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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Solutions with compact support of the porous medium equation in arbitrary dimensions


Author: Michiaki Watanabe
Journal: Proc. Amer. Math. Soc. 103 (1988), 149-152
MSC: Primary 35K55; Secondary 35K65
MathSciNet review: 938660
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Abstract: Compactness of the support is discussed of a solution $ u$ to the Cauchy problem for the porous medium equation $ {u_t} = \Delta \phi (u),t > 0$, in $ {R^N}$ of arbitrary dimension $ N \geq 1$, where $ \phi $ is a nondecreasing function on $ {R^1}$. It is shown that if $ u(0,x) = 0$ for $ \left\vert x \right\vert \geq R,R > 0$, then for all $ t \geq 0$

$\displaystyle u(t,x) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\left\vert x \right\vert \geq R + C{t^{1/2}}$

with a constant $ C$ depending on $ \phi $ and $ u(0, \cdot )$.

The result is well known when $ N = 1$, but the study for $ N > 1$ has somehow been neglected.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0938660-3
PII: S 0002-9939(1988)0938660-3
Keywords: Cauchy problem for porous medium equation, solution with compact support
Article copyright: © Copyright 1988 American Mathematical Society