Asymptotic behavior of a class of integrodifferential equations with diffusion. (English) Zbl 0848.35015

Murakami, Satoru

Methods Appl. Anal. 2, No. 2, 237-247 (1995).

The paper deals with systems of integrodifferential equations of the form \[ {\partial u_i\over \partial t}(t, x)= k_i \Delta u_i(t, x)+ u_i(t, x)\Biggl\{ a_i- b_i u_i(t, x)- \sum^N_{j= 1} \int^t_0 u_j(t- \tau, x) df_{ij}(\tau)\Biggr\},\tag{S} \] \(t> 0\), \(x\in \Omega\), \(i= 1,\dots, N\), where \(\Omega\) is a bounded region in the space \(\mathbb{R}^m\). Initial value conditions are attached to (S), as well as boundary value conditions of the type of Neumann. Under several assumptions concerning the data, including inequalities involving the constants occurring in (S), one shows that the solutions of (S) tend asymptotically (as \(t\) tends to \(\infty\)) toward the solutions of a system of ordinary differential equations of Volterra-Lotka type. In particular, no restrictions are imposed on the \(k_i> 0\), a feature that is encountered often in the existing literature.

Reviewer: C.Corduneanu (Arlington)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
45K05	Integro-partial differential equations
35K50	Systems of parabolic equations, boundary value problems (MSC2000)
35K55	Nonlinear parabolic equations

Keywords:

integro-differential equation; asymptotic behavior of solutions