Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations

Ackleh, Azmy; Ke, Lan

doi:10.1090/S0002-9939-00-05912-8

Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations
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by Azmy S. Ackleh and Lan Ke PDF

Proc. Amer. Math. Soc. 128 (2000), 3483-3492 Request permission

Abstract:

We establish existence and uniqueness of solutions for a general class of nonlocal nonlinear evolution equations. An application of this theory to a class of nonlinear reaction-diffusion problems that arise in population dynamics is presented. Furthermore, conditions on the initial population density for this class of problems that result in finite time extinction or persistence of the population is discussed. Numerical evidence corroborating our theoretical results is given.

References

A. S. Ackleh and B. G. Fitzpatrick, Estimation of time dependent parameters in general parabolic evolution systems, J. Math. Anal. Appl. 203 (1996), no. 2, 464–480. MR 1410934, DOI 10.1006/jmaa.1996.0391
Azmy S. Ackleh and Simeon Reich, Parameter estimation in nonlinear evolution equations, Numer. Funct. Anal. Optim. 19 (1998), no. 9-10, 933–947. MR 1656403, DOI 10.1080/01630569808816867
H. T. Banks, Simeon Reich, and I. G. Rosen, An approximation theory for the identification of nonlinear distributed parameter systems, SIAM J. Control Optim. 28 (1990), no. 3, 552–569. MR 1047423, DOI 10.1137/0328033
H. T. Banks, Simeon Reich, and I. G. Rosen, Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems, Appl. Math. Optim. 24 (1991), no. 3, 233–256. MR 1123788, DOI 10.1007/BF01447744
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843, DOI 10.1007/978-94-010-1537-0
J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.
M. Chipot and B. Lovat, Existence and Uniqueness Results for a Class of Nonlocal Elliptic and Parabolic Problems, Dynamics of Discrete Continuous and Impulsive Systems, to appear. Manuscript can be obtained from internet: http://www.math.unizh.ch/~chipot/Quenching.ps.
M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94. MR 300166, DOI 10.1007/BF02761448
Ben G. Fitzpatrick, Analysis and approximation for inverse problems in contaminant transport and biodegradation models, Numer. Funct. Anal. Optim. 16 (1995), no. 7-8, 847–866. MR 1355277, DOI 10.1080/01630569508816650
R. A. Freeze and J. Cherry, Groundwater, Prentice-Hall, Englewodd Cliffs, NJ, 1979.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
Kazuaki Taira, Analytic semigroups and semilinear initial-boundary value problems, London Mathematical Society Lecture Note Series, vol. 223, Cambridge University Press, Cambridge, 1995. MR 1369247, DOI 10.1017/CBO9780511662362