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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonradial solvability structure of super-diffusive nonlinear parabolic equations


Authors: Panagiota Daskalopoulos and Manuel del Pino
Journal: Trans. Amer. Math. Soc. 354 (2002), 1583-1599
MSC (1991): Primary 35K15, 35K55, 35K65; Secondary 35J40
Published electronically: December 4, 2001
MathSciNet review: 1873019
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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 $m < 0$ in ${\bf R}^2$, with $u(x,0)= f(x)$ 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 $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= \vert x\vert$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ 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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Additional Information

Panagiota Daskalopoulos
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
Email: pdaskalo@math.uci.edu

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Email: delpino@dim.uchile.cl

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02888-4
PII: S 0002-9947(01)02888-4
Received by editor(s): March 25, 1998
Published electronically: December 4, 2001
Additional Notes: The first author was partially supported by The Sloan Foundation and by NSF/Conicyt-Chile grant INT-9802406
The second author was partially supported by grants Lineas Complementarias Fondecyt 8000010 and FONDAP
Article copyright: © Copyright 2001 American Mathematical Society