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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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