The Cauchy problem for a coupled semilinear parabolic system

Abstract

We consider a semilinear parabolic coupled system u t= Δu+f( ∇u, ∇v),v t= Δv+g( ∇u, ∇v) in R n× R + where the maximum principle and the minimum principle fail for the solution itself and also for its first derivatives with respect to x. Under the assumption that f and g present a subquadratic growth near the origin, we prove the global existence and uniqueness of the solutions to the Cauchy problem, and the existence of nonnegative solutions. The maximum principle and the minimum principle are replaced by the existence of some invariant regions in R 2n+2 for ( u, v, ∂u/ ∂x 1, ∂v/ ∂x 1,…, ∂u/ ∂x n , ∂v/ ∂x n ).