Nonradial solvability structure of super-diffusive nonlinear parabolic equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Nonradial solvability structure of super-diffusive nonlinear parabolic equations

Authors:	Panagiota Daskalopoulos Manuel del Pino

Institution:	Department of Mathematics, University of California at Irvine, Irvine, California 92697 ; Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo~3, Santiago, Chile

Abstract:	We study the solvability of the Cauchy problem for the nonlinear parabolic equation $\begin{displaymath}\frac {\partial u}{\partial t} = \mbox{div}\, (u^{m-1}\nabla u)\end{displaymath}$ when in ${\bf R}^2$ , with a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$ , $r=\vert x\vert$ for the spherical averages of is critical for local solvability, in particular ensuring it if is radially symmetric. We show that if the initial data behaves in polar coordinates like $r^{-2/1-m} g(\theta )$ , for large $r= \vert x\vert$ with nonnegative and $2\pi$ -periodic, then the following holds: If vanishes on some interval of length $l^* = \frac {(m-1)\pi}{2m} >0$ , then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.

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