Study on asymmetry and anisotropy in blow-up theory for nonlinear parabolic equations

点击翻译按钮获取中文摘要

Blow-up phenomenon of positive solutions to the Cauchy problem for nonlinear heat equation of Fujita type is studied. The blow-up phenomenon was first observed and proved by Hiroshi Fujita about half a century ago. There is a large literature on the subject, especially on the formation of blow-up solutions in configuration space up to the blow-up time. We have proved that the blow-up time is characterized by means of the spherical average of the Cauchy data. We removed the assumptions of uniformity and isotropy on the Cauchy data, which were necessary in the former works by Lee and Ni (Trans. Ams, 1992) and Gui and Wang (JDE 1995). We described how the ODE structure controls the blow-up phenomenon.