Well–posedness and Asymptotic Behaviour for a Non-classical and Non-autonomous Diffusion Equation with Delay

In this paper, it is analyzed a non-classical non-autonomous diffusion equation with delay. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined 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 difficulty 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-Arzelà theorem to prove the compactness.