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Asymptotic periodicity of the Volterra equation with infinite delay. (English) Zbl 1133.35004
Asymptotic periodic properties are obtained for special Volterra delay differential equations. As an example the following equation is considered
\begin{alignedat}{2} \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2} &= u\left[ a-bu-c\int^{\infty}_{0}u(t-\tau,x)\,d\mu(\tau)\right], &\quad &(t,x)\in (0,\infty)\times (0,1),\\ u(t,0)&= u(t,1)=0, &\quad &t\in (0,\infty),\\ u(t,x)&=\varphi(t,x), &\quad &(t,x)\in (-\infty,0]\times[0,1]. \end{alignedat}

##### MSC:
 35B10 Periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 35R10 Partial functional-differential equations 45K05 Integro-partial differential equations
##### Keywords:
Volterra equation; asymptotic periodicity, delay
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##### References:
  Gopalsamy, K.; He, X.Z., Dynamics of an almost periodic logistic integro-differential equation, Bulletin of the institute of mathematics. academia sinica, 20, 3, 267-284, (1992)  Henriquez, H.R., Regularity of solutions of abstract retarded functional differential equations with unbounded delay, Nonlinear analysis, 28, 513-531, (1997) · Zbl 0864.35112  Hess, P., ()  Lu, X., Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonlinear analysis, 27, 6, 699-709, (1996) · Zbl 0862.35134  Pao, C.V., Coupled nonlinear parabolic systems with time delays, Journal of mathematical analysis and applications, 196, 237-265, (1995) · Zbl 0854.35122  Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044  Ruan, S.G.; Wu, J.H., Reaction – diffusion equations with infinite delay, The Canadian applied mathematics quarterly, 2, 485-550, (1994) · Zbl 0836.35158  Redlinger, R., Existence theorems for semi-linear parabolic systems with functionals, Nonlinear analysis, 8, 6, 667-682, (1984) · Zbl 0543.35052  Shi, B.; Chen, Y., A prior bounds and stability of solutions for a Volterra reaction – diffusion equation with infinite delay, Nonlinear analysis, 44, 97-121, (2001) · Zbl 0981.35095  Smoller, J., Shock waves and reaction – diffusion equations, (1983), Springer · Zbl 0508.35002  Volterra, V., Lecons sur la theorie mathematique de la lutte pour la vie, (1931), Gauthier-Villars Paris · JFM 57.0466.02  Wu, J.H., Theory and applications of partial functional differential equations, (1996), Springer New York  Wang, J.L.; Zhou, L., Existence and uniqueness of periodic solution of delayed logistic equation and its asymptotic behavior, Journal of partial differential equations, 16, 4, 1-13, (2003)  Wang, J.L.; Zhou, L.; Tang, Y.B., Asymptotic periodicity of a food-limited diffusive population model with time-delay, Journal of mathematical analysis and applications, 313, 2, 381-399, (2006) · Zbl 1096.35123  Zhou, L.; Fu, Y.P., Existence and stability of periodic quasi-solutions in nonlinear parabolic systems with discrete delays, Journal of mathematical analysis and applications, 250, 139-161, (2000) · Zbl 0970.35004
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