Well--posedness and asymptotic behaviour for a non-classical and non-autonomous diffusion equation with delay

Abstract

In this paper, it is analyzed a non-classical non-autonomous di_usion equation with delay. First, the well-posedness and the existence of a local solution is proved by using a _xed point theorem. Then, the existence of solutions de_ned globally in future is ensured. The asymptotic behaviour of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of non-autonomous dynamical systems. One di_culty in the case of delays concerns the phase space that one needs to consider to construct the evolution process. This yields to the necessity of using a version of the Ascoli-Arzel_a theorem to prove the compactness.