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báo cáo hóa học:" Research Article Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation Yaojun Ye"

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: Research Article Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation Yaojun Ye | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 357404 12 pages doi 10.1155 2010 357404 Research Article Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation Yaojun Ye Department of Mathematics and Information Science Zhejiang University of Science and Technology Hangzhou 310023 China Correspondence should be addressed to Yaojun Ye yeyaojun2002@yahoo.com.cn Received 14 December 2009 Revised 21 May 2010 Accepted 4 August 2010 Academic Editor Tocka Diagana Copyright 2010 Yaojun Ye. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The initial boundary value problem for a class of hyperbolic equations with strong dissipative term utt - n 1 d dxi du dxi p-2 du dxi - aầut b u r-2u in abounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in W0fp Of and showing the exponential decay of the energy of global solutions through the use of an important lemma of V. Komornik. 1. Introduction We are concerned with the global solvability and exponential asymptotic stability for the following hyperbolic equation in a bounded domain utt - àpu - aàut b u r-2u x e Q t 0 1.1 with initial conditions u x 0 u0 x ufx 0 u1 x x e Q 1.2 and boundary condition u x tf 0 x e dQ t 0 1.3 2 Advances in Difference Equations where Q is a bounded domain in Rn with a smooth boundary dQ a b 0 and r p 2 are real numbers and Ap 2n 1 d dxi d dxi p-2 d dxi is a divergence operator degenerate Laplace operator with p 2 which is called a p-Laplace operator. Equations of type 1.1 are used to describe longitudinal motion in viscoelasticity mechanics and can also be seen as field equations governing the longitudinal motion of a viscoelastic configuration obeying the nonlinear Voight model 1-4 . For b 0 it is well .