Existence of Bounded Solutions for a Nonlinear Parabolic System with Nonlinear Gradient Term

Abstract

In this note we show the existence of bounded solutions of the nonlinear parabolic system
(u₁)_t + A₁u₁ = a₁(x) ∣∇u₁∣^p1 + f₁(x, u₁, u₂)
(u₂)_t + A₂u₂ = a₂(x) ∣∇u₂∣^p2 + f₂(x, u₁, u2)
where A_i is the pseudo-Laplacian operator and a_i, f_i are given functions, i = 1, 2. We prove the existence of weak bounded solutions using a priori L^∞-estimate and the theory of nonlinear parabolic approximation.

Key words and phrases. Nonlinear parabolic systems; nonlinear gradients terms; p-Laplacian; existence and bounded solutions.

2000 Mathematics Subject Classification. 35K65, 35L05, 35K55, 65M15, 65M60