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Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems. (English) Zbl 1128.37055

The author studies the following lattice dynamical system of a partly dissipative reaction-diffusion system: \[ \begin{cases} \dot u_i+\nu(Au)_i+f_1 (u_i)+g_1(u_i)+\alpha h_1(u_i,v_i)=q_{1,i}\\ \dot v_i+f_2(v_i)+g_2 (v_i)-\beta h_2 (u_i,v_i)=q_{2,i},\end{cases}\tag{1} \] \(i=(i_1,\dots,i_r)\in \mathbb{Z}^r\), \(t>0\), with the initial conditions \[ u_i(0)=u_{i,0},\quad v_i (0)=v_{i,0},\tag{2} \] where \(\nu\) is a positive constant, \(\alpha,\beta\in \mathbb{R}\), \(\alpha\beta>0\), the operator \(A:\ell^2\to\ell^2\) is defined by, for all \(u=(u_i)_{i\in\mathbb{Z}}\in \ell^2\), \(i=(i_1,\dots,i_r),\) \((Au)_i=2 ru_{(i_1,\dots,i_r)}-u_{(i_1-1,i_2,\dots, i_2)}-u_{(i_1,i_2-1,\dots,i_r)}\dots -u_{(i_1,\dots,i_{r-1})}-u_{(i_1,\dots,i_{r-1})}-u_{(i_1+1,i_2,\dots, c_r)}-\dots u_{(i_1, \dots,i_r+1)}\) and for \(j=1,2,s,s_1,s_2\in\mathbb{R}\) \[ q_j= (q_{j,i})_{i \in\mathbb{Z}^r}\in\ell^2,\;f_j,g_j\in C^1(\mathbb{R},\mathbb{R}),h_j(s_1,s_2)\in C^1 (\mathbb{R}^2,\mathbb{R}) \] The author studies the existence of a global attractor and its upper semicontinuity for (1)–(2).

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
35K57 Reaction-diffusion equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
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