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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.

MSC:
35R10 Partial functional-differential equations
92D25 Population dynamics (general)
35B35 Stability in context of PDEs
45K05 Integro-partial differential equations
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References:
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