| Contents |
| We address the asymptotic behavior, as $\varepsilon \downarrow 0$, of the solutions to the (Cauhy problem for the) gradient flow equation
\begin{equation}
\varepsilon u'(t) + \mathrm{D} \mathcal{E} (t,u(t)) \ni 0
\qquad \text{in } \mathscr{H}, \quad t \in (0,T),
\end{equation}
where $\mathscr{H}$ is a (separable) Hilbert space, $\mathcal{E} : (0,T) \times \mathscr{H} \to (-\infty,+\infty]$ is a time-dependent energy functional with $u \mapsto \mathcal{E}(t,u)$ possibly nonconvex.
The main difficulty attached to the analysis as $\varepsilon \downarrow 0$ for a family of solutions (u_\varepsilon)_\varepsilon$ resides in the lack of estimates for $u_\varepsilon'$.
We develop a variational approach to this problem, based on the study of the limit of the energy identity
\[
\frac{\varepsilon}2 \int_s^t |u_\varepsilon'(r)|^2 \, \mathrm{d}r + \frac1{2\varepsilon}
\int_s^t |\mathrm{D} \mathcal{E} (r,u_\varepsilon(r))|^2 \, \mathrm{d}r +
\mathcal{E} (t,u_\varepsilon(t)) = \mathcal{E} (s,u_\varepsilon(s)) +\int_s^t
\partial_t \mathcal{E} (r,u_\varepsilon(r) \, \mathrm{d}r
\]
for all $0 \leq s \leq t \leq T$, and on a fine analysis of the asymptotic properties of the quantity
\[
\int_s^t |u_\varepsilon'(r)||\mathrm{D} \mathcal{E} (r,u_\varepsilon(r))|\,
\mathrm{d}r.
\]
In this context, the crucial hypothesis is that for every $t \in (0,T)$ the critical points of $\mathcal{E}(t,\cdot)$ are isolated, a condition of which we discuss the genericity. |
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