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We consider a class of semi-linear dissipative hyperbolic equations in whichthe operator associated to the linear part has a nontrivial kernel.  Under appropriate assumptions on the nonlinear term, we prove that allsolutions decay to 0, as t -> +infinity, at least as fast as a suitablenegative power of t. Moreover, we prove that this decay rate is optimal in thesense that there exists a nonempty open set of initial data for which thecorresponding solutions decay exactly as that negative power of t.  Our results are stated and proved in an abstract Hilbert space setting, andthen applied to partial differential equations.

Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations