A mixed type boundary problem describing the propagation of disturbances in viscous media (Q1174561)

The following mixed problem is studied: \[ u_{tt}=Au_ t+b(x,t,u_ x,u_{xx},u_{xt},u_ t), \] \[ u(x,0)=u_ 0(x), u_ t(x)=u_ 1(x), x\in Q; \;u(x,t)=\psi (x,t) \text{ on } S_ T=\partial\Omega\times[0,T], \] where \(Q_ T=\Omega\times[0,T]\), \(\Omega\) bounded domain in \(R^ n\), \(\partial\Omega\in H_{2+\alpha}\), \(0<\alpha<1\), \[ A=\sum^ n_ ia_{ij}(x,t)u_{x_ ix_ j}(x,t). \] Existence, uniqueness, continuous dependence of the solutions are proved under suitable hypotheses.