Summary: One considers the nonlinear evolution equation with source and damping terms
where 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 P. Martinez in [ESAIM, Control Optim. Calc. Var. 4, 419–444 (1999; Zbl 0923.35027)], when the frictional damping mechanism on the manifold does not necessarily have a polynomial growth near the origin.