Approximation of solutions to non-linear integrodifferential parabolic equations in $L^p$-spaces

Abstract

This paper is concerned with non-linear parabolic integrodifferential equations arising in continuum mechanics, phase-field models, and elsewhere where memory terms are important. More precisely, we will consider a non-linear initial-value problem depending on a small parameter and related to a uniformly elliptic second-order differential operator $A$. The integrodifferential character of our problem is expressed by a convolution term multiplying $A$, the scalar kernel approximating a delta-type function. Consequently, the limit problem reduces to a nonlinear parabolic differential problem involving a multiple of operator $A$. Two different non-linearities, which are locally Lipschitz-continuous in suitable metrics, are considered. The basic aim of the paper consists in determining the rate of convergence of the approximating solutions to the exact one. The results proved in the linear case (cf. [10]) are used here as starting points to solve our problems.