A Unified Approach to Nonlinear Integro-Differential Inverse Problems of Parabolic Type

Abstract

We give a unified approach to a class of nonlinear parabolic inverse problems involving kernels of convolution type. Our main tools are optimal regularity results, in Sobolev and H¨older spaces, for parabolic equations and analytic semigroup theory. We apply the main abstract results (Theorems 2.1-2.2) to a model of population dynamics, to the theory of combustion of a material with memory and, finally, to a parabolic equation with elliptic part of order $2m$, which for $m$ = 1 is the heat equation with memory and with non-linearity containg derivatives up to order $2m–1$.