Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T18:52:01.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic convergence to dipole solutions in nonlinear parabolic equations

Published online by Cambridge University Press:  14 November 2011

Victor A. Galaktionov ,
Sergey A. Posashkov  and
Juan Luis Vazquez
Victor A. Galaktionov
Affiliation:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Sq. 4, 125047 Moscow, Russia
Sergey A. Posashkov
Affiliation:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Sq. 4, 125047 Moscow, Russia
Juan Luis Vazquez
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

We study the asymptotic behaviour as t → ∞ of the solution u = u(x, t) ≧ 0 to the quasilinear heat equation with absorption ut = (um)xx − f(u) posed for t > 0 in a half-line I = { 0 < x < ∞}. For definiteness, we take f(u) = up but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p >m. We impose u = 0 on the lateral boundary {x = 0, t > 0}, and consider a non-negative, integrable and compactly supported function uo(x) as initial data. This problem is equivalent to solving the corresponding equation in the whole line with antisymmetric initial data, uo(−x) = −uo(x).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 877 - 900
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronson, D. G. and Vazquez, J. L.. The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 4 (1987), 203–30.Google Scholar
2Barenblatt, G. I. and Zel'dovich, Ya. B.. On dipole-type solutions in problems of non-stationary filtration of gas under polytropic regime. Prikl. Mat. Mekh. 21 (1957), 718–20.Google Scholar
3Bertsch, M., Nanbu, T. and Peletier, L. A.. Decay of solutions of a degenerate nonlinear diffusion equation. Nonlinear. Anal. 6 (1982), 539–54.Google Scholar
4Brezis, H., Peletier, L. A. and Terman, D.. A very singular solution of the heat equation with absorption. Arch. Rational Mech. Anal. 95 (1986), 185–209.Google Scholar
5Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A.. On asymptotic ‘eigenfunctions’ of the Cauchy problem for a nonlinear parabolic equation. Mat. Sb. 126 (1985), 435–72; English translation: Math. USSR-Sb. 54 (1986), 421–55.Google Scholar
6Galaktionov, V. A. and Posashkov, S. A.. Asymptotics of nonlinear heat conduction with absorption under the critical exponent (preprint, Keldysh Inst. Appl. Math., Acad. Sci. USSR 71, Moscow, 1986).Google Scholar
7Galaktionov, V. A. and Vazquez, J. L.. Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach. J. Fund. Anal. 100 (1991), 435–62.CrossRefGoogle Scholar
8Kalashnikov, A. S.Some problems on the qualitative theory of non-linear degenerate second-order parabolic equations. Uspekhi Mat. Nauk 42 (1987), 135–76; English translation: Russian Math. Surveys 42 (1987), 169–222.Google Scholar
9Kamin, S.. Similar solutions and the asymptotics of filtration equations. Arch. Rational Mech. Anal. 60(1975/1976), 171–83.Google Scholar
10Kamin, S. and Vazquez, J. L.. Asymptotic behaviour of solutions of the porous medium equation with changing sign. SI AM J. Math. Anal. 22 (1991), 34–45.Google Scholar
11Kamin, S. and Vazquez, J. L.. Singular solutions of some nonlinear parabolic equations. J. Analyse Math. 59(1992), 51–74.Google Scholar
12Kamin, S. and Véron, L.. Existence and uniqueness of the very singular solution of the porous medium equation with absorption. J. Analyse Math. 51 (1988), 245–58.Google Scholar
13Peletier, L. A. and Terman, D.. A very singular solution of the porous media equation with absorption. J. Differential Equations 65 (1986), 396–410.CrossRefGoogle Scholar
14Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P.. Blow-up in Quasilinear Parabolic Equations (Moscow: Nauka, 1987; English translation: Walter de Gruyter, Berlin, 1995).Google Scholar
15Vazquez, J. L.. Singular solutions and asymptotic behaviour of nonlinear parabolic equations. In International Conference on Differential Equations (Equadiff 91), Barcelona, Spain, Vol. 1, 234–49 (Singapore: World Scientific, 1993).Google Scholar