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Maximum principles and comparison theorems for semilinear parabolic systems and their applications

Published online by Cambridge University Press:  14 November 2011

G. Lu  and
B. D. Sleeman
G. Lu
Affiliation:
Department of Mathematics and Computer Science, The University, Dundee DD1 4HN, Scotland
B. D. Sleeman
Affiliation:
Department of Mathematics and Computer Science, The University, Dundee DD1 4HN, Scotland
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Synopsis

A fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 5 , 1993 , pp. 857 - 885
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Arnol'd, V. I. and , co-workers. Some unsolved problems in the theory of differential equations and mathematical physics. Uspekhi Mai. Nauk 44 (1989), 191–202.Google Scholar
2Aronson, D. G. and Weinberger, H. F.. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1978), 33–76.CrossRefGoogle Scholar
3Escobedo, M. and Herrero, M. A.. Boundedness and blow up for a semilinear reaction-diffusion system. Differential Equations 89 (1991), 176–202.CrossRefGoogle Scholar
4Escobedo, M. and Levine, H. A.. Explosion et existence globale pour un système faiblement couple d'equations de réaction diffusion. C.R. Acad. Sci. Paris t.134 Serie 1 (1992), 737–739.Google Scholar
5Fife, P. C.. Mathematical Aspects of Reaction and Diffusing Systems (Berlin: Springer, 1978).Google Scholar
6Fife, P. C. and Tang, M. M.. Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances. Differential Equations 40 (1981), 168–185.CrossRefGoogle Scholar
7Fife, P. C. and Tang, M. M.. Corrigendum. J. Differential Equations 51 (1984), 442–447.Google Scholar
8Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, N.J.: Prentice-Hall, 1964).Google Scholar
9Friedman, A., Partial Differential Equations. (New York: Holt, Rinehart and Winston, 1969).Google Scholar
10Friedman, A. and Giga, Y.. A single point blow-up for solutions of semilinear parabolic systems. J. Fac. Sci. Univ. Tokyo, Sec. IA 34 (1987), 65–79.Google Scholar
11Fujita, H.. On the blowing up of solutions of the Cauchy problem for J. Fac. Sci. Univ. Tokyo. Sec. I 13 (1966), 109–124.Google Scholar
12Fujita, H.. On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Proc. Symp. Pure. Math. (1970), 105–113.CrossRefGoogle Scholar
13Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A.. A parabolic system of quasilinear equations I. Differential Equations 19 (1984), 1558–1574.Google Scholar
14Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A.. A parabolic system of quasilinear equations II. Differential Equations 21 (1986), 1049–1062.Google Scholar
15Grindrod, P. and Sleeman, B. D.. Comparison principles in the analysis of reaction-diffusion systems modelling unmyelinated nerve fibres. IMA J. Math. Appl. Med. Biol. 1 (1984), 343–363.CrossRefGoogle ScholarPubMed
16Hadeler, K. P. and Glas, D.. Quasimonotone systems and convergence to equilibrium in a population genetic model. J. Math. Anal. Appl. 95 (1983), 297–303.CrossRefGoogle Scholar
17Henry, D.. Geometric Theory of Semilinear Parabolic Equations (Berlin: Springer, 1981).CrossRefGoogle Scholar
18Hirch, M. W.. Systems of differential equations which are competitive or cooperative I: limit sets. SIAM J. Math. Anal. 13 (1982), 167–179.CrossRefGoogle Scholar
19Hofbauer, J. and Sigmund, K.. The Theory of Evolution and Dynamical Systems (Cambridge: Cambridge University Press, 1984).Google Scholar
20Istratescu, V. I.. Fixed Point Theory (Dordrecht: D. Reidel, 1981).CrossRefGoogle Scholar
21Lakshmikanthan, V. and Leela, S.. Differential and Integral Inequalities, Vol. I (New York: Academic Press, 1969).Google Scholar
22Levine, H. A.. The role of critical exponents in blow up theorems. SIAM Rev. 32 (1990), 262–288.CrossRefGoogle Scholar
23Levine, H. A.. A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations Z.A.M.P. 42 (1992) 408–430.Google Scholar
24Lu, G. and Sleeman, B. D.. Sub and super solutions to systems of parabolic equations with Applications to Generalised Fujita Type Systems. Math. Methods in the Appl. Sciences (to appear).Google Scholar
25Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (Englewood Cliffs, N.J.: Prentice-Hall, 1967).Google Scholar
26Sperb, R.. Maximum Principles and Their Applications (New York: Academic Press, 1981).Google Scholar
27Stokes, A. P.. The application of a fixed-point theorem to a variety of nonlinear stability problems. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 231–235.CrossRefGoogle ScholarPubMed
28Terman, D.. Comparison theorems for reaction-diffusion systems defined in an unbounded domain (University of Wisconsin, Madison Mathematics Research Center, Technical Summary Report 2374).Google Scholar
29Terman, D.. Threshold phenomena for a reaction-diffusion system. J. Differential Equations 47 (1983), 406–443.CrossRefGoogle Scholar