Nonlocal Problems for Parabolic Inclusions

Let be a an open bounded domain in R , N 2; with a smooth boundary @ ; and T > 0. De…ne D = (0; T ) and = @ [0; T ]: In this paper we are concerned with the existence of solutions of the following parabolic inclusion ut + Lu 2 F (x; t; u), (x; t) 2 D; u(x; t) = 0; (x; t) 2 subjected to the nonlocal condition u(x; 0) = R T 0 g(x; t; u(x; t))dt for x 2 : We provide su¢ cient conditions on L; F; g that guarantee the existence of at least one solution. Our technique is based on the Green’s function for linear parabolic partial di¤erential equations, …xed point theorems for multivalued maps. Keywords. parabolic problems; integral representation of solutions; multivalued maps; nonlocal conditions; …xed point theorems. AMS (MOS) Subject Classi…cation: 35K20; 35K55; 35K60; 35C15; 35A05; 35B50; 35R05; 35R70 1 Introduction Let be a an open bounded domain in R , N 2; with a smooth boundary @ : We denote the norm (usually the Euclidean norm) of x 2 by kxk : Let T be a positive real number. De…ne D = (0; T ) and = @ [0; T ]: Our objective is to investigate the existence of solutions of the following parabolic problem with a