Regresar

Existence and stability of bounded solutions for a system of parabolic equations

Abstract:

In this paper we study the existence and the stability of bounded solutions of the following non-linear system of parabolic equations with homogeneous Dirichlet boundary conditions: ut = D Δu + f(t, u), t ≥ 0, u ∈ ℝn, u = 0 on ∂Ω, where f ∈ C1 (ℝ × ℝn), D = diag(d1, d2,..., dn) is a diagonal matrix with di > 0, i = 1, 2,..., n, and Ω is a sufficiently regular bounded domain in ℝN (N = 1, 2, 3). Roughly speaking we shall prove the following result: if f is globally Lipschitz with constant L, 3/4 < α < 1 and (λ1d)1-α/Γ(1 - α) > 6ML, then the system has a bounded solution on ℝn which is stable, where 2d = min{di: i = 1, 2,..,n}, (λjdit αe-λj(di/2)t < M, λi are the eigenvalues of - Δ, and Γ(·) is the well-known gamma function. Also, we prove that for a large class of functions f this bounded solution is almost periodic. © 2003 Published by Elsevier Science (USA).