On the initial value problem for parabolic with memory quantity containing fractional type

Abstract

In this paper, we are interested in considering Cauchy problem for parabolic equation with memory term.  The memory component contains a fractional Laplacian operator term. First, we represent the mild solution as a Fourier series. Next, we consider the well-posedness  of the Cauchy problem when the initial data and source function  are in Gevrey spaces.  Under  appropriate given data, we also investigate the continuity of the solution according to the parameter $k.$  The another results of this paper is to show the ill-posedness in the sense of Hadamard and give some regularized methods. In the homogeneous case, we use the quasi-boundary value method to regularize the problem and obtain  the error estimate when the observation data in $L^2$.  In the case of inhomogeneous source term, we use the truncation method to approximate  the problem with observed data in $L^p.$