\) has a unique blow-up. For \(x\in \mathcal S_\mu\), let
\[
D_x:=\{y\in\mathbb R^n:(\tau_{y,1})_\#\mu_x=\mu_x\}
\]
be the invariant linear subspace of the blow-up, and for \(k\in\{0,1,\dots,n\}\) define
\[
\mathcal S_\mu^k:=\{x\in\mathcal S_\mu:\dim D_x=k\}.
\]
Which statement holds for every signed Radon measure \(\mu\) on \(\mathbb R^n\) and every \(k\in\{0,1,\dots,n\}\)?

The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs, and for \(\mathcal H^k\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) is exactly \(D_x\).

Choices:
B. The set \(\mathcal S_\mu^k\) is contained in a countable union of \((k+1)\)-dimensional Lipschitz graphs, and for \(\mathcal H^{k+1}\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) contains \(D_x\).
C. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs.
D. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs, and for every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) exists and is exactly \(D_x\).
E. The set \(\mathcal S_\mu^k\) is contained in a finite union of \(k\)-dimensional Lipschitz graphs, and for \(\mathcal H^k\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) is a \(k\)-plane parallel to \(D_x\).

The stem defines the objects and hypotheses but does not explicitly reveal the conclusion. There is no obvious lexical or structural cue that singles out choice A.

The item is largely a theorem-selection question: it asks for the correct structural conclusion under the stated assumptions. However, it is not a pure tautology because the options include weaker and stronger nearby variants that must be distinguished.

Some reasoning is needed to compare rectifiability, graph structure, dimension, and the a.e. versus everywhere tangent-plane conclusion. Still, the problem primarily tests recognition or recall of the precise theorem rather than substantial generative derivation.

The distractors are mathematically plausible and target common failure modes: wrong dimension, weaker true statement, overstrong everywhere claim, and unjustified finiteness/exactness upgrades. They are distinct and well aligned with the theorem’s subtleties.

A solid advanced-theory MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than genuine generative reasoning.


Question:
Let \(\mu\) be a signed Radon measure on \(\mathbb{R}^n\). For \(x\in \operatorname{supp}\mu\), say that \(\mu\) has a unique blow-up at \(x\) if there exists a Radon measure \(\mu_x\), with either \(|\mu_x|(B_1(0))=1\) or \(\mu_x=0\), such that for every sequence \(r_j\to 0^+\) there are a subsequence and a constant \(c\neq 0\) for which
\[
\mu_{x,r_j}:=\frac{1}{|\mu|(B_{r_j}(x))}(\tau_{x,r_j})_\#\mu \overset{*}{\rightharpoonup} c\mu_x,
\qquad \tau_{x,r}(y)=\frac{y-x}{r}.
\]
Let \(\mathcal S_\mu\) be the set of points where \(\mu\) has a unique blow-up. For \(x\in \mathcal S_\mu\), let
\[
D_x:=\{y\in\mathbb R^n:(\tau_{y,1})_\#\mu_x=\mu_x\}
\]
be the invariant linear subspace of the blow-up, and for \(k\in\{0,1,\dots,n\}\) define
\[
\mathcal S_\mu^k:=\{x\in\mathcal S_\mu:\dim D_x=k\}.
\]
Which statement holds for every signed Radon measure \(\mu\) on \(\mathbb R^n\) and every \(k\in\{0,1,\dots,n\}\)?

Choices:
B. The set \(\mathcal S_\mu^k\) is contained in a countable union of \((k+1)\)-dimensional Lipschitz graphs, and for \(\mathcal H^{k+1}\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) contains \(D_x\).
C. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs.
D. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs, and for every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) exists and is exactly \(D_x\).
E. The set \(\mathcal S_\mu^k\) is contained in a finite union of \(k\)-dimensional Lipschitz graphs, and for \(\mathcal H^k\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) is a \(k\)-plane parallel to \(D_x\).

The stem gives only definitions and setup; it does not state or strongly hint at the full conclusion in choice A. The correct answer is not leaked explicitly or trivially.

This is not tautological from the definitions, but it is essentially a theorem-identification item: the correct option states a precise structural result, while the others are nearby variants. So it is more than a restatement of the stem, but still close to recall of a known theorem.

Selecting A requires some care with subtle distinctions such as countable vs finite cover, k-dimensional vs (k+1)-dimensional graphs, and almost-everywhere vs every-point tangent identification. However, the task is still primarily theorem recognition rather than substantial derivation.

The distractors are strong: one is a weaker true statement, others are plausible overstatements or dimension/quantifier mistakes that reflect common mathematical failure modes. They are distinct and discriminating.

A solid high-level theorem-recognition MCQ with no answer leakage and strong distractors, though it tests recall/comparison more than genuinely generative reasoning.


\begin{theorem}\label{Teo1}
	Assume that the potential $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}. Then, for every $\varepsilon > 0$ and $a_0 > 0$, there exists $\delta > 0$ such that for all $a \in (0, a_0]$, and any solution $u_a$ of \eqref{Equation} with initial data satisfying
	$$
	\|u_a(0)-u_*\| + \|\dot{u}_a(0)\| < \delta,
	$$
	the following hold:
	\begin{itemize}
		\item[\emph{(a)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le \varepsilon$;
		\item[\emph{(b)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	If, additionally, \hyperlink{W2}{$(W_2)$} and \hyperlink{W3}{$(W_3)$} hold with $\delta = \lambda = +\infty$, then there exists $R > 0$ such that, for all $a \in (0, a_0]$, and any solution $u_a$ of \eqref{Equation},
	\begin{itemize}
		\item[\emph{(c)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le R$;
		\item[\emph{(d)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and  $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	Moreover, under the additional assumption \hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C(a) e^{-\gamma(a) t}, \quad \forall\, t \ge 0.
	\end{equation*}
	In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$ such that
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C_* e^{-\gamma_* t},\quad \forall\, t \ge 0,\ \forall\, a\in[a_1,a_2].
	\end{equation*}
\end{theorem}

Second-order dissipative systems of the form
\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
arise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \eqref{Equation} also models the motion of a \emph{damped simple pendulum}.
Let $u=\theta$ be the pendulum angle, take $W(\theta)=\frac{g}{\ell}(1-\cos\theta)$, then
$$
\ddot{\theta}(t)+ a\,\dot{\theta}(t)+\dfrac{g}{\ell}\sin\theta(t)=0,
$$
which is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.

Second-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\varphi(t) \ge c\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \eqref{Equation} and on models where the damping parameter depends on time, such as \cite{1,2,3,4,6,7}.

The term $a\dot{u}$ in \eqref{Equation} represents a damping force, while $\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \eqref{Equation}, where $a > 0$ is a damping coefficient, $u\colon \mathbb{R} \to \mathbb{R}^N$ is the unknown trajectory, and $W\colon \mathbb{R}^N \to \mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:
\begin{itemize}
	\item[\hypertarget{W1}{($W_1$)}] $W\in C^2(\mathbb{R}^N; \mathbb{R})$.

	\item[\hypertarget{W2}{($W_2$)}] There exists $\delta > 0$ and $u_* \in \mathbb{R}^{N}$ such that
	$$
	W(u_*) = 0 \quad \text{and} \quad W(u) > 0 \quad \text{for all} \quad u \in B_\delta(u_*) \setminus \{u_*\}.
	$$

	\item[\hypertarget{W3}{($W_3$)}] There exists $\lambda > 0$ such that
	$$
	\langle \nabla W(u), u - u_* \rangle > 0 \quad \text{for all} \quad u \in B_\lambda(u_*) \setminus \{u_*\}.
	$$

	\item[\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,
	$$
	\lim_{\|u\| \to +\infty} W(u) = +\infty.
	$$

	\item[\hypertarget{W5}{($W_5$)}] There exist constants $\alpha, \beta, \mu > 0$ such that, for every $x \in B_\lambda(u_*)$,
	$$
	\alpha\|x - u_*\|^2 \le W(x) \le \beta\|x - u_*\|^2 \quad \text{and} \quad \langle \nabla W(x), x - u_* \rangle \ge \mu\|x - u_*\|^2.
	$$
\end{itemize}

A simple example satisfying \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential
$$
W(u)=\frac{1}{2}\|u\|^2.
$$
In this case, \eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking
$$
W(u)=\Phi(u-u_*),
$$
where $\Phi\colon\mathbb{R}^N\to\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\|u-u_*\|^p$, with $p\ge 2$, and the exponential potential
$$
W(u)=\frac12\big(e^{\|u-u_*\|^2}-1\big),
$$
which is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential
$$
W(u)=\frac14\left(\|u\|^2-1\right)^2,
$$
whose set of global minima is the unit sphere $\{u\in\mathbb{R}^N : \|u\| = 1\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential
$$
W(u)=c_1\|u\|^{4}-c_2\|u\|^{2}+c_3,\quad c_1,c_2,c_3>0,
$$
whose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \eqref{Equation}, see, for instance, \cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

Second-order dissipative systems of the form
\begin{equation}\label{Equation}
 \ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
arise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \eqref{Equation} also models the motion of a \emph{damped simple pendulum}.
Let $u=\theta$ be the pendulum angle, take $W(\theta)=\frac{g}{\ell}(1-\cos\theta)$, then
$$
\ddot{\theta}(t)+ a\,\dot{\theta}(t)+\dfrac{g}{\ell}\sin\theta(t)=0,
$$
which is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.

The term $a\dot{u}$ in \eqref{Equation} represents a damping force, while $\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \eqref{Equation}, where $a > 0$ is a damping coefficient, $u\colon \mathbb{R} \to \mathbb{R}^N$ is the unknown trajectory, and $W\colon \mathbb{R}^N \to \mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:
\begin{itemize}
 \item[\hypertarget{W1}{($W_1$)}] $W\in C^2(\mathbb{R}^N; \mathbb{R})$.

\item[\hypertarget{W2}{($W_2$)}] There exists $\delta > 0$ and $u_* \in \mathbb{R}^{N}$ such that
 $$
 W(u_*) = 0 \quad \text{and} \quad W(u) > 0 \quad \text{for all} \quad u \in B_\delta(u_*) \setminus \{u_*\}.
 $$

\item[\hypertarget{W5}{($W_5$)}] There exist constants $\alpha, \beta, \mu > 0$ such that, for every $x \in B_\lambda(u_*)$,
 $$
 \alpha\|x - u_*\|^2 \le W(x) \le \beta\|x - u_*\|^2 \quad \text{and} \quad \langle \nabla W(x), x - u_* \rangle \ge \mu\|x - u_*\|^2.
 $$
\end{itemize}

A simple example satisfying \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential
$$
W(u)=\frac{1}{2}\|u\|^2.
$$
In this case, \eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking
$$
W(u)=\Phi(u-u_*),
$$
where $\Phi\colon\mathbb{R}^N\to\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\|u-u_*\|^p$, with $p\ge 2$, and the exponential potential
$$
W(u)=\frac12\big(e^{\|u-u_*\|^2}-1\big),
$$
which is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential
$$
W(u)=\frac14\left(\|u\|^2-1\right)^2,
$$
whose set of global minima is the unit sphere $\{u\in\mathbb{R}^N : \|u\| = 1\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential
$$
W(u)=c_1\|u\|^{4}-c_2\|u\|^{2}+c_3,\quad c_1,c_2,c_3>0,
$$
whose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \eqref{Equation}, see, for instance, \cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

In this section we discuss the dynamics of the conservative system
\begin{equation}\label{EqConservative}
 \ddot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
which arises as the limit of \eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative
case, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\mathbb{R}^{2N}$ by setting $x=u$ and $y=\dot{u}$:
\begin{equation}\label{S0}
 \begin{cases}
 \dot{x} = y,\\[2mm]
 \dot{y} = -\nabla W(x).
 \end{cases}
\end{equation}
The natural energy associated with \eqref{EqConservative},
$$
E(x,y) := \frac12\|y\|^2 + W(x), \quad (x,y)\in B_\eta(0)\times \mathbb{R}^N,
$$
where $\eta=\min\{\delta,\lambda\}$ with $\delta$ and $\lambda$ given in \hyperlink{W2}{$(W_2)$}-\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \eqref{S0}, then deriving
$E(x(t),y(t))$ along the trajectory yields
$$
\frac{d}{dt}E(x(t),y(t))= \langle y(t),\dot{y}(t)\rangle + \langle \nabla W(x(t)),\dot{x}(t)\rangle.
$$
Using $\dot{x}=y$ and $\dot{y}=-\nabla W(x)$, we immediately obtain
$$
\frac{d}{dt}E(x(t),y(t))= -\langle y(t),\nabla W(x(t))\rangle+ \langle \nabla W(x(t)),y(t)\rangle= 0.
$$
Thus, if $u(t)$ is any solution of \eqref{EqConservative}, then the energy is conserved in time, that is,
\begin{equation}\label{EnergyConservation}
 \frac12\|\dot{u}(t)\|^2 + W(u(t)) = \tau,\quad \forall\, t\in\mathbb{R},
\end{equation}
for some $\tau\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that
$(0,0)$ is stable: for every $\varepsilon>0$ there exists
$\delta>0$ such that
$$
\|u(0)\| + \|\dot{u}(0)\| < \delta\quad\Longrightarrow\quad\sup_{t\ge 0} \big( \|u(t)\| + \|\dot{u}(t)\| \big) \le \varepsilon.
$$
However, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\dot{u}(t))$ were to converge to $(0,0)$ as $t\to+\infty$, then \eqref{EnergyConservation} would force $\tau=0$, which implies that $u(t)\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.

We now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \ref{Sec2}. Assume that $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W5}{$(W_5)$}. The conservative system \eqref{EqConservative} arises as the limit of the dissipative equation \eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \to 0$, and let $u_n$ be the solution of the perturbed problem
\begin{equation}\label{EqDamped-n}
 \ddot u_n(t) + a_n \dot u_n(t) + \nabla W(u_n(t)) = 0,
\end{equation}
with fixed initial data $(u_n(0),\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\neq(0,0)$ and $\|(z_0,z_1)\|<\sigma$. Here $\sigma>0$ depends on a prescribed $\varepsilon\in(0,\delta)$, with $\delta>0$ given in \hyperlink{W2}{\((W_2)\)}, and is provided by the uniform stability result of Lemma \ref{L5}. In particular, Lemma \ref{L5} ensures that
$$
\lim_{t\to+\infty} (u_n(t),\dot u_n(t)) = (0,0)\quad\text{and}\quad\|u_n(t)\| + \|\dot u_n(t)\| \le \varepsilon \quad\text{for all } t\ge 0\text{ and }n\in\mathbb{N}.
$$
Moreover, by Lemma \ref{L7}, each $u_n$ satisfies the exponential decay estimate
\begin{equation}\label{Exp}
 \|u_n(t)\| + \|\dot u_n(t)\|\le C e^{-\gamma(a_n) t}, \qquad t\ge 0,
\end{equation}
where $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\mathrm{loc}}([0,+\infty))$ to a limit
function $u_0$, which solves the conservative system
$$
\ddot u_0 + \nabla W(u_0)=0
$$
with initial data $u_0(0)=z_0$ and $\dot u_0(0)=z_1$, and satisfying
\begin{equation*}
 \sup_{t\ge 0}\left(\|u_0(t)\|+\|\dot{u}_0(t)\|\right) \leq \epsilon.
\end{equation*}
Although each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that
$$
\lim_{t\to+\infty} u_0(t)=0\quad\text{and}\quad
\lim_{t\to+\infty} \dot u_0(t)=0.
$$
Since $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\to 0$ and $\|\dot u_0(t)\|\to 0$. Inserting these limits into the conservation law \eqref{EnergyConservation}, we obtain $\tau=0$. But $\tau=0$ implies
$$
\dot u_0(t)\equiv 0\quad\text{and}\quad W(u_0(t))\equiv 0,
$$
so that $u_0(t)= 0$ for all $t\in\mathbb{R}$. This contradicts the fact that $(u_0(0),\dot u_0(0))\neq(0,0)$. Therefore,
$$
\lim_{t \to +\infty} (u_0(t),\dot{u}_0(t)) \neq (0,0).
$$
This argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\gamma(a)\to 0$ as $a\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.

\begin{lemma}\label{asd}
 Let $B\subset \mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.
 Then there exists a bounded set $K\subset \mathbb{R}^{2N}$ such that
 $$
 T(t) B \subset K,\quad \forall\, t\ge 0.
 $$
 In other words, $K$ serves as a absorbing set for the family of semigroup $\{T(t)\}_{t\ge 0}$.
\end{lemma}
\begin{proof}
Fix $a>0$ and let $z_0=(x_0,y_0)\in B$. Since $B$ is bounded in $\mathbb{R}^{2N}$ and $W$ is continuous by \hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that
$$
\sup_{z_0\in B} E(z_0)\le R_B.
$$
Therefore, by the monotonicity property of the energy in \eqref{decreasing-property}, we obtain
$$
W(x_a(t))\leq R_B\quad\text{and}\quad\|y_a(t)\|^2\leq 2R_B\quad\forall \, t\geq 0,
$$
where $x_a$ and $y_a$ denote the components of the trajectory $\varphi_a(t;z_0)$. Next, by the coercivity assumption \hyperlink{W4}{$(W_4)$}, the sublevel set
$$
\mathcal{S}_{B} \coloneqq \{ x \in \mathbb{R}^{N}:W(x)\le R_B\}
$$
is bounded. Consequently, we may define the positive real number
$$
C_B \coloneqq \sup \left\{\|x\|^{2} : x \in \mathcal{S}_{B}\right\} < \infty.
$$
In this way, we have
$$
\Vert T(t)z_0 \Vert_{\ast}^2 = \Vert (x_a(t), y_a(t))\Vert_{\ast}^2 = \Vert x_a(t) \Vert^2 + \Vert y_a(t) \Vert^2 \leq C_B + 2R_B,\quad \forall\, t\ge 0,
$$
Now, defining
$$
K\coloneqq\Big\{(x,y)\in\mathbb{R}^{2N}:\|(x,y)\|_{\ast}^{2}\le \sqrt{C_B + 2R_B}\Big\}
$$
it follows the inclusion $T(t)B \subset K$ for all $t\geq 0$, which finishes our proof.
\end{proof}

The Theorem \ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\to +\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system
\begin{equation}\label{S-intro}
	\begin{cases}
		\dot{x}=y,\\[1mm]
		\dot{y}=-a\,y-\nabla W(x),
	\end{cases}
\end{equation}
which encodes the full dynamics of \eqref{Equation} in the phase space $\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional
\begin{equation}
	V_a(x,y):=\frac{1}{2}\|y\|^2+2W(x)+\frac{1}{2}\|y+a(x-u_*)\|^2,
\end{equation}
which can be interpreted as a smooth perturbation of the mechanical energy
$$
E(x,y):=\frac12\|y\|^2+W(x).
$$
The functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \hyperlink{W1}{($W_1$)}-\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\to 0^{+}$, since the corresponding rate $\gamma(a)$ necessarily satisfies $\gamma(a)\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.

We emphasize that the conclusions of Theorem \ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.

When $a=0$, the equation \eqref{Equation} reduces to the conservative system
$$
\ddot u(t) + \nabla W(u(t))=0,
$$
corresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\ddot{u}(t)$ its acceleration, and $\nabla W(u)$ the force derived from $W$. Under assumptions \hyperlink{W1}{($W_1$)}-\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\mathbb{R}^{N}$, that is,
$$
\sup_{t\in\mathbb{R}}\left(\|u(t)\|+\|\dot u(t)\|\right)<\infty.
$$
Thus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.

In addition to the study of the asymptotic stability of \eqref{Equation}, two additional components complement our study. In Section \ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\mathcal{A}$ contains the entire set of equilibria of the system, that is,
$$\{(x,0)\in\mathbb{R}^{2N}:\nabla W (x)=0\}\subset \mathcal{A}.
$$
For readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \cite{Livro-Alexandre,Hale,Temam}. Finally, Section \ref{Sec5} presents numerical simulations for some cases of \eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \emph{R}. The simulations rely on a functional-programming framework supported by \texttt{tidyverse} (data manipulation), \texttt{deSolve} (time-integration of differential equations), and \texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \emph{R} scripts at the end of Section \ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.

The plan of the paper is as follows. Section \ref{Sec2} contains the proof of Theorem \ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\to 0^{+}$. Section \ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \ref{Sec2}. Finally, Section \ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.

To prove Theorem~\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \eqref{Equation} written as the phase-space system \eqref{S-intro}. The argument introduces
\[
V_a(x,y):=\tfrac12\|y\|^2+2W(x)+\tfrac12\|y+a(x-u_*)\|^2,
\]
viewed as a “smooth perturbation of the mechanical energy” \(E(x,y)=\frac12\|y\|^2+W(x)\). The functional \(V_a\) is used to obtain “stability estimates that are uniform with respect to \(a\)”. The assumptions \hyperlink{W1}{\((W_1)\)}–\hyperlink{W4}{\((W_4)\)} are used to ensure that \(V_a\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \(t\to+\infty\)). Under the additional condition \hyperlink{W5}{\((W_5)\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \(a\to0^+\) (the text notes “\(\gamma(a)\to0\)”).

To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of
\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
written as the phase-space system \eqref{S-intro}. The argument introduces
\[
V_a(x,y):=\tfrac12\|y\|^2+2W(x)+\tfrac12\|y+a(x-u_*)\|^2,
\]
viewed as a “smooth perturbation of the mechanical energy” \(E(x,y)=\frac12\|y\|^2+W(x)\). The functional \(V_a\) is used to obtain “stability estimates that are uniform with respect to \(a\)”. The assumptions \hyperlink{W1}{\((W_1)\)}–\hyperlink{W4}{\((W_4)\)} are used to ensure that \(V_a\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \(t\to+\infty\)). Under the additional condition \hyperlink{W5}{\((W_5)\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \(a\to0^+\) (the text notes “\(\gamma(a)\to0\)”).

\label{Teo1}
	Assume that the potential $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}. Then, for every $\varepsilon > 0$ and $a_0 > 0$, there exists $\delta > 0$ such that for all $a \in (0, a_0]$, and any solution $u_a$ of
	\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
	with initial data satisfying
	$$
	\|u_a(0)-u_*\| + \|\dot{u}_a(0)\| < \delta,
	$$
	the following hold:
	\begin{itemize}
		\item[\emph{(a)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le \varepsilon$;
		\item[\emph{(b)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	If, additionally, \hyperlink{W2}{$(W_2)$} and \hyperlink{W3}{$(W_3)$} hold with $\delta = \lambda = +\infty$, then there exists $R > 0$ such that, for all $a \in (0, a_0]$, and any solution $u_a$ of the equation above,
	\begin{itemize}
		\item[\emph{(c)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le R$;
		\item[\emph{(d)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and  $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	Moreover, under the additional assumption \hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C(a) e^{-\gamma(a) t}, \quad \forall\, t \ge 0.
	\end{equation*}
	In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$ such that
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C_* e^{-\gamma_* t},\quad \forall\, t \ge 0,\ \forall\, a\in[a_1,a_2].
	\end{equation*}

Universal–Existential
Asymptotic or Limit

Question:
Let $u_a:\mathbb{R}\to\mathbb{R}^N$ solve the damped second-order system
$$
\ddot u(t)+a\dot u(t)+\nabla W(u(t))=0,
$$
where $a>0$ and $W:\mathbb{R}^N\to\mathbb{R}$ is a potential. Assume $W\in C^2(\mathbb{R}^N)$, there exist $u_*\in\mathbb{R}^N$ and radii $\delta,\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\in B_\delta(u_*)\setminus\{u_*\}$, $\langle \nabla W(x),x-u_*\rangle>0$ for every $x\in B_\lambda(u_*)\setminus\{u_*\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate
$$
\alpha\|x-u_*\|^2\le W(x)\le \beta\|x-u_*\|^2,
\qquad
\langle \nabla W(x),x-u_*\rangle\ge \mu\|x-u_*\|^2
$$
for all $x\in B_\lambda(u_*)$, for some $\alpha,\beta,\mu>0$. Which statement holds about the stability and decay of solutions?

For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally (that is, with $\delta=\lambda=+\infty$), then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and again
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist constants $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a\in[a_1,a_2].
$$

Choices:
B.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exist constants $R>0$, $\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0.
$$
C.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
D.
For every $\varepsilon>0$ there exists $\rho>0$ such that for every $a>0$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, there exist constants $\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a>0.
$$
E.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and every bounded trajectory converges to some equilibrium point of $W$ as $t\to+\infty$.
$$
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$

extra global exponential estimate without (W5)

dropped the global-positivity/global-monotonicity conclusion and the uniform-in-$[a_1,a_2]$ exponential constants

uniformity in $a$ down to $a=0$ for stability radius and exponential rate

limit identified only as some equilibrium rather than the distinguished $u_*$ under global assumptions

The stem does not explicitly state the conclusion; it gives hypotheses and asks which asymptotic claim follows. Although the assumptions are detailed, they do not directly reveal the exact correct choice, especially because the options differ in subtle quantifier and uniformity claims.

The item is close to a theorem-recall question: the correct option is essentially a full theorem statement under the listed assumptions. Still, it is not a pure restatement, since the distractors alter important logical features such as convergence clauses, global hypotheses, and uniformity in the damping parameter.

Selecting the correct option requires meaningful reasoning about which conclusions are justified: local vs global attraction, dependence on bounds for a, and when exponential decay can be uniform. The solver must distinguish valid conclusions from stronger but false variants and a weaker true variant.

The distractors are mathematically plausible and reflect common failure modes: overclaiming uniformity in a, dropping necessary radial monotonicity for global attraction, omitting convergence, or extending exponential estimates to a→0. They are distinct and well-targeted.

A strong, technically sophisticated MCQ with high-quality distractors and genuine logical discrimination, though it remains somewhat theorem-statement-driven rather than fully generative.


Question:
Let $u_a:\mathbb{R}\to\mathbb{R}^N$ solve the damped second-order system
$$
\ddot u(t)+a\dot u(t)+\nabla W(u(t))=0,
$$
where $a>0$ and $W:\mathbb{R}^N\to\mathbb{R}$ is a potential. Assume $W\in C^2(\mathbb{R}^N)$, there exist $u_*\in\mathbb{R}^N$ and radii $\delta,\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\in B_\delta(u_*)\setminus\{u_*\}$, $\langle \nabla W(x),x-u_*\rangle>0$ for every $x\in B_\lambda(u_*)\setminus\{u_*\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate
$$
\alpha\|x-u_*\|^2\le W(x)\le \beta\|x-u_*\|^2,
\qquad
\langle \nabla W(x),x-u_*\rangle\ge \mu\|x-u_*\|^2
$$
for all $x\in B_\lambda(u_*)$, for some $\alpha,\beta,\mu>0$. Which statement holds about the stability and decay of solutions?

Choices:
B.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exist constants $R>0$, $\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0.
$$
C.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
D.
For every $\varepsilon>0$ there exists $\rho>0$ such that for every $a>0$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, there exist constants $\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a>0.
$$
E.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and every bounded trajectory converges to some equilibrium point of $W$ as $t\to+\infty$.
$$
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$

The stem does not state the conclusion outright. It gives hypotheses and asks which global/local stability and decay statement is valid; the correct answer is not leaked explicitly, though the extra quadratic assumptions do hint that some exponential decay statement may be relevant.

This is close to theorem recognition: the correct option is essentially a full theorem statement assembled from the listed assumptions. However, it is not a pure verbatim restatement, since the alternatives differ in subtle quantifiers, global-vs-local assumptions, and uniformity claims.

Moderate reasoning is required to compare the strength of the claims, especially the dependence on a, the need for global positivity/monotonicity, and when exponential rates can be uniform. Still, the task is mainly discriminating among theorem variants rather than generating a conclusion from scratch.

The distractors are mathematically plausible and target realistic failure modes: unjustified uniformity as a -> 0, overly strong global exponential decay, omission of global conclusions, and convergence only to some equilibrium. They are distinct and well aligned with common misunderstandings.

A solid MCQ with strong distractors and little answer leakage, but it remains largely a theorem-identification item rather than a deeply generative reasoning question.