The attractor of a parabolic–elliptic system

Following Coclite, Holden and Karlsen [G.M. Coclite, H. Holden and K.H. Karlsen, Well-posedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst. 13 (3) (2005) 659–682] and Tian and Fan [Lixin Tian, Jinling Fan, The attractor on viscosity Degasperis-Procesi equation, Nonlinear Analysis: Real World Applications, 2007], we study the dynamical behaviors of the parabolic–elliptic system ut+(f(t,x,u))x+g(t,x,u)+Px−εuxx=0ut+(f(t,x,u))x+g(t,x,u)+Px−εuxx=0 and −Pxx+P=h(t,x,u,ux)+k(t,x,u)−Pxx+P=h(t,x,u,ux)+k(t,x,u) with initial data u|t=0=u0.u|t=0=u0. The existence of global solution to the parabolic–elliptic system in L2L2 under the periodic boundary condition is discussed. We also establish the existence of the global attractor of semi-group to solutions on the parabolic–elliptic system in H2H2.