Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T04:14:20.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of mixed boundary conditions in some reaction–diffusion systems

Published online by Cambridge University Press:  14 November 2011

Robert H. Martin  and
Michel Pierre
Robert H. Martin
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 28695, U.S.A.
Michel Pierre
Affiliation:
Institut Elie Cartan, Université de Nancy I, UMR CNRS 9973, Inria-Lorraine, ProjectNUMATH, B.P. 239, 54506-Vandoeuvre, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

We analyse global existence of solutions to a system of two reaction–diffusion equations for whicha ‘balance’ law holds. The main aim is to make clear the influence of different combinations ofboundary conditions on global existence under the assumption that the nonlinearities satisfy polynomial growth estimates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 1053 - 1066
Copyright
Copyright © Royal Society of Edinburgh 1997

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.. L p bounds of solutions of reaction–diffusion equations. Comm. Partial Differential Equations 4 (1979), 827–68.CrossRefGoogle Scholar
2Baras, P. and Pierre, M.. Problèmes paraboliques semi-linéaires avec données mesures. Appl. Anal. 18 (1984), 111–49.CrossRefGoogle Scholar
3Bebernes, J. and Lacey, A.. Finite time blow-up for a particular parabolic system. SIAM J. Math. Anal. 21 (1990), 1415–25.CrossRefGoogle Scholar
4Bebernes, J. and Lacey, A.. Finite time blow-up for semilinear reactive–diffusive systems. J. Differential Equations 95 (1992), 105–29.CrossRefGoogle Scholar
5Bénilan, Ph. and Labani, H.. Existence of attractors for the Brusselator. In Recent Advances in Nonlinear Elliptic and Parabolic Problems, eds Benilan, Ph., Chipot, M., Evans, L. C. and Pierre, M., Pitman Research Notes in Mathematics 208, 139–51 (Harlow: Longman 1989).Google Scholar
6Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order (Berlin: Springer, 1977).CrossRefGoogle Scholar
7Haraux, A. and Youkana, A.. On a result of K. Masuda concerning reaction–diffusion equations. Tohoku Math. J. 40 (1988), 159–63.CrossRefGoogle Scholar
8Hollis, S. L.. Globally Bounded Solutions of Reaction–Diffusion systems (PhD. Thesis, North Carolina State University, 1986).Google Scholar
9Hollis, S. L.. On the question of global existence for reaction–diffusion systems with mixed boundary conditions. Quart. Appl. Math. 51 (2) (1993).CrossRefGoogle Scholar
10Hollis, S. L., Martin, R. H. and Pierre, M.. Global existence and boundedness in reaction diffusion systems. SIAM J. Math. Anal. 18 (1987) 744–61.CrossRefGoogle Scholar
11Kanel, J. I.. Cauchy's problem for semilinear parabolic equations with balance condition. Differential'nye Uravneniya 20 (1984), 1753–60.Google Scholar
12Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N.. Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23 (Providence, R.I.: American Mathematical Society, 1968).CrossRefGoogle Scholar
13Manteuffel, T. and Parter, S.. Preconditioning and boundary conditions. SIAM J. Numer. Anal. 27 (1990), 656–94.CrossRefGoogle Scholar
14Martin, R. H. and Pierre, M.. Nonlinear reaction–diffusion systems. In Nonlinear Equations in the Applied Sciences, eds Ames, W. F. and Rogers, C.(Boston: Academic Press, 1992).Google Scholar
15Masuda, K.. On the global existence and asymptotic behavior of reaction–diffusion equations. Hokkaido Math. J. 12 (1983) 360–70.CrossRefGoogle Scholar
16Morgan, J.. Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20 (1989), 1128–44.CrossRefGoogle Scholar
17Morgan, J.. On a question of blow-up for semilinear parabolic systems. Differential Integral Equations (to appear).Google Scholar
18Rothe, F.. Global solutions of reaction–diffusion systems, Lecture Notesin Mathematics 1072 (Berlin: Springer 1984).Google Scholar
19Solonnikov, V.. Boundary value problems of mathematical physics, III. In Proceedings of the Steklov Institute of Mathematics, ed. Ladyzenskaja, O. A. (Providence, R.I.: American Mathematical Society, 1967).Google Scholar