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 of the Solutions of Nonlinear Wave Equation | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 42954 51 pages doi 2007 42954 Research Article Blow up of the Solutions of Nonlinear Wave Equation Svetlin Georgiev Georgiev Received 14 March 2007 Accepted 26 May 2007 Recommended by Peter Bates We construct for every fixed n 2 the metric gs h1 r dt2 - h2 r dr2 - k1 w dw2 - - kn-1 w dwn-1 where h1 r h2 r ki te 1 i n - 1 are continuous functions r x for which we consider the Cauchy problem utt - Au gs f u g x where x e Rn n 2 u 1 x uo x e L2 Rn ut 1 x u1 x e H 1 Rn where f e 1 R1 f 0 0 a u f u blul g e R g r 0 r x a and b are positive constants. When g r 0 we prove that the above Cauchy problem has a nontrivial solution u t r in the form u t r v t tt r for which limt 0 u L2 0 o 00. When g r 0 we prove that the above Cauchy problem has a nontrivial solution u t r in the form u t r v t u r for which limM l u L2 0 o o. Copyright 2007 Svetlin Georgiev Georgiev. 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 In this paper we study the properties of the solutions of the Cauchy problem utt - àu gs f u g x x e Rn n 2 u 1 x uo x e L2 Rn ut 1 x u1 x e H-1 Rn where gs is the metric gs h1 r dt2 - h2 r dr2 - k1 w dw2 - - kn-1 w dw2-1 1 2 2 Boundary Value Problems the functions h1 r h2 r satisfy the conditions h1 r h2 r e 1 0 to h1 r 0 h2 r 0 Vr e 0 to - ịh2 s p h2 r 0 V h1 s s h1 r h2 s fto I h1 T h2 r drds 0 V h1 s s v to Í A B í a isc1 c2 h1 r h2 T dT ds dr to 0 r h1 s s vv h1 T 1 u 22 c1 c2 are arbitrary nonnegative constants 7 Í m ÍTO I rSC1 C h1 T h2 r drẦdsỸdr to 0 r V h1 s s AV h1 r c1 c2 are arbitrary nonnegative constants i1 max Jh1 r h2 r to re 0 TO v TO ir Ĩ7I r Ĩ7Idd 2dr TO 0 r h1 s Js Vh1 T 7 -TO f TO n A c TO 1 2 2 h2 s h1 r h2 r dT ds dr to 0 r h1 s s Bĩ r S dr 1 2d7 2dr to r h1 s s I1 t lịTặds dr
