On a parabolic equation with slow and fast diffusions. (English) Zbl 0840.35052
The authors consider the initial value problem \(u_t= \alpha(u)_{xx}\), \(u(0, x)= u_0(x)\), where \(\alpha(u)= u^n\) if \(u\geq 0\) and \(|u|^{m- 1} u\) if \(u\leq 0\) with \(0< n\leq 1< m\), \(u_0\in L^1(\mathbb{R})\). Under additional assumptions on supports of positive and negative parts of the initial function they prove that the solution \(u(t, x)\) of the problem is positive if and only if \(x< a(t)\), where \(a(t)\) is determined via the negative part of the solution. Also, it is proved that \(u(t, x)\) becomes positive on \(x\in \mathbb{R}\) for sufficiently large \(t\) provided that \(\int_{\mathbb{R}} u_0(x) dx> 0\).
Reviewer: P.B.Dubovskij (Obninsk)
MSC:
35K65 | Degenerate parabolic equations |
76S05 | Flows in porous media; filtration; seepage |
35K15 | Initial value problems for second-order parabolic equations |
35K55 | Nonlinear parabolic equations |
Keywords:
positivity of solutionsReferences:
[1] | Aronson, D. G., The porous medium equation, (Fasano, A.; Primicerio, M., Some Problems in Nonlinear Diffusion. Some Problems in Nonlinear Diffusion, Lecture Notes in Maths, Vol. 1224 (1986), Springer: Springer Kyoto) · Zbl 0626.76097 |
[2] | Peletier, L., The porous medium equation, (Amann, H.; etal., Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman: Pitman Berlin) · Zbl 0497.76083 |
[3] | Kamin S. & Vázquez J.E., Asymptotic behavior of solutions of the porous medium equation with changing sign, preprint.; Kamin S. & Vázquez J.E., Asymptotic behavior of solutions of the porous medium equation with changing sign, preprint. · Zbl 0755.35011 |
[4] | Béniean, PH.; Bouieeet, J. E., An equation of diffusion type whose singularities depend on the sign of the solution, (Communication to the XXXIX Reunion Anual (October (1989)), Union Matemática Argentina: Union Matemática Argentina New York) |
[5] | Atkinson, C.; Bouillet, J. E., A generalized diffusion equation: radial symmetries and comparison theorems, J. math. Analysis Applic., 95, 1, 37-68 (1983 .) · Zbl 0536.35036 |
[6] | Barenblatt, G. I., On self-similar motions of compressible fluids in porous media, Prik. Mat. Mekh., 16, 679-698 (1952), (In Russian.) · Zbl 0047.19204 |
[7] | Béniean, PH.; Crandall, M. G., Regularizing effects of homogeneous evolution equations, (Contributions to Analysis and Geometry (1980), Johns Hopkins University Press: Johns Hopkins University Press Rosario), 23-29 |
[8] | Kamin, S., The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14, 76-87 (1973) · Zbl 0254.35054 |
[9] | Vázquez, J. L., Symétrisation pour \(u_t = Δ (u)\) et applications, C. r. Acad. Sci. Paris, 295, 71-74 (1982) · Zbl 0501.35015 |
[10] | Bénilan, Ph., The continuous dependence onof solutions of \(u_t-\) Δ \((u) = 0\), Indiana Univ. Math., 30, 161-177 (1981) · Zbl 0482.35012 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.