Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping

Abstract

This paper deals with a class of quasilinear wave equations with structural damping or strong damping. By virtue of the improved Faedo–Galerkin method and some technical efforts, we first establish the local well-posedness of weak solutions, especially the continuity of weak solutions with respect to time in the natural phase space. Then we investigate the global existence, asymptotic behavior and finite time blow-up of weak solutions with subcritical or critical initial energy. As for the supercritical initial energy case, we show that the weak solutions may blow up in finite time with arbitrarily high initial energy. Last but not least, the upper and lower bounds of the blow-up time for blow-up solutions are derived.