Pullback attractors for nonclassical diffusion equations with a delay operator

Abstract

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $\ddot{I}†(t,u_t)$ contains some hereditary characteristics and the external force $k\in L_{loc}^2\left(\mathbb{R};L^2(\hat{I}\text{c◯})\right)$. First, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{{\mathcal{H}}_t(\hat{I}\text{c◯})}$ and $C_{{\mathcal{H}}_t^1(\hat{I}\text{c◯})}$, respectively.

31 pages