Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems

Abstract

. This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation u t − M ( R Ω φu d x )div ( A ( x, t, u ) ∇ u ) = g ( x, t, u ) in Ω × (0 , T ), where Ω is a bounded domain of R n ( n > 1), T > 0 is a positive number, A ( x, t, u ) is an n × n matrix of variable coefficients depending on u and M : R → R , φ : Ω → R , g : Ω × (0 , T ) × R → R are given functions. We consider two different assumptions on g . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if A ( x, t, u ) = a ( x, t ) depends only on the variable ( x, t ), we investigate two uniqueness theorems and give a continuity result depending on the initial data.