Approximation of solutions to non-linear integrodifferential parabolic equations in \(L^p\)-spaces. (English) Zbl 1212.34174

Summary: This paper is concerned with non-linear parabolic integrodifferential equations arising in continuum mechanics and phase-field models where memory terms are important. More precisely, we 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 the 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 are used here as starting points to solve our problems.


34G20 Nonlinear differential equations in abstract spaces
41A35 Approximation by operators (in particular, by integral operators)