Existence and asymptotic stability for evolution problems on manifolds with damping and source terms ✩

One considers the nonlinear evolution equation with source and damping terms ut t +Au + g(ut ) = |u|ρu on Γ ×(0,∞), where Γ is a compact manifold. We prove the global existence of solutions making use of the potential well method. Furthermore, we study the asymptotic behaviour of solutions adapting the ideas introduced by Patrick Martinez in ESAIM Control Optim. Calc. Var. 4 (1999) 419–444 when the frictional damping mechanism g on the manifold does not necessarily have a polynomial growth near the origin.  2003 Elsevier Inc. All rights reserved.