I am currently investigating some finite speed propagation property for nonlinear wave equations and I am asking myself if there is a way of proving such a property just using energy estimates like in the linear wave equation where one localized the energy to prove such a property.

In L.C Evans's partial differential book, he treated some semilinear case by this method, but for a fully linear setting, for example for the equation : $\partial ^2 _t u = \Delta (u^3)$ id does not seem to work the same (I asked the question here for this example but I am interested in any equation where finite speed propagation can be shown by an energy method).

If someone has seen some article dealing with finite speed propagation and proving it by energy methods, please feel free to give a reference. I personally did'nt find anything interesting.

Thank you.

I am currently investigating some finite speed propagation property for nonlinear wave equations, and I am asking myself if there is a way of proving such a property just using energy estimates like in the linear wave equation, where one localized the energy to prove such a property.

In L.C Evans's partial differential book, he treated some semilinear cases by this method, but for a fully linear setting, for example for the equation : $\partial ^2 _t u = \Delta (u^3)$ it does not seem to work the same (I asked the question here for this example but I am interested in any equation where finite speed propagation can be shown by an energy method).

Are there any articles dealing with finite speed propagation and proving it by energy methods?

Thank you.