TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON RN

We study the longtime dynamics of non-autonomous wave equations with strong damping in the case of critical nonlinearity. First of all, when 1 <= p <= p(*)=N+2/(N-2)+, we get the well-posedness of strong damped equation with dime-dependent decay coefficient in H-t=H-1(R-N)xL(2)(R-N), and prove the quasi-stability of weak solution in H-t,H--1=H-1(R-N)xH(-1)(R-N). Then the time-dependent attractor is proved in Ht. Finally, by using the quasi-stability of weak solution, we establish the existence the pullback exponential attractor for non-autonomous dynamical system (U(t,tau),H-t,H-t,H--1).