Stability of planar diffusion wave for nonlinear evolution equation Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010  

作　　者：HE Cheng HUANG FeiMin YONG Yan 

机构地区：[1]National Natural Science Foundation of China,Beijing 100085,China [2]Institute of Applied Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China [3]College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China

出　　处：《Science China Mathematics》2012年第1期337-352,共16页中国科学：数学（英文版）

基　　金：Acknowledgements He＇s research is supported in part by National Basic Research Program of China （Grant No. 2006CB805902）. Huang＇ research is supported in part by National Natural Science Foundation of China for Distinguished Youth Scholar （Grant No. 10825102）, NSFC-NSAF （Grant No. 10676037） and 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, （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 ∈ R^n. 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 ＋ ut - △f（u） = 0, x ∈ R^n. 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：O411.1[理学—理论物理] G311[理学—物理]