Nonclassical diffusion with memory lacking instantaneous damping

Abstract

We consider the nonclassical diffusion equation with hereditary memory \begin{document}$ u_t-\Delta u_t -\int_0^\infty \kappa(s)\Delta u(t-s)\,{{\rm{d}}} s +f(u) = g $\end{document} on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form $ -\Delta u $, contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity.

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