Averaging of Nonclassical Diffusion Equations with Memory and Singularly Oscillating Forces

Abstract

We consider for $\rho \in [0,1)$ and $\epsilon >0$, the following nonclassical diffusion equations with memory and singularly oscillating external force

together with the averaged equation

formally corresponding to the limiting case $\epsilon =0$. Under suitable assumptions on the nonlinearity and on the external force, we prove the uniform (w.r.t. $\epsilon$) boundedness as well as the convergence of the uniform attractor ${\mathcal{A}}^{\epsilon }$ of the first equation to the uniform attractor ${\mathcal{A}}^0$ of the second equation as $\epsilon \to 0^{+}$.