Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation

被引:7
|
作者
Polacik, Peter [2 ]
Yanagida, Eiji [1 ]
机构
[1] Tohoku Univ, Math Inst, Sendai, Miyagi 9808578, Japan
[2] Univ Minnesota, Sch Math, Minneapolis, MN 55455 USA
基金
日本学术振兴会;
关键词
Semilinear parabolic equation; Critical exponent; Anisotropic decay; Quasi-convergence; Self-similar solution; PARABOLIC EQUATION; ELLIPTIC-EQUATIONS; POSITIVE SOLUTIONS; CAUCHY-PROBLEM; STEADY-STATES; INSTABILITY; THEOREMS; BEHAVIOR; BLOWUP;
D O I
10.1007/s10884-009-9136-7
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.
引用
收藏
页码:329 / 341
页数:13
相关论文
共 50 条