### A boundary value problem for the wave equation.

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In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.

We consider the wave equation damped with a locally distributed nonlinear dissipation. We improve several earlier results of E. Zuazua and of M. Nakao in two directions: first, using the piecewise multiplier method introduced by K. Liu, we weaken the usual geometrical conditions on the localization of the damping. Then thanks to some new nonlinear integral inequalities, we eliminate the usual assumption on the polynomial growth of the feedback in zero and we show that the energy of the system decays...

We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction...

The goal of this paper is to work out a thermodynamical setting for nonisothermal phase-field models with conserved and nonconserved order parameters in thermoelastic materials. Our approach consists in exploiting the second law of thermodynamics in the form of the entropy principle according to I. Müller and I. S. Liu, which leads to the evaluation of the entropy inequality with multipliers. As the main result we obtain a general scheme of phase-field models which involves an...