2014/09/11 by M. L. Oliveira, Oliveira, M. L., A. J. R. Feitosa +3
Mathematics · #35L05 #35L20 #35L70 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L05 #msc:35L20 #msc:35L70
paper · pdf · doi:10.48550/arxiv.1409.3448
arxiv created 2014/09/11 · arxiv updated 2014/09/12
This paper is concerned with the existence and decay of solutions of the following Timoshenko system: ‖u"-μ(t)Δu+α1 ∑i=1n\frac∂ v∂ xi=0, ∈ Ω× (0, ∞),
v"-Δv-α2 ∑i=1n\frac∂ u∂ xi=0, ∈ Ω× (0, ∞), . subject to the nonlinear boundary conditions, ‖u=v=0 in Γ0× (0, ∞),
(∂ u)/(∂ ν) + h1(x,u')=0 in Γ1× (0, ∞),
(∂ v)/(∂ ν) + h2(x,v')+σ(x)u=0 in Γ1× (0, ∞), . and the respective initial conditions at t=0. Here Ω is a bounded open set of ℝn with boundary Γ constituted by two disjoint parts Γ0 and Γ1 and ν(x) denotes the exterior unit normal vector at x∈ Γ1. The functions hi(x,s), (i=1,2) are continuous and strongly monotone in s∈ 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 appropriate Lyapunov functional and the multiplier method.