ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR HYPERBOLIC EQUATIONS WITH TIME-DEPENDENT MEMORY KERNELS

This paper gives a detailed study of long-time dynamics generated by a wave equation with time-dependent speed of propagation epsilon(t) and time-dependent memory kernel h(t)(.). Within the recent theory of process on time-dependent spaces, we first establish the global well-posedness result in extended time-dependent memory space H-t, then we develop further some new time-space (energy) estimates to overcome the mixed difficulties from epsilon(t) and h(t)(.), and obtain the dissipativity and regularity of the dynamical process. Furthermore, we study the existence and regularity of the time-dependent global attractor. And, when epsilon(t) -> 0 and the time-dependent rescaled kernel approaches a multiple of the Dirac measure as t -> infinity, show that the asymptotic structure of time-dependent attractor converges to the attractor of a nonclassical diffusion equation.