A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations

Nikos I. Karachalios, Nikos B. Zographopoulos

We consider the semilinear reaction diffusion equation ${\partial }_t\varphi -\nu \Delta \varphi -V(x)\varphi +f(\varphi )=0$, $\nu >0$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$. We assume the standard Allen-Cahn-type nonl…