Comparison and Stability Results for Parabolic Integro-Differential Equations

Abstract

The author considers the semilinear parabolic system (1) u˙k+Lkuk=fk(t,x,u,Hu,Ku), Bkuk=hk(t,x,u,H1u,K1u), k=1,⋯,n, where the Lk are elliptic operators in a bounded domain Ω, the Bk are Dirichlet, Neumann or mixed boundary operators, H is a linear nonlocal operator, K is a nonlocal memory operator, and H1, K1 are operators of the same type acting on the boundary of Ω. The comparison principle for a slightly more general system is given. This makes possible the use of a monotone scheme to prove existence and uniqueness for (1), provided the globally Lipschitz functions fk, gk are quasimonotone and lower and upper solutions exist. The method of vector-valued Lyapunov functions and the comparison principle yield the stability of the trivial solution to (1). Three examples demonstrate these stability results.