Propriety of the mixed problem for a degenerate hyperbolic equation with arbitrary type of degeneracy (Q1103107)

There is proved the well-posedness of the problem \[ \phi (t)u_{tt}- \sum_{i,j}(a_{ij}u_{x_ i})_{x_ j}+au_ t+bu=f\quad in\quad D\times [0,T],\quad u|_{t=0}=u|_{\partial D\times [0,T]}=0, \] where \(a_{ij}=a_{ji}\in C^ 2\), \(a,b\in C^ 1\), \((a_{ij})\) is uniformly positive matrix, \(\partial D\in C^ 2\), \(\phi(0)=0\), \(\phi '(t)>0\) for \(t>0\), lim \(t^{\alpha} \phi'(t)/\phi (t)=+\infty\forall \alpha \in [0,1)\), \[ \iint_{D\times [0,T]}(f^ 2+f^ 2_ t)\phi^{-1} \exp \phi^{-1} dx dt<\infty. \]