×

Dirichlet problem for the diffusive Nicholson’s blowflies equation. (English) Zbl 0923.35195

The equation considered in this paper is the diffusive Nicholson’s blowflies equation \[ \partial u(t,x)/\partial t= d\Delta u(t,x)- 2\tau u(t,x)+ \beta\tau u(t- 1,x) e^{-u(t-1,x)},\tag{1} \] where \((t,x)\in D\equiv (0,\infty)\times\Omega\), \(\Omega\subseteq \mathbb{R}^n\) \((n\geq 1)\) being a bounded domain with smooth boundary \(\partial\Omega\); \(\beta\), \(\tau\), \(d\) are positive constants.
The aim of this paper is to study the asymptotic behavior of solutions of the equation (1) under Dirichlet condition \[ u\equiv 0\quad\text{on }\Gamma= (0,\infty)\times \partial\Omega\tag{2} \] and initial condition \[ u(\theta,x)= u_0(\theta, x)\quad\text{in }D_1\equiv [-1,0]\times\overline\Omega.\tag{3} \] A new approach is developed to study the global attractivity of the positive solution of the problem \[ d\Delta\phi(x)- \tau\phi(x)+ \beta\tau\phi(x) \exp(-\phi(x))= 0\quad\text{for }x\in\Omega,\quad \phi(x)= 0\quad\text{for }x\in\partial\Omega.\tag{4} \] A main result is the following: If \(e\leq \beta\leq e^2\) and \(u(t,x)\) is a nontrivial and nonnegative solution of (1)–(2) and \(\phi(x)\) is the positive solution of (4), then \[ \lim_{t\to\infty} \| u(t,.)- (.)\|_{C(\overline\Omega)}= 0\quad\text{and} \quad \lim_{t\to\infty} \| u(t,.)- \phi(.)\|_{L^2(\Omega)}= 0. \]

MSC:

35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Periodic solutions of semilinear parabolic equations, Nonlinear Analysis: A collection of papers in honor of E. H. Rothe (1978), Academic Press: Academic Press New York, p. 1-29 · Zbl 0464.35050
[2] K. L. Cooke, W. Huang, Dynamics and global stability for a class of population models with delay and diffusion effects, 1992; K. L. Cooke, W. Huang, Dynamics and global stability for a class of population models with delay and diffusion effects, 1992
[3] Engler, H., Functional differential equations in Banach space: Growth and decay of solutions, J. Reine Angew. Math., 322, 53-73 (1981) · Zbl 0436.34058
[4] Fitzgibbon, W. E., Semilinear functional differential equations in Banach space, J. Differential Equations, 29, 1-14 (1978) · Zbl 0392.34041
[5] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs · Zbl 0144.34903
[6] Friedman, A., Partial Differential Equations (1976), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York
[7] Friesecke, G., Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Diff. Eqns., 5, 89-103 (1993) · Zbl 0798.35150
[8] Green, D.; Stech, H. W., Diffusion and Hereditary effects in a class of population models, (Busenberg, S. N.; Cooke, K. L., Differential equation and applications in ecology, epidemics and population problems (1981), Academic Press: Academic Press San Diego), 19-28 · Zbl 0526.92012
[9] Gurney, W. S.C.; Blythe, S. P.; Nisbet, R. M., Nicholson’s blowflies revisited, Nature, 287, 17-21 (1980)
[10] Heard, M. L.; Rankin, S. M., A semilinear parabolic Volterra integrodifferential equations, J. Differential Equations, 71, 201-233 (1988) · Zbl 0642.45006
[11] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., 840 (1981), Springer-Verlag: Springer-Verlag Berlin/Heidelberg · Zbl 0456.35001
[12] Hess, P., On uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z., 154, 17-18 (1977) · Zbl 0352.35046
[13] Huang, W., On asymptotic stability for linear delay equations, Diff. Integ. Eqns., 4, 1303-1310 (1991) · Zbl 0737.34054
[14] Inoue, A.; Miyakawa, T.; Yoshida, K., Some properties of solutions for semilinear heat equations with time lag, J. Diff. Eqns., 24, 383-396 (1977) · Zbl 0314.35051
[15] Karakostas, G.; Philos, Ch. G.; Sficas, Y. G., Stable steady state of some population models, J. Dynam. Diff. Eqns., 4, 161-190 (1992) · Zbl 0744.34071
[16] Kuang, Y., Global Attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology, Japan J. Indust. Appl. Math., 9, 205-238 (1992) · Zbl 0758.34065
[17] Kulenovic, M. R.S.; Ladas, G., Linearized oscillations in population dynamics, Bull. Math. Bio., 49, 615-627 (1987) · Zbl 0634.92013
[18] Martin, R. H.; Smith, H. L., Abstract functional differential equations and reaction- diffusion systems, Trans. Amer. Math. Soc., 321, 1-44 (1990) · Zbl 0722.35046
[19] Nicholson, A. J., An outline of the dynamics of animal populations, Aust. J. Zool., 2, 9-65 (1954)
[20] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[21] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0153.13602
[22] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, 979-1000 (1972) · Zbl 0223.35038
[23] Schiaffino, A.; Tesei, A., Monotone methods and attracting results for Volterra integro-partial differential equations, Proc. Royal Soc. Edingburgh, 89A, 135-142 (1981) · Zbl 0474.45004
[24] Smith, H. L., Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems (1995), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0821.34003
[25] So, J. W.-H.; Yu, J. S., Global attractivity and uniform persistence in Nicholson’s blowflies, Diff. Eqns. Dynam. Syst., 2, 11-18 (1994) · Zbl 0869.34056
[26] Travis, C. C.; Webb, G. F., Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200, 395-418 (1974) · Zbl 0299.35085
[27] Travis, C. C.; Webb, G. F., Existence, stability, and compactness in the α-norm for partial functional differential equations, Trans. Amer. Math. Soc., 240, 129-143 (1978) · Zbl 0414.34080
[28] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0870.35116
[29] Yamada, Y., Asymptotic behavior of solutions for semilinear Volterra diffusion equations, Nonl. Anal. TMA, 21, 227-239 (1993) · Zbl 0806.35096
[30] Y. Yang, J. W.-H. So, Dynamics for the diffusive Nicholson blowflies equation, Proceedings of the International Conference on Dynamical Systems and Differential Equations, held in Springfield, Missouri, U.S.A. May 29-June 1, 1996; Y. Yang, J. W.-H. So, Dynamics for the diffusive Nicholson blowflies equation, Proceedings of the International Conference on Dynamical Systems and Differential Equations, held in Springfield, Missouri, U.S.A. May 29-June 1, 1996
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.