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Báo cáo hóa học: " Research Article Global Existence, Uniqueness, and Asymptotic Behavior of Solution for p-Laplacian Type Wave Equation"

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 Global Existence, Uniqueness, and Asymptotic Behavior of Solution for p-Laplacian Type Wave Equation | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 216760 15 pages doi 10.1155 2010 216760 Research Article Global Existence Uniqueness and Asymptotic Behavior of Solution for p-Laplacian Type Wave Equation Caisheng Chen Huaping Yao and Ling Shao Department of Mathematics Hohai University Nanjing Jiangsu 210098 China Correspondence should be addressed to Caisheng Chen cshengchen@hhu.edu.cn Received 10 May 2010 Accepted 13 July 2010 Academic Editor Michel C. Chipot Copyright 2010 Caisheng Chen et al. 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. We study the global existence and uniqueness of a solution to an initial boundary value problem for the nonlinear wave equation with the p-Laplacian operator utt -div Vu p-2Vu - Nut g x ù f x . Further the asymptotic behavior of solution is established. The nonlinear term g likes g x u a x u a-1u - b x u p-1u with appropriate functions a x and b x where a p 1. 1. Introduction This paper is concerned with the global existence uniqueness and asymptotic behavior of solution for the nonlinear wave equation with the p-Laplacian operator uff - div Vu p-2Vu - ầuf g x u f Jx in Q X 0 to 1.1 u x 0 u0 x uf x 0 u1 x in Q u x f 0 on dQ X 0 to 1.2 where 2 p n and Q is a boundary domain in Rn with smooth boundary ÔQ. The assumptions on f g u0 and u1 will be made in the sequel. Recently Ma and Soriano in 1 investigated the global existence of solution u f for the problem 1.1 - 1.2 under the assumptions p n g u u 0 g u Cy exp p u n n 1J u e R. 1.3 2 Journal of Inequalities and Applications Moreover if f 0 and ug u G u then there exist positive constants c and Y such that E f cexp -pt t 0 if n 2 1.4 Et c 1 t -n n-2 t 0 if n 3 1.5 where Et 1 Wudt W2 1IIVu t nn Í G x u ty dx 1.6 2 n Q with G x u fU f x s ds. Gao and Ma in 2 also .