GLOBAL SOLUTIONS AND DECAY PROPERTY WITH REGULARITY-LOSS FOR QUASI-LINEAR HYPERBOLIC SYSTEMS WITH DISSIPATION

Abstract

We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. The decay property for this system exhibits a loss of regularity, and we establish a global existence theory together with sharp decay estimates, under a smallness condition and under sufficient regularity on the initial data. We combine together various tools of analysis, especially energy estimates in Fourier space, the time-weighted energy method, and the semi-group method.