Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Arbitrary decays for a viscoelastic equation | Wu Boundary Value Problems 2011 2011 28 http content 2011 1 28 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Arbitrary decays for a viscoelastic equation Shun-Tang Wu Correspondence stwu@ General Education Center National Taipei University of Technology Taipei 106 Taiwan Springer Abstract In this paper we consider the nonlinear viscoelastic equation ut putt - Au - Autt f0g t - s Au s ds u pu 0 in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential polynomial type which is a necessary condition for the exponential polynomial decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata Q Appl Math 64 499-513 2006 and the work of Tatar J Math Phys 52 013502 2010 with necessary modification imposed by our problem. Mathematical Subject Classification 2010 35B35 35B40 35B60 Keywords Viscoelastic equation Kernel function Exponential decay Polynomial decay 1 Introduction It is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences their dynamics are of great importance and interest. From the mathematical point of view their memory effects are modeled by an integro-differential equations. Hence questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. Many authors have focused on this problem for the last two decades and several results concerning existence decay and blow-up have been obtained see 1-28 and the reference therein. In 3 Cavalcanti et al. studied the following problem t ut putt Au Autt Ịg t s Au s ds yAut 0 in X 0 to u x 0 u0 x ut x 0 u1 x x e u x t
