# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Aronson, D.G, Regularity properties of flows through porous media: the interface, Arch. rational mech. anal., 37, 1-10, (1970) · Zbl 0202.37901 [2] Aronson, D.G, Density dependent interaction-diffusion systems, () · Zbl 0529.35046 [3] Barenblatt, G.I, On some unsteady motions in a liquid or a gas in a porous medium, Prikl. mat. mekh., 16, 637-732, (1952) [4] Brezis, H; Friedman, A, Nonlinear parabolic equations involving measures as initial conditions, J. math. pures appl., 62, 73-97, (1983) · Zbl 0527.35043 [5] Brezis, H; Peletier, L.A; Terman, D, A very singular solution of the heat equation with absorption, Arch. rational mech. anal., (1986) · Zbl 0627.35046 [6] Caffarelli, L.A; Friedman, A, Continuity of the density of a gas flow in a porous medium, Trans. amer. math. soc., 252, 99-113, (1979) · Zbl 0425.35060 [7] \scM. Escobedo and O. Kavian, Asymptotic behaviour of positive solutions of a nonlinear heat equation, to appear. · Zbl 0666.35046 [8] Galaktionov, V.A; Kurdjumov, S.P; Samarskii, A.A, On asymptotic stability of similarity solutions of the heat equation with nonlinear absorption, Dokl. acad. nauk SSSR, 281, 23-28, (1985), [in Russian] [9] Guckenheimer, J; Holmes, P.J, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001 [10] Kamin, S; Peletier, L.A, Source-type solutions of degenerate diffusion equations with absorption, Israel J. math., 50, 219-230, (1985) · Zbl 0581.35035 [11] Kamin, S; Peletier, L.A, Singular solutions of the heat equation with absorption, () · Zbl 0607.35046 [12] \scL. Oswald, Classification des solutions positives de l’equation de la chaleur Nonlineare avec derivee initiale singuliere en 1 point, preprint. [13] Peletier, L.A, The porous media equation, (), 229-241 · Zbl 0497.76083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.