Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-12T22:38:24.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption*

Published online by Cambridge University Press:  14 November 2011

F. Bernis
F. Bernis
Affiliation:
Departamento de Matemáticas, Universidad Politécnica de Cataluńa, Apdo. 30002, 08034 Barcelona, Spain
Get access Rights & Permissions [Opens in a new window]

Synopsis

The “energy solutions” to the equation

have finite speed of propagation if l < q < 2 or l < r < 2. If 1 <r <2 (Vq <1) support u(· t) is uniformly bounded for t >0 (localisation property) and if q<2 ≦ r, sharp upper bounds of the interface (or free boundary) are obtained. We use a weighted energy method, the weights being powers of the distance to a variable half-space. We also study decay rates as t→∞ and extinction in finite time for bounded and unbounded domains (with null Dirichlet boundary conditions). Our equation includes the porous media equation with absorption. Analogous results hold if (−Δ)m is replaced by an appropriate quasilinear elliptic operator.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 1 - 19
Copyright
Copyright © Royal Society of Edinburgh 1986

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

1Alikakos, N. D. and Rostamian, R.. Decay rates of solutions of the equation u1 = δ(∣u∣m−u − ∣u∣p−1u(to appear).Google Scholar
2Alt, H. W. and Luckhaus, S.. Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341.Google Scholar
3Aronson, D. G. and Peletier, L. A.. Large time behaviour of solutions of the porous media equation in bounded domains. J. Differential Equations 39 (1981), 378–412.CrossRefGoogle Scholar
4Bernis, F.. Qualitative properties for some nonlinear higher order degenerate parabolic equations. IMA Preprint 184, University of Minnesota (1985).Google Scholar
5Bernis, F.. Existence results for “doubly” nonlinear higher order parabolic equations on unbounded domains. IMA Preprint 226, University of Minnesota (1986).Google Scholar
6Bernis, F.. Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n. Comm. Partial Differential Equations 9 (1984), 271–312.Google Scholar
7Bertsch, M., Kersner, R. and Peletier, L. A.. Sur le comportement de la frontière libre dans une équation en théorie de la filtration. C. R. Acad. Sci. Paris Sér. 1 Math. 295 (1982), 63–66.Google Scholar
8Bertsch, M., Kersner, R. and Peletier, L. A.. Positivity versus localization in degenerate diffusion equations. Nonlinear Anal, (to appear).Google Scholar
9Bertsch, M., Nambu, T. and Peletier, L. A.. Decay of solutions of a degenerate nonlinear diffusion equation. Nonlinear Anal. 6 (1982), 539–554.Google Scholar
10Bertsch, M. and Peletier, L. A.. A positivity property of solutions of nonlinear diffusion problems. Differential Equations (to appear).Google Scholar
11I, J.. Díaz and Herrero, M. A.. Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 249–258.Google Scholar
12Díaz, J. I. and Véron, L.. Compacité du support des solutions d'équations quasi-linéaires elliptiques ou paraboliques. C. R. Acad. Sci. Paris Sér. 1 Math. 297 (1983), 149–152.Google Scholar
13Grange, O. and Mignot, F.. Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires. J. Fund. Anal. 11 (1972), 77–92.Google Scholar
14Gurtin, M. E. and MacCamy, R. C.. On the diffusion of biological populations. Math. Biosci. 33 (1977), 35–49.CrossRefGoogle Scholar
15Herrero, M. A.. On the growth of the interfaces of a nonlinear degenerate parabolic equation. In Contributions to nonlinear partial differential equations, ed. Bardos, C. et al. Research Notes in Mathematics 89 (London: Pitman, 1983, 218–224).Google Scholar
16Herrero, M. A. and Vázquez, J. L.. The one-dimensional nonlinear heat equation with absorption: regularity of solutions and interfaces. SIAM J. Math. Anal, (to appear).Google Scholar
17Kalashnikov, A. S.. The propagation of disturbances of nonlinear heat conduction with absorption. (English translation) USSR Comput. Math, and Math. Phys. 14 (1974), 70–85.Google Scholar
18Kalashnikov, A. S.. On the differential properties of generalized solutions of equations of the nonsteady filtration type. (English translation) Moscow Univ. Math. Bull. 29 (1974), 48–53.Google Scholar
19Kamin, S. and Peletier, L. A.. Large time behaviour of solutions of the porous media equation with absorption. Report 14, Mathematical Institute, University of Leiden, 1985.Google Scholar
20Kersner, R.. Some properties of generalized solutions of quasi-linear degenerate parabolic equations. Acta Math. Acad. Sci. Hungar. 32 (1978), 301–330.Google Scholar
21Kersner, R.. On the behaviour when t grows to infinity of generalized solutions of a degenerate quasi-linear parabolic equation. Acta Math. Acad. Sci. Hungar. 34 (1979), 157–163.Google Scholar
22Kersner, R.. Nonlinear heat conduction with absorption: space localization and extinction in finite time. SIAM J. Appl. Math. 43 (1983), 1274–1285.Google Scholar
23Knerr, B. F.. The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension. Trans. Amer. Math. Soc. 249 (1979), 409–424.Google Scholar
24Knops, R. J. and Straughan, B.. Decay and nonexistence for sublinearly forced systems in Continuum Mechanics. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II, eds. Brezis, H. and Lions, J. L., pp. 251–264. Research Notes in Mathematics 60 (London: Pitman, 1982).Google Scholar
25Martinson, L. K. and Pavlov, K. B.. The problem of the three-dimensional localization of thermal perturbations in the theory of nonlinear heat conduction. (English translation) USSR Comput. Math, and Math. Phys. 12 (1972), 261–268.Google Scholar
26Nirenberg, L.. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13 (1959), 115–162.Google Scholar
27Payne, L. E.. Improperly posed problems in partial differential equations. SIAM Regional Conference Series in Applied Mathematics 22 (1975).Google Scholar
28Straughan, B.. Instability, nonexistence and weighted energy methods in fluid dynamics and related theories. Research Notes in Mathematics 74 (London: Pitman, 1982).Google Scholar