Long-time Dynamics Of A Quasi-linear Strongly Damped Membrane Equation

In this paper,we consider the global well-posedness and attractor for the quasi-linear membrane equations with strong damping:utt+?2u+?2ut+??(?u)=g(x).where(?)is a bounded domain with smooth boundary ??.? is a nonlinear term with the growth exponent p.g?L2(?).In space H=H3?W02,p+1×L2,when 1?p<p**:=(N+4)/((N-4)+),we prove the global well posedness of the weak solution of the problem.When N=1,the existence of global attractor and exponential attractor for the solution operator semigroups in phase space H=(H3?H01)×L2.When N?2,we consider the existence of exponential attractor in phase space H1=W02,p+1 × L2.