Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms  

作　　者：SHI Peihu & WANG MingxinDepartment of Mathematics, Southeast University, Nanjing 210018, China Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China 

出　　处：《Science China Mathematics》2004年第5期659-674,共16页中国科学：数学（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China(Grant No.19831060);the"333"Project of Jiangsu Province.

摘　　要：This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣ p withm ≥ 1,p,q > 0 andp +q >m. Form = 1, it is shown that the very singular self-similar solution exists if and only ifnq + (n +1)p <n + 2, and in case of existence, such solution is unique. Form > 1, it is shown that very singular self-similar solutions exist if and only if 1 <m < 2 andnq + (n + 1)p < 2 +mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.This paper is devoted to study the classification of self-similar solutions to the m ≥ 1,p,q ＞ 0 and p + q ＞ m. For m = 1, it is shown that the very singular self-similar solution exists if and only if nq + (n + 1)p ＜ n + 2, and in case of existence, such solution is unique. For m ＞ 1, it is shown that very singular self-similar solutions exist if and only if 1 ＜ m ＜ 2 and nq + (n + 1)p ＜ 2 + mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.

关 键 词：QUASILINEAR PARABOLIC equations  SELF-SIMILAR solutions  very SINGULAR solution  classification. 

分 类 号：O171[理学—数学]