ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 1, pp. 72–80. c  Pleiades Publishing, Ltd., 2012. Original Russian Text c  M.M. Aptsiauri, T.A. Jangveladze, Z.V. Kiguradze, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 1, pp. 70–78. INTEGRAL EQUATIONS Asymptotic Behavior of the Solution of a System of Nonlinear Integro-Differential Equations M. M. Aptsiauri, T. A. Jangveladze, and Z. V. Kiguradze Ilia State University, Tbilisi, Georgia I. Vekua Institute of Applied Mathematics, Tbilisi, Georgia I. Javakhishvili Tbilisi State University, Tbilisi, Georgia Received November 13, 2009 Abstract— We study the asymptotic behavior as time tends to infinity of the solution of an initial–boundary value problem for a system of nonlinear integro-differential equations that arises in the mathematical modeling of penetration of electromagnetic field into a medium whose electric conductivity substantially depends on temperature. Both homogeneous and in- homogeneous boundary conditions are considered. The exponential stabilization of the solution is established. DOI: 10.1134/S0012266112010089 1. INTRODUCTION Integro-differential equations arising in mathematics and physics often contain derivatives with respect to several variables; therefore, these equations are referred to as partial integro-differential equations. Numerous publications in the 20th century deal with the study of integro-differential equations of various kinds (see the bibliography in [1–4]). Numerous scientific papers deal with the investigation of nonlinear integro-differential equations of the parabolic type. Such an integro-differential model appears, for example, in the mathematical modeling of penetration of electromagnetic field into a medium whose electric conductivity sub- stantially depends on temperature. On the basis of the Maxwell system of differential equations [5], a general statement of the above-mentioned diffusion problem was given in [6], where the prob- lem was also reduced to an integro-differential model, the corresponding initial–boundary value problems were posed, and the uniqueness and existence of their global solutions were considered. The above-mentioned integro-differential model has the form [6] ∂W ∂t + rot  a  t  0 | rot W | 2 dτ  rot W  =0, div W =0. (1.1) The model (1.1) is complicated and can be studied only in special cases. An analysis of the existence and uniqueness of a solution of initial–boundary value problems for equations and systems of the form (1.1) was carried out in [6–13]. In particular, an existence theorem was proved in [6] for a generalized solution of the first boundary value problem for the equation corresponding to system (1.1) in the spatially one-dimensional case with a one-component magnetic field and with a(S )=1+ S , and uniqueness was proved there for more general cases. A similar case with a(S )= (1 + S ) p ,0 <p ≤ 1, was considered in [7]. Existence theorems for a solution of the first boundary value problem were proved in [6–11] with the use of the compactness method [2, pp. 118–132; 14]. An analysis of spatially multidimensional cases was carried out for the first time in [8–10] and then in [11–13, 15] and other papers. 72