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

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October 1995, 1(4): 521-546. doi: 10.3934/dcds.1995.1.521

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

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

Received June 1995 Published August 1995

Abstract
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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.

Keywords: Cauchy problem, porous medium equation., Radon measure.

Mathematics Subject Classification: 76S05, 35K1.

Citation: Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521

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