Averaging of nonclassical diffusion equations with memory and singularly oscillating forces (Q1656239)

Summary: We consider for \(\rho \in [0,1)\) and \(\epsilon > 0\), the following nonclassical diffusion equations with memory and singularly oscillating external force \[ u_t -\Delta u_t - \Delta u - \int_0^\infty \kappa (s) \Delta u(t-s)ds+ f(u) = g_0(t)+\epsilon^{- \rho}g_1(\frac{t}{\epsilon}), \] together with the averaged equation \[ u_t - \Delta u_t - \Delta u - \int_0^\infty \kappa (s) \Delta u(t-s)ds+ f( u) = g_{0}( t) \] 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 \(\epsilon \rightarrow 0^+\).