Decay of solutions of a degenerate hyperbolic equation

This article studies the asymptotic behavior of solutions to the damped, non-linear wave equation $$ ddot u +gamma dot u -m(|abla u|^2)Delta u = f(x,t),, $$ which is known as degenerate if the greatest lower bound for $m$ is zero, and non-degenerate if the greatest lower bound is positive. For the non-degenerate case, it is already known that solutions decay exponentially, but for the degenerate case exponential decay has remained an open question. In an attempt to answer this question, we show that in general solutions can not decay with exponential order, but that $|dot u|$ is square integrable on $[0, infty)$. We extend our results to systems and to related equations