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 Remark on the Blowup of Solutions to the Laplace Equations with Nonlinear Dynamical Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 203248 11 pages doi 2010 203248 Research Article A Remark on the Blowup of Solutions to the Laplace Equations with Nonlinear Dynamical Boundary Conditions Hongwei Zhang and Qingying Hu Department of Mathematics Henan University of Technology Zhengzhou 450001 China Correspondence should be addressed to Hongwei Zhang wei661@ Received 24 April 2010 Revised 19 July 2010 Accepted 7 August 2010 Academic Editor Zhitao Zhang Copyright 2010 H. Zhang and Q. Hu. 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 present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type. 1. Introduction Let Q be abounded domain of Rn N 1 with a smooth boundary ÔQ S S1 uS2 where S1 and S2 are closed and disjoint and S1 possesses positive measure. We consider the following problem -Au 0 in Q X 0 T d2M du 0f2 k n g u S1 X 0 T a tt- bu 0 on S2 X 0 T on u x 0 u0 ut x 0 u1 on S1 where a 0 b 0 a b 1 and k 0 are constants A is the Laplace operator with respect to the space variables and d dn is the outer unit normal derivative to boundary S. u0 u1 are given initial functions. For convenience we take k 1 in this paper. 2 Boundary Value Problems The problem - can be used as models to describe the motion of a fluid in a container or to describe the displacement of a fluid in a medium without gravity see 1-5 for more information. In recent years the problem has attracted a great deal of people. Lions 6 used the theory of maximal monotone operators to solve the existence of solution of the following problem Aw 0 in Q X 0 T du kfn f u uipu 0 on S X 0 T u x 0 u0 ut x 0 u1 on S. Hintermann 2 used the theory of semigroups in .
