A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x 0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).