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Zbl 1073.35130
Fan, Huijun
Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure.
(English)
[J] Acta Math. Sin., Engl. Ser. 20, No. 4, 663-682 (2004). ISSN 1439-8516; ISSN 1439-7617/e

The author studies the existence of nonnegative solutions of the Cauchy problem $u_t=\nabla\cdot(\vert \nabla u^m\vert ^{p-2}\nabla u^m)-u^q$ in $\Bbb R^N\times(0,\infty)$, $u(\cdot,0)=\mu$ in $\Bbb R^N$, where $p>1$, $m>0$, $q>0$ and $\mu$ is a nonnegative bounded Radon measure. The existence is proved provided $m(p-1)+p/N>1$ and $q<m(p-1)+p/N$. It is also shown that these conditions are essentially optimal. The existence proof is based on the approximation of $\mu$ by smooth functions.
[Pavol Quittner (Bratislava)]
MSC 2000:
*35K65 Parabolic equations of degenerate type
35R05 PDE with discontinuous coefficients or data
35K15 Second order parabolic equations, initial value problems

Keywords: degenerate parabolic equation; measure data; Cauchy problem; nonnegative bounded Radon measure

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