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Báo cáo hóa học: " Research Article Systems of Quasilinear Parabolic Equations with Discontinuous Coefficients and Continuous Delays"

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 Systems of Quasilinear Parabolic Equations with Discontinuous Coefficients and Continuous Delays | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 925173 25 pages doi 10.1155 2011 925173 Research Article Systems of Quasilinear Parabolic Equations with Discontinuous Coefficients and Continuous Delays Qi-Jian Tan Department of Mathematics Sichuan College of Education Chengdu 610041 China Correspondence should be addressed to Qi-Jian Tan tanqjxxx@yahoo.com.cn Received 24 December 2010 Accepted 3 March 2011 Academic Editor Jin Liang Copyright 2011 Qi-Jian Tan. 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. This paper is concerned with a weakly coupled system of quasilinear parabolic equations where the coefficients are allowed to be discontinuous and the reaction functions may depend on continuous delays. By the method of upper and lower solutions and the associated monotone iterations and by difference ratios method and various estimates we obtained the existence and uniqueness of the global piecewise classical solutions under certain conditions including mixed quasimonotone property of reaction functions. Applications are given to three 2-species Volterra-Lotka models with discontinuous coefficients and continuous delays. 1. Introduction Reaction-diffusion equations with time delays have been studied by many researchers see 1-8 and references therein . However all of the discussions in the literature are devoted to the equations with continuous coefficients. In this paper we consider a weakly coupled system of quasilinear parabolic equations where the coefficients are allowed to be discontinuous and the reaction functions may depend on continuous infinite or finite delays. To describe the problem we first introduce some notations. Let Q be a bounded domain with the boundary dQ in Rn n 1 . Suppose that Q consists of a finite number of domains Qk k 1 . K separated by