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Báo cáo toán học: " General decay for a wave equation of Kirchhoff type with a boundary control of memory type"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: General decay for a wave equation of Kirchhoff type with a boundary control of memory type | Wu Boundary Value Problems 2011 2011 55 http www.boundaryvalueproblems.eom content 2011 1 55 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access General decay for a wave equation of Kirchhoff type with a boundary control of memory type Shun-Tang Wu Correspondence stwu@ntut.edu.tw 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 boundary as follows utt M Vu 2 Au l t h Vu Aut a x f u 0 in X 0 k 1.1 u 0 on r0 X 0 k 1.2 t du u J g t s M Vu s 2 dv s dut -A- s ds 0 on r1 X 0 k dv 1.3 u x 0 u0 x ut x 0 u1 x in 1.4 where o is a bounded domain with smooth boundary do r0 u r1. The partition r0 and r1 are closed and disjoint with meas r0 0 V represents the unit normal vector directed towards the exterior of o u is the transverse displacement and g is the relaxation function considered positive and nonincreasing belonging to W1 2 O . From the physical point of view we know that the memory effect described in integral equation 1.3 can be caused by the interaction with another viscoelastic element. In fact the boundary condition 1.3 signifies that o is composed of a material which is clamped in a rigid body in the portion r0 of its boundary and is clamped in a body with viscoelastic properties in the portion of r 1. When r1 j problem 1.1 has its