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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A blow up result for viscoelastic equations with arbitrary positive initial energy | Ma et al. Boundary Value Problems 2011 2011 6 http www.boundaryvalueproblems.eom content 2011 1 6 0 Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access A blow up result for viscoelastic equations with arbitrary positive initial energy Jie Ma Chunlai Mu and Rong Zeng Correspondence ma88jie@163. com College of Mathematics and Statistics Chongqing University Chongqing 401331 PR China SpringerOpen0 Abstract In this paper we consider the following viscoelastic equations futt Au fỹg t T Au t dT ut a1 v q 1 u p 1 u vtt Av fog t T Av t dT vt a2 u p 1 v q 1 v with initial condition and zero Dirichlet boundary condition. Using the concavity method we obtained sufficient conditions on the initial data with arbitrarily high energy such that the solution blows up in finite time. Keywords viscoelastic equations blow up positive initial energy 1 Introduction In this work we study the following wave equations with nonlinear viscoelastic term utt Au fgg t T Au t dT ut a vỊI ỊuỊI 1u x t e X 0 to Vtt Av J0tg t T Av t dr vt a2 u p 1 v q 1 v x t e Q X 0 to 1 1 u x 0 u0 x ut x 0 u1 x v x 0 v0 x vt x 0 v1 x x e S2 u x t 0 v x t 0 x e 9Q where o is a bounded domain of Rn with smooth boundary 90 p 1 q 1 and g is a positive function. The wave equations 1.1 appear in applications in various areas of mathematical physics see 1 . If the equations in 1.1 have not the viscoelastic term f0yg t T dT the equations are known as the wave equation. In this case the equations have been extensively studied by many people. We observe that the wave equation subject to nonlinear boundary damping has been investigated by the authors Cavalcanti et al. 2 3 and Vitillaro 4 5 . It is important to mention other papers in connection with viscoelastic effects such as Aassila et al. 6 7 and Cavalcanti et al. 8 . Furthermore related to blow up of the solutions of equations with nonlinear damping and source terms acting in the domain we can cite the work of Alves and Cavalcanti 9 Cavalcanti and .
