The attractor of a parabolic–elliptic system Original Research Article

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 Turn MathJax on and −Pxx+P=h(t,x,u,ux)+k(t,x,u)−Pxx+P=h(t,x,u,ux)+k(t,x,u) Turn MathJax on with initial data u|t=0=u0.u|t=0=u0. Turn MathJax on 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.