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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 $\Omega = [0,1]$: $$ \begin{cases} \partial^2_{tt} u - \partial_{xx}^2 u = f(u), & t>0, \quad x \in [0,1],\\ u(0,x) = u_0(x), & x \in [0,1], \\ \partial_t u(0,x) = u_1(x), & x \in [0,1], \\ \partial_x u(t,0) = \partial_x u(t,1) = 0, \qquad & t>0. \end{cases} $$

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