Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$

Abstract

We consider a family of non-autonomous
reaction-diffusion equations
$$
u_t=\sum_{i,j=1}^N a_{ij}(\omega
t)\partial_i\partial_j u+f(\omega t,u)+ g(\omega t,x),
\quad x\in\mathbb R^N
\tag{$\text{\rm E}_\omega$}
$$
with almost periodic,
rapidly oscillating principal part and nonlinear interactions. As
$\omega\to \infty$, we prove that the solutions of
$(\text{\rm E}_\omega)$
converge to the solutions of the averaged equation
$$
u_t=\sum_{i,j=1}^N \overline a_{ij}\partial_i\partial_j u+\overline f(u)+ \overline g(x),
\quad x\in\mathbb R^N.
\tag{$\text{\rm E}_\infty$}
$$
If $f$ is
dissipative, we prove existence and upper-semicontinuity of
attractors for the family $(E_\omega)$ as $\omega\to\infty$.