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

In this paper we study the existence and the stability of bounded solutions of the following nonlinear system of parabolic equations with homogeneous Dirichlet boundary conditions: ut = DΔu + 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}, (λj di t)αe−λj (di/2)t