zbMATH — the first resource for mathematics

A prior bounds and stability of solutions for a Volterra reaction-diffusion equation with infinite delay. (English) Zbl 0981.35095
A simple model of evolution of a single species population has been considered by V. Volterra [Leçons sur la théorie mathématique de la lutte pour la vie (1931; Zbl 0002.04202)] \[ dx(t)/dt=x(t) \Biggl(b-cx(t)-\int_{0}^{\infty }k(\tau)x(t-\tau) d\tau \Biggr) \quad \text{for }t\in \mathbb{R}^+, \] where \(x(t)\) is the population size, \(b\) and \(c\) are positive constants. The integral part means a hereditary term concerning the effect of the past history on the present growth rate. There are two results of R. K. Miller [SIAM J. Appl. Math. 14, 446-452 (1966; Zbl 0161.31901)] and R. Redlinger [SIAM J. Math. Anal. 16, 135-142 (1985; Zbl 0593.92014)] for the existence of positive solutions and their asymptotic behavior as \(t\to\infty \). Recently, Gopalsamy and He proved that there exists a solution \(x(t)\) that satisfies \(\varliminf_{t\to\infty }x(t)\geq m\) (\(m\) is a certain constant) and that any two solutions \(x_1(t)\) and \(x_2(t)\) satisfy \(\lim_{t\to\infty }(x_1(t)-x_2(t))=0\).
The authors of this paper extend the above results to the case of Volterra reaction-diffusion equations with variable coefficients, i.e., \(b\) and \(c\) are replaced by \(b(t,x)\) and \(C(t,x)\), respectively, and a factor \(d(t,x)\) before the integral term is introduced. The question for stability characteristics of the solutions is discussed as well.

35R10 Partial functional-differential equations
92D25 Population dynamics (general)
35B35 Stability in context of PDEs
45K05 Integro-partial differential equations
Full Text: DOI
[1] R.P. Agarwal, Difference Equations and Inequalities, TMA, Marcel Dekker, New York, 1992.
[2] Gopalsamy, K.; He, X.Z., Dynamics of an almost periodic logistic integrodifferential equation, Methods appl. anal., 2, 1, 38-66, (1995) · Zbl 0835.45004
[3] Gopalsamy, K.; Weng, P.X., On the stability of a neutral integro-partial differential equation, Bull. inst. math. acad. sinica, 20, 3, 267-284, (1992) · Zbl 0759.45007
[4] Henriquez, H.R., Regularity of solutions of abstract retarded functional differential equations with unbounded delay, Nonlinear anal., 28, 513-531, (1997) · Zbl 0864.35112
[5] W.G. Kelly, A.C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991.
[6] Miller, R.K., On Volterra’s population equation, SIAM J. appl. math., 14, 446-452, (1966) · Zbl 0161.31901
[7] Murakami, S., Asymptotic behavior in a class of integrodifferential equations with diffusion, Methods appl. anal., 2, 2, 237-247, (1995) · Zbl 0848.35015
[8] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1984. · Zbl 0549.35002
[9] Redlinger, R., Existence theorems for semilinear parabolic systems with functionals, Nonlinear anal., 8, 6, 667-682, (1984) · Zbl 0543.35052
[10] Redlinger, R., On Volterra’s population equation with diffusion, SIAM J. math. anal., 16, 1, 135-142, (1985) · Zbl 0593.92014
[11] Ruan, S.G.; Wu, J., Reaction-diffusion equations with infinite delay, Canad. appl. math. quart., 2, 485-550, (1994) · Zbl 0836.35158
[12] B. Shi, A Prior bounds of solutions of nonlinear Volterra integrodifferential systems with diffusion and infinite delay, Acta Math. Appl. Sinica, to appear.
[13] B. Shi, Existence and uniqueness of positive solutions and bounded positive solutions of systems of Volterra reaction-diffusion equations with infinite delay, Acta Math. Sinica, to appear. · Zbl 1125.35363
[14] V. Volterra, Lecons sur la théorie mathematique de la lutte pour la vie, Gauthier-Villars, Paris, 1931. · Zbl 0002.04202
[15] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. · Zbl 0870.35116
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.