*Accepted Paper*

**Inserted:** 20 oct 2005

**Journal:** ESAIM: COCV

**Year:** 2005

**Abstract:**

This paper addresses the Cauchy problem for the
gradient flow equation in a Hilbert space $H$
$$
u'(t)+\partial_{\ell\phi}(u(t))\ni f(t)\quad
\hbox{a.e.\ in }(0,T),\quad
u(0)=u_{0,
}
$$
where $\phi:H\to (-\infty,+\infty]$ is a proper,
lower semicontinuous functional which is not supposed to be a
(smooth perturbation of a) convex functional and $\partial_\ell\phi$ is
(a suitable limiting version of) its subdifferential.
We will present some new existence results for the solutions of the
equation by exploiting a *variational approximation*
technique, featuring some ideas from the theory of *Minimizing Movements*
and of *Young measures*.

Our analysis
is also motivated by some models describing phase transitions
phenomena, leading to
systems of evolutionary PDEs which have a *common
underlying gradient flow structure*:
in particular, we will focus on
*quasistationary* models, which exhibit
highly non convex Lyapunov functionals.

**Keywords:**
Gradient flows, minimizing movements, Quasistationary phase field models

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