SUMMARYThis paper is concerned with the non-linear viscoelastic equationWe prove global existence of weak solutions. Furthermore, uniform decay rates of the energy are obtained assuming a strong damping ut acting in the domain and provided the relaxation function decays exponentially.

We establish, subject to some natural additional assumptions imposed on the relation between the source and the damping, both well-posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation. The theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources.

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The nonlinear and damped extensible plate (or beam) equation is considered [Formula: see text] where Ω is any bounded or unbounded open set of Rn, α>0 and f, g are power like functions. The existence of global solutions is proved by means of the Fixed Point Theorem and continuity arguments. To this end we avoid handling the nonlinearity M(∫Ω|∇u|2dx) in the a priori estimates of energy. Furthermore, uniform decay rates of the energy are also obtained by making use of the perturbed energy method for domains with finite measure.

The wave equation with a source term is considered u tt À Du ¼ juj r u in O Â ð0; þNÞ:We prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback bðu t Þ acting on the boundary provided that b has necessarily not a polynomial growth near the origin. To obtain the existence of global solutions we make use of the potential well method combined with the Faedo-Galerkin procedure and constructing a special basis. Furthermore, we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of Lasiecka and Tataru (Differential Integral Equations 6 (3) (1993) 507) and Patrick Martinez (ESAIN: Control, Optimisation Calc. Var. 4 (1999) 419). The resulting problem generalizes Martinez results and complements the works of Lasiecka and

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We consider the nonlinear model of the wave equation ytt − ∆y + f0 (∇y) = 0 subject to the following nonlinear boundary conditionsWe show existence of solutions by means of Faedo-Galerkin method and the uniform decay is obtained by using the multiplier technique.

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