General decay for a wave equation of Kirchhoff type with a boundary control of memory type

Wu

Description

A nonlinear wave equation of Kirchhoff type with memory condition at the boundary in a bounded domain is considered. We establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow certain relaxation functions which are not necessarily of exponential and polynomial decay. This improves earlier results in the literature. MSC: 35L05; 35L70; 35L75; 74D10.

Sujets

• Wave equation
• Relaxation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 12
• Langue: English

Extrait

WuBoundary Value Problems2011,2011:55 http://www.boundaryvalueproblems.com/content/2011/1/55

R E S E A R C HOpen Access General decay for a wave equation of Kirchhoff type with a boundary control of memory type ShunTang Wu

Correspondence: [e-mail] General Education Center, National Taipei University of Technology, Taipei 106, Taiwan

Abstract A nonlinear wave equation of Kirchhoff type with memory condition at the boundary in a bounded domain is considered. We establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow certain relaxation functions which are not necessarily of exponential and polynomial decay. This improves earlier results in the literature. MSC:35L05; 35L70; 35L75; 74D10. Keywords:general decay, wave equation, relaxation, memory type, Kirchhoff type, nondissipative

1 Introduction In this article, we study the asymptotic behavior of the energy function related to a nonlinear wave equation of Kirchhoff type subject to memory condition at the bound ary as follows:   2 || u tt −M||∇u 2 u+l(t)h(∇u)−u t +a(x)f(u) = 0 in×(0,∞), (1:1)

u= 0 on 0 ×(0,∞),

(1:2)

  t   ∂u∂u t 2 u+g(t−s)M||∇u(s)|| (s) +(s)ds= 0 on 1 ×(0,∞), (1:3) 2 0 ∂ν ∂ν

u(x, 0)=u 0 (x),u t (x, 0)=u 1 (x) in, (1:4) whereΩis a bounded domain with smooth boundary∂Ω=Γ 0 ∪Γ 1 . The partitionΓ 0 andΓ 1 are closed and disjoint, withmeas(Γ 0 )>0,νrepresents the unit normal vector directed towards the exterior ofΩ,uis the transverse displacement, andgis the 1,2 relaxation function considered positive and nonincreasing belonging toW(Ω). From the physical point of view, we know that the memory effect described in inte gral equation (1.3) can be caused by the interaction with another viscoelastic element. In fact, the boundary condition (1.3) signifies thatΩis composed of a material which is clamped in a rigid body in the portionΓ 0 of its boundary and is clamped in a body with viscoelastic properties in the portion ofΓ 1 . WhenΓ 1 =j, problem (1.1) has its origin in describing the nonlinear vibrations of an elastic string. More precisely, we have

© 2011 Wu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.