--- abstract: 'In this paper, we construct the global attractor of the problem with periodic boundary conditions and positive damping coefficient on each boundary of the domain. The main result obtained is the existence of the global attractor, and its topological structure. We also prove a weak global attractor theorem, and show that the global attractor of the problem (\[3.1\]) is the $\omega$-limit set of the finite dimensional attractor.' address: - 'Department of Applied Mathematics, Kyungpook National University, Daegu, South Korea.' - 'Department of Applied Mathematics, Kyungpook National University, Daegu, South Korea.' - 'Department of Mathematical Sciences, Seoul National University, Seoul, South Korea.' - 'Department of Mathematical Sciences, Seoul National University, Seoul, South Korea.' author: - 'M. A. Safonov' - 'S.-O. Londen' - 'K. S. Zahorjan' - 'G.-J. Sun' title: On the existence of the global attractor for a reaction diffusion equation with convective term --- Nonlinear evolution equation, reaction-diffusion equation, domain with one and two sides, existence of the attractor, global attractor, weak global attractor. 35B41, 35K59, 35K55. Introduction ============ In this paper, we study the non-autonomous parabolic problem on the infinite cylinder, with periodic boundary conditions, on the interior of which the equation is dissipative. We are interested in the existence of a global attractor for this problem, and the regularity and structure of its fractal dimension. Our motivation comes from the work [@MNR], which deals with the case when the external force is a constant function (i.e., the case of null space in $L_2$). It was proved that a bounded global attractor exists for the model problem with positive damping coefficient, even when a constant external force is present. A detailed analysis of this case is presented in [@MNR]. A similar situation arises in practice, for example, when a solid body is located at a certain point of the channel through which a fluid passes. This situation is modelled by the parabolic equation with nonconstant coefficient in the Navier-Stokes system for incompressible viscous fluid (see [@LR]), the nonlinear Schrödinger equation with nonconstant coefficient in plasma physics (see [@FJ]), and the Cahn-Hilliard equation for description of the evolution of the orientation field of a non-oriented material (see [@FH]). Another related equation arises in the magnetohydrodynamics, the nonlinear MHD equations. The evolution problem with dissipation leads to the damped wave equation for magnetohydrodynamics (see [@YOKO; @YU]). In the case of the Cahn-Hilliard equation with bounded right-hand side in [@SZh1; @SZh], a set-valued global attractor, and a finite dimensional global attractor (i.e., a compact attractor with finite fractal dimension) were constructed. Moreover, the structure of the fractal dimension of the global attractor was described in detail. We remark that the case of problem with bounded right-hand side is different from the case of problem with compactly supported right-hand side studied in [@FJK]. In this paper, we consider a non-compact domain and a non-negative and integrable right-hand side of this problem (see the details below). To describe the model problem we consider in this paper, we set $M=\{0,\pm1,\ldots,\pm m\}$ and consider the initial-boundary value problem (BVP) for reaction-diffusion equations in an infinite cylinder $G\times M$, $$\label{1.1} \begin{cases} \frac{\partial u}{\partial t}=d_1\Delta u + f(t,x,u), \\ u\left(\cdot,0\right)=u^0(\cdot), \\ u\left(\cdot,\pm m\right)=u\left(\cdot,\mp1\right), \\ u \text{ satisfies the periodic boundary conditions}, \\ \end{cases}$$ where $d_1>0$, $d_2>0$ are the parameters of the model, and the right-hand side $f=f(t,x,u)$ contains the time-varying parameter $t$. We assume that $f$ is continuous and $t$-periodic in $t$, $x$ and $u$. We also consider the boundary conditions $u(t,\pm1,t)=u(t,\pm m,t)$. The main purpose of this paper is to show the existence of the global attractor for the BVP , and to determine its topological structure, using methods of $C_0$-semigroups and theory of attractors. In the theory of attractors, we use a new method of approximation, namely, taking the limit when $n\to \infty$ of the sequence of attractors of the problem $$\label{1.2} \begin{cases} \frac{\partial u}{\partial t}=\Delta u - d_1 \Delta u_t + f(t,x,u), \\ u(t,0)=u^0(t), \\ u \text{ satisfies the periodic boundary conditions}, \\ \end{cases}$$ where $d_1>0$. We note that similar BVP, but for another equation, namely, for an initial boundary value problem for the hyperbolic equation, were studied in [@MNR; @Zh_hyperb; @Zh]. Moreover, this equation is an example of a model for the motion of a solid body at a certain point of the channel, where the fluid flows. It was noted in [@MNR; @Zh_hyperb; @Zh] that for all right-hand sides $f(t,x,u)$ such that $|f(t,x,u)|\leq K(|u|^2+1)$, where $K$ is a positive constant, for which the solution of BVP for the hyperbolic equation exists, has a global attractor. However, this solution can not be obtained from the solution of the damped wave equation for magnetohydrodynamics. The equation is also similar to equation considered in [@JG; @MNR1; @MNR2], where the global attractor is established for the case when the boundary conditions for the equation in infinite cylinder are not necessarily periodic. In this case, the solutions of the initial boundary value problems for the equation in a bounded domain of ${\bf R}^N$ are also taken into account. In this paper, we consider the global attractor of the equation as a global attractor for the non-autonomous initial-boundary value problem and analyze its topological structure using techniques of semigroups and attractors. We note that for system with $L_2$-valued right-hand side, for which the solutions of equation are considered, existence of global attractors is not studied yet. We remark that if we take into account the conditions of , it is not possible to construct a finite dimensional global attractor. In this case, the condition means that the solution of the non-autonomous problem exists, and it is necessary to prove that it is differentiable for all $t>0$ at each point $x$, where $u(t,x,u^0)$ is a solution of the BVP , when $u^0\in L^2(G)$. The condition of the non-autonomous problem is sufficient for the existence of the solution. Indeed, as a rule, the non-autonomous problems are considered when the solution $u(t,x,u^0)$ of the problem is not unique. Statement of the problem ======================== In what follows, we consider a general infinite cylinder $G\times M$, where $M=\{0,\pm1,\ldots,\pm m\}$, and $G$ is a closed bounded subset of ${\bf R}^N$, $N\geq1$, and $m\geq2$. Let $P=\{(x,t):\, x\in G,\, 0