| Abstract: |
| We prove existence and uniqueness of solutions
to a doubly-nonlinear Cahn-Hilliard system with nonlinear viscosity
of the form
\begin{align*}
\partial_t u- \Delta\mu = 0 \qquad&\text{in } \Omega\times(0,T)\,,\
\mu\in\varepsilon\partial_t u + \beta(\partial_t u) - \delta\Delta u + \psi`(u) - g
\qquad&\text{in } \Omega\times(0,T)\,,\
\mu=0\,, \quad \partial_{\bf n} u =0 \qquad&\text{in } \partial\Omega\times(0,T)\,,\
u(0)=u_0 \qquad&\text{in } \Omega
\end{align*}
on a smooth bounded domain $\Omega\subseteq\mathbb{R}^3$,
where $\varepsilon,\delta$ are positive parameters,
$\beta$ is a maximal monotone graph in $\mathbb{R}\times\mathbb{R}$,
$\psi:(a,b)\to\mathbb{R}$ is a so-called double-well potential and $g$ is a given source.
Existence of solutions is proved by approximating the problem and passing to the limit
through monotonicity and compactness arguments, under some growth
assumptions either on $\beta$ or on $\psi$. Moreover,
the asymptotic behaviour of the solutions as $\delta\searrow0$
and $\varepsilon\searrow0$ (separately) is shown.
This study is based on a joint work with Elena Bonetti (Universit\`a degli Studi di Milano),
Pierluigi Colli (Universit\`a degli Studi di Pavia) and Giuseppe Tomassetti
(Universit\`a degli Studi ``Roma Tre``). |
|