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Localization for a porous medium type equation in high dimensions

Authors:	Changfeng Gui Xiaosong Kang

Institution:	Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269 ; The Fields institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1

Abstract:	We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla(u^ {\sigma} \nabla u)+u^ \beta$ , $x \in \mathbf{R}^ N$ , , $\beta >\sigma +1$ , $\sigma>0,$ in the case of arbitrary compactly supported initial functions . We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname{supp}u_0$ and the blow-up time . Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.

Keywords:	Porous medium type equation with source localization property blow-up self-similar solutions comparison

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