# Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $$f$$ is discontinuous and in $$L^\infty$$?

If $$\Omega$$ is bounded, assume Neumann conditions at the boundary. I'd be interested even in only the $$1$$-dimensional case where $$\Omega = [0,1]$$: $$\begin{cases} \partial^2_{tt} u - \partial_{xx}^2 u = f(u), & t>0, \quad x \in [0,1],\\ u(0,x) = u_0(x), & x \in [0,1], \\ \partial_t u(0,x) = u_1(x), & x \in [0,1], \\ \partial_x u(t,0) = \partial_x u(t,1) = 0, \qquad & t>0. \end{cases}$$