Global existence and decay estimates for the semilinear nonclassical-diffusion equations with memory in $\mathbb{R}^{n}$

Abstract

In this paper, we study the initial value problem for a semi-linear nonclassical diffusion equations with fading memory in $\mathbb{R}^{n}$. Under smallness conditions on the initial data, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the pointwise decay estimates of the solution in the Fourier space and a fixed point-contraction mapping argument.