Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problemu¨(t)+g(u˙(t))+M(u(t))=0,t∈R+, where MM is the gradient operator of a non-negative functional and g is a non-linear damping operator, under some conditions relating the Łojasiewicz exponent of the functional and the growth of the damping around the origin. The main result is applied to non-linear wave or plate equations, in some cases direct constructive proofs of the Łojasiewicz gradient inequality are given, applicable to some non-analytic functionals in presence of multiple critical points. At the end similar results are obtained when a fast decaying source term is added in the right-hand side.