Nonlinear boundary stabilization for Timoshenko beam system

This paper is concerned with the existence and decay of solutions of the following Timoshenko system:‖u″−μ(t)Δu+α1∑i=1n∂v∂xi=0,inΩ×(0,∞),v″−Δv−α2∑i=1n∂u∂xi=0,inΩ×(0,∞), subject to the nonlinear boundary conditions:‖u=v=0inΓ0×(0,∞),∂u∂ν+h1(x,u′)=0inΓ1×(0,∞),∂v∂ν+h2(x,v′)+σ(x)u=0inΓ1×(0,∞), and the respective initial conditions at t=0t=0. Here Ω is a bounded open set of RnRn with boundary Γ constituted by two disjoint parts Γ0Γ0 and Γ1Γ1 and ν(x)ν(x) denotes the exterior unit normal vector at x∈Γ1x∈Γ1. The functions hi(x,s)hi(x,s)(i=1,2)(i=1,2) are continuous and strongly monotone in s∈Rs∈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.