Nonlinear boundary stabilization for Timoshenko beam system

This paper is concerned with the existence and decay of solutions of the following Timoshenko system: parallel to u ''-mu(t)Delta u+alpha(1) Sigma i=1n partial derivative u/partial derivative x(i) = 0, in Omega x (0,infinity), v '' - Delta v - alpha(2)Sigma(n)(i=1) partial derivative u/partial derivative x(i) = 0, in Omega x (0,infinity), subject to the nonlinear boundary conditions: parallel to u = v = 0 in Gamma(0) x (0, infinity), partial derivative u/partial derivative v+h(1)(x,u')=0 in Gamma(1) x (0,infinity), partial derivative v/partial derivative v+h(2)(x,v') + sigma(x)u = 0 in Gamma(1) x (0,infinity), and the respective initial conditions at t = 0. Here Omega is a bounded open set of R-n with boundary Gamma constituted by two disjoint parts Gamma(0) and Gamma(1) and v(x) denotes the exterior unit normal vector at x is an element of Gamma(1). The functions h(i)(x, s) (i = 1,2) are continuous and strongly monotone in s is an element of R. The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using the multiplier and the Zuazua method. (C) 2015 Elsevier Inc. All rights reserved.