On some class of problems with nonlocal source and boundary flux

Abstract

In this paper we study a nonlocal, semilinear, parabolic problem. The existence and uniqueness of a maximal solution is proved for bounded domains, in arbitrary dimensions, using the Schauder fixed-point theorem. In the one-dimensional case, we give a result of positivity and a comparison principle for the integral of the solution. The proofs are based on the decomposition of the solutions in an appropriate spectral basis.