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

摘要

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{double-struck}_+, where M is the gradient operator of a non-negative functional and g is a non-linear damping operator, under some conditions relating the ?ojasiewicz 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 ?ojasiewicz 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.