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Localization of singularities for nonlinear parabolic equations with a potential Souplet, Philippe


We consider nonlinear parabolic equations of the form $$u_t= \Delta u+V(x)f(u),$$ where $V\ge 0$ is a potential and $f$ is a nonlinearity which can be either of blow-up or of quenching type, typically $$f(u)=(1+u)^p\quad (p>1) \quad\hbox{ or }\quad f(u)=(1-u)^{-p}\quad (p>0).$$ We are interested in the localization of finite time singularities. A main question is: can one rule out the occurence of singularities at zeros of the potential ? More generally, can one show that singularities concentrate at points where $V$ is large in a certain sense ? We will present positive and negative results, which indicate that the issue is delicate. The techniques involve combinations of various kinds of maximum principle arguments, as well as Liouville type theorems. Joint works in collaboration with Jong-Shenq Guo and with Carlos Esteve.

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