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báo cáo hóa học:" Research Article A Quasilinear Parabolic System with Nonlocal Boundary Condition"

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 A Quasilinear Parabolic System with Nonlocal Boundary Condition | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 750769 18 pages doi 10.1155 2011 750769 Research Article A Quasilinear Parabolic System with Nonlocal Boundary Condition Botao Chen 1 Yongsheng Mi 1 2 and Chunlai Mu2 1 College of Mathematics and Computer Sciences Yangtze Normal University Fuling Chongqing 408100 China 2 College of Mathematics and Physics Chongqing University Chongqing 401331 China Correspondence should be addressed to Chunlai Mu chunlaimu@yahoo.com.cn Received 8 May 2010 Revised 25 July 2010 Accepted 11 August 2010 Academic Editor Daniel Franco Copyright 2011 Botao 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 investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. 2009 which considered the special case m1 m2 1 p1 0 q2 0 and Wang et al. 2007 which studied the single equation. 1. Introduction In this paper we deal with the following degenerate parabolic system ut àum1 up1 vq1 vt àvm2 vp2uq2 x e Q t 0 1.1 with nonlocal boundary condition u x f J f x y u y f dy v x f ị g x y v y f dy x e dQ f 0 1.2 and initial data u x 0 uo x v x 0 v0 x x e Q 1.3 2 Boundary Value Problems where mi pi qi 1 i 1 2 and Q c RN is a bounded connected domain with smooth boundary. f x y 0 and g x y 0 for the sake of the meaning of nonlocal boundary are nonnegative continuous functions defined for x dQ and y Q while the initial data v0 u0 are positive continuous functions and satisfy the compatibility conditions Uo x JQ f x y u0 y dy and v0 x JQ g x y v0 y dy for x ÔQ .