Abstract
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 + , where M 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.
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