Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T20:13:49.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Positivity for elliptic and parabolic systems

Published online by Cambridge University Press:  14 November 2011

J. F. G. Auchmuty
J. F. G. Auchmuty
Affiliation:
Fluid Mechanics Research Institute, University of Essex, and Department of Mathematics, Indiana University, Bloomington
Get access Rights & Permissions [Opens in a new window]

Synopsis

The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 3-4 , 1978 , pp. 183 - 191
Copyright
Copyright © Royal Society of Edinburgh 1978

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

1Aris, R.The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, 2 Vols. (London: Oxford Univ. Press, 1975).Google Scholar
2Auchmuty, J. F. G. and Nicolis, G.Bifurcation Analysis of Nonlinear Reaction-Diffusion Equations, I. Bull. Math. Biol. 37 (1975), 323–365.Google Scholar
3Crandall, M. G. and Rabinowitz, P. H.Nonlinear Sturm-Liouville Eigenvalue Problems and Topological Degree. J. Math. Mech. 19 (1970), 1083–1102.Google Scholar
4Dancer, E. N.Global Solution Branches for Positive Mappings. Arch. Rational Mech. Anal. 52 (1973), 181–192.Google Scholar
5Gavalas, G. R.Nonlinear Differential Equations of Chemically Reacting Systems (New York: Springer, 1969).Google Scholar
6Gierer, A. and Meinhardt, H.A Theory of Biological Pattern Formation. Kiberneticka 12 (1972), 30–39.Google Scholar
7Goldbeter, A.Patterns of Spatiotemporal Organization in an Allosteric Enzyme Model. Proc. Nat. Acad. Sci. U.S.A. 70 (1973), 3255–3259.CrossRefGoogle Scholar
8Martin, R. H.Differential Equations on Closed Subsets of Banach Space. Trans. Amer. Math. Soc. 179 (1973), 339–414.Google Scholar
9Mlak, W.Differential Inequalities of Parabolic Type. Ann. Polon. Math. 3 (1957), 349–354.Google Scholar
10Nagumo, M.Uber die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proc. Phys-Math. Soc. Japan 24 (1942), 551–559.Google Scholar
11Nussbaum, R. D. and Stuart, C. A. A Singular Bifurcation Problem. Battelle Inst. Report (1975).Google Scholar
12Protter, M. H. and Weinberger, H. F.Maximum Principles in Differential Equations (Englewood Cliffs: Prentice Hall, 1967).Google Scholar
13Rabinowitz, P. H.Nonlinear Sturm-Liouville Problems for Second-Order Ordinary Differential Equations. Comm. Pure and Appl. Math. 23 (1970), 939–961.Google Scholar
14Rabinowitz, P. H.Some Global Results for Nonlinear Eigenvalue Problems. J. Functional Analysis 7 (1971), 487–513.Google Scholar
15Szarski, J.Sur la limitation et l'unicité des solutions d'un système non-linéaire d'équations parabolique aux dérivées partielles du second ordre. Ann. Polon. Math. 2 (1955), 237–249.Google Scholar
16Szarski, J.Sur un système non-linéaire d'inégalités différentielles paraboliques. Ann. Polon. Math. 15 (1964), 15–22.CrossRefGoogle Scholar
17Szarski, J.Differential Inequalities (Warsaw: Polish Sci. Pub., 1965).Google Scholar
18Turner, R. E. L.Transversality and Cone Maps. Arch. Rational Mech. Anal. 58 (1975), 151–179.Google Scholar
19Weinberger, H. F.Invariant Sets for Weakly Coupled Parabolic and Elliptic Systems. Rend. Mat. 8 (1975), 295–310.Google Scholar
20Yorke, J. A.Differential Inequalities and Non-Lipschitz Scalar Functions. Math. Systems Theory 4 (1970), 140–153.Google Scholar