Boundary value and mixed problems for homogeneous hyperbolic equations in a complete scale of spaces of Sobolev type (Q912289)

In the region \(R^ 3_+=\{(t,x,y)|\) \(t,x,y>0\}\) is considered the following initial boundary value problem: \((1)\quad u_{tt}-u_{xx}- u_{yy}-p_ 0(t,x,y)u_ t-p_ 1(t,x,y)u_ x-p_ 2(t,x,y)u_ y-p_ 3(t,x,y)u=f(t,x,y),(2)\quad u_ t-a(t,y)u_ x-b(t,y)u_ y+c(t,y)u=f_ 2(t,y),\quad x=0,\quad (t,y)\in R^ 2_+,(3)\quad u_ t-\alpha (t,x)u_ y-\beta (t,x)u_ x+\gamma (t,x)u=f_ 2(t,x),\quad y=0,\quad (t,x)\in R^ 2_+,(4)\quad u=\phi (x,y),\quad u_ t=\psi (x,y),\quad t=0,\quad (x,y)\in R^ 2_+.\)It is obtained an a-priori estimate for the mixed problem (1)-(4) by the help of which a theorem on existence and uniqueness can be proved.