On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data

Kazuhiro Ishige

doi:10.3934/dcds.1995.1.521

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1995, Volume 1, Issue 4: 521-546. Doi: 10.3934/dcds.1995.1.521

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On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data

Kazuhiro Ishige^1,

1.
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-77

Received: May 31, 1995

Published: September 1995

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Abstract

For any nonnegative Radon measure $\mu$, we prove the existence of solutions for the Cauchy problem:

$ u_t =\Delta\phi(u)\qquad\text{in}\quad R^N\times(0,T);\qquad u(\cdot,0) =\mu(\cdot)\ge 0\quad \text{in}\quad R^N, $

where $\phi'(s)$ ~ $\log^m s$, $m<-1$, as $s\to\infty$. On the other hand, for the case $m\ge -1$, we give a sufficient condition for the solvability of the Cauchy problem.

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Mathematics Subject Classification: 76S05, 35K10.

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