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A very singular solution of the porous media equation with absorption. (English) Zbl 0617.35115
For the Cauchy problem (1) \(u_ t=\Delta (u^ m)-u^ p\), (2) \(u>0\), (3) \(u(x,0)=c\delta (x)\) in \(R^ n\times (0,\infty)\) it has been shown by H. Brezis and the authors [Arch. Ration. Mech. Anal. 95, 185-209 (1986)] that when \(m=1\) there exists a ”very singular” solution W(x,t) of (1), (2) which is smooth except at (0,0) and is more singular at (0,0) than the Barenblatt-Pattle solution [G. I. Barenblatt, Prikl. Mat. Mekh. 16, 679-698 (1952; Zbl 0047.192)]. This very singular solution plays an important role in the description of both the short- and long- time behaviour of solutions of (1), (2). Here, it is shown that if \(m>1\), \(m<p<m+(2/n)\) such a very singular solution also exists.
Reviewer: F.Brauer

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35B99 Qualitative properties of solutions to partial differential equations
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