Stability of planar diffusion wave for nonlinear evolution equation  

作　　者：He Cheng Huang FeiMin Yong Yan 

机构地区：[1]Shanghai Univ Sci & Technol, Coll Sci, Shanghai 200093, Peoples R China [2]Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100190, Peoples R China [3]Natl Nat Sci Fdn China, Beijing I00085, Peoples R China

出　　处：《Science China Mathematics》2012年第2期337-352,共16页中国科学：数学（英文版）

基　　金：supported in part by National Basic Research Program of China (GrantNo. 2006CB805902);supported in part by National Natural Science Foundation of China for Distinguished Youth Scholar (Grant No. 10825102);NSFC-NSAF (Grant No. 10676037);National Basic Research Program of China (Grant No. 2006CB805902)

摘　　要：It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.

关 键 词：STABILITY PLANAR DIFFUSION WAVE nonlinear EVOLUTION EQUATION 

分 类 号：O175.2[理学—数学]