I'm looking for results on this kind of problems: $$ \partial_{tt}^2 u - \partial_x(a(x) \partial_x) = f,$$ $$u(t=0) = u_0, \quad \partial_t u(t=0) = u_1,$$ where $a$ changes sign: $a(x)= -c^2 < 0$ for $x \in \Omega$ a bounded open set of $\mathbb R^d$ and $a(x) = 1 > 0$ for $x \in \mathbb R^d \setminus \overline \Omega$.
Thus it is the classical wave equation outside $\Omega$ and becomes elliptic (Laplace equation) in $\Omega$.
I couldn't find any result for this kind of problems, maybe I don't know the right key-words...
Any help would be appreciated !
