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Monotone methods and attractivity results for Volterra integro-partial differential equations

Published online by Cambridge University Press:  14 November 2011

Andrea Schiaffino  and
Alberto Tesei
Andrea Schiaffino
Affiliation:
Istituto Matematico “G. Castelnuovo”, Città Universitaria, Rome, Italy
Alberto Tesei
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, Rome, Italy
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Synopsis

A Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 1-2 , 1981 , pp. 135 - 142
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Cushing, J. M.. Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics 20 (Berlin: Springer, 1977).Google Scholar
2Grossman, S. I. and Miller, R. K.. Perturbation theory for Volterra integrodifferential systems. J. Differential Equations 8 (1970), 457–474.CrossRefGoogle Scholar
3Miller, R. K.. On Volterra's population equation. SIAM J. Appl. Math. 14 (1966), 446–452.CrossRefGoogle Scholar
4Miller, R. K.. Asymptotic stability and perturbations for linear Volterra integrodifferential systems. In: Delay and Functional Differential Equations and Their Applications (New York: Academic Press, 1972).Google Scholar
5Miller, R. K.. Some Volterra integral equations of growth rate type. Zagadn. Drgan Nieliniow. 15 (1974), 125–137.Google Scholar
6Puel, J. P.. Existence, comportement à l'infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d'ordre 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 89–119.Google Scholar
7Sattinger, D. H.. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979–1000.CrossRefGoogle Scholar
8Schiaffino, A. and Tesei, A.. On the asymptotic stability for abstract Volterra integro-differential equations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 67 (1979), 67–74.Google Scholar
9Schiaffino, A. and Tesei, A.. Asymptotic stability properties for nonlinear diffusion Volterra equations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 67 (1979), 227–232.Google Scholar
10Schiaffino, A.. On a diffusion Volterra equation. Nonlinear Anal: TMA 3 (1979), 595–600.CrossRefGoogle Scholar
11Solonnikov, V. A.. On boundary value problems for linear parabolic systems of differential equations of general type. Trudy Mat. Inst. Steklov, 83 (1965). English transl. in: Boundary Value Problems of Mathematical Physics (Providence, R.I.: Amer. Math. Soc., 1967).Google Scholar
12Stakgold, I. and Payne, L. E.. Nonlinear problems in nuclear reactor analysis. In: Nonlinear Problems in the Physical Sciences and Biology. Lecture Notes in Mathematics 322 (Berlin: Springer, 1973).Google Scholar
13Stewart, H. B.. Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974), 141–162.CrossRefGoogle Scholar
14Tesei, A.. Stability properties for partial Volterra integrodifferential equations. Ann. Mat. Pura Appl. 126 (1980), 103–115.CrossRefGoogle Scholar
15Volterra, V.. Leçons sur la Théorie Mathématique de la Lutte pour la Vie (Paris: Gauthier-Villars, 1931).Google Scholar