Let \(c,\mu >0\) Consider the wave equation with nonlinear fric- tional dissipation \begin{equation} \tag {1} \begin {cases} u_{tt}-c^2\Delta u+\mu u_t^3=f & \text {in } (0,T]\times \mathbb {R}^d,\\ (u,u_t)=(\phi ,\psi ) & \text {on } \{t=0\}\times \mathbb {R}^d. \end {cases} \end{equation}
Explain the notion of domain of dependence in the context of wave-type PDEs. Show finite speed of propagation for this PDE by showing, using an energy method, a natural upper bound on the domain of dependence of some point \((t_0,x_0)\). Hint: You do not need to concoct a complicated energy to get perfect energy conservation, just use a simple and standard energy where you can prove energy dissipation (i.e. monotone decreasing).