Long Time Behavior Of A Class Of Nonlinear Kirchhoff-type Coupled Equations

This paper investigate the global well-posedness and the longtime dynamics of solutions for a class of Kirchhoff-type coupled equations with nonlinear:utt-M(??u?2 +??v?2)?u-??ut+g1(u,v)=f1(x),(1.1)vtt-M(??u?2 +??v?2)?u-??vt+g2(u,v)=f2(x),(1.2)u(x,O)=uo(x),ut(jx,O)=u1(x),x ? ?,(1.3)v(x,0)=vo(x),vt(x,O)=v1(x),x??,(1.4)u?? = v?? =O,(1.5)where ? is a bounded domain in R2 with the smooth boundary ?,?>Ois a constant.M(s)? C1 and M(s)is a nonnegative function,-?ut,and-?uvt,are strongly damping,g1(u,v)andg2(u,v)are nonlinear source term,f1(x)are f2(x)given forcing function.The existence and uniqueness of the global solution of the above equation are proved by prior estimation and Galerkin method.Then the solution semigroup of the equation is obtained,and the bounded absorption set and compact global attractor of the understanding semigroup are constructed.Secondly,the Hausdorff dimension of the global attractor are estimated.Finally,the existence of inertia universality of the above equation is discussed by using the graph norm method.