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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

Kazuhiro Ishige 1,

1. 

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

Received  June 1995 Published  August 1995

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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