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 problem ¨ u(t)+ g ˙ u(t) +M u(t) = 0, t∈ R+, whereMis 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. © 2011 Elsevier Inc. All rights reserved.