# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] 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) [2] Henriquez, H.R., Regularity of solutions of abstract retarded functional differential equations with unbounded delay, Nonlinear analysis, 28, 513-531, (1997) · Zbl 0864.35112 [3] Hess, P., () [4] 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 [5] Pao, C.V., Coupled nonlinear parabolic systems with time delays, Journal of mathematical analysis and applications, 196, 237-265, (1995) · Zbl 0854.35122 [6] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York · Zbl 0780.35044 [7] Ruan, S.G.; Wu, J.H., Reaction – diffusion equations with infinite delay, The Canadian applied mathematics quarterly, 2, 485-550, (1994) · Zbl 0836.35158 [8] Redlinger, R., Existence theorems for semi-linear parabolic systems with functionals, Nonlinear analysis, 8, 6, 667-682, (1984) · Zbl 0543.35052 [9] 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 [10] Smoller, J., Shock waves and reaction – diffusion equations, (1983), Springer · Zbl 0508.35002 [11] Volterra, V., Lecons sur la theorie mathematique de la lutte pour la vie, (1931), Gauthier-Villars Paris · JFM 57.0466.02 [12] Wu, J.H., Theory and applications of partial functional differential equations, (1996), Springer New York [13] 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) [14] 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 [15] 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
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.