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 Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 691496 9 pages doi 2009 691496 Research Article Blow-Up Results for a Nonlinear Hyperbolic Equation with Lewis Function Faramarz Tahamtani Department of Mathematics Shiraz University Shiraz 71454 Iran Correspondence should be addressed to Faramarz Tahamtani tahamtani@ Received 17 February 2009 Accepted 28 September 2009 Recommended by Gary Lieberman The initial boundary value problem for a nonlinear hyperbolic equation with Lewis function in a bounded domain is considered. In this work the main result is that the solution blows up in finite time if the initial data possesses suitable positive energy. Moreover the estimates of the lifespan of solutions are also given. Copyright 2009 Faramarz Tahamtani. 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. 1. Introduction Let Q be a bounded domain in R with smooth boundary ÔQ. We consider the initial boundary value problem for a nonlinear hyperbolic equation with Lewis function a x which depends on spacial variable afx utt - pàut - div Vu m 2Vu f u x e Q t 0 u dQ 0 x e dQ t 0 u x 0 uo x ufx 0 u1 x x e Q where afx 0 p 0 m 2 and f is a continuous function. The large time behavior of solutions for nonlinear evolution equations has been considered by many authors for the relevant references one may consult with 1-14 . In the early 1970s Levine 3 considered the nonlinear wave equation of the form Putt Au h u 2 Boundary Value Problems in a Hilbert space where P are A are positive linear operators defined on some dense subspace of the Hilbert space and h is a gradient operator. He introduced the concavity method and showed that solutions with negative initial energy blow up in finite time. This method was later improved by Kalantarov and Ladyzheskaya 4 to .
