A problem with data on a characteristic for a linearized equation of transonic gas dynamics (Q1357992)

The equation \(u_{xt}+ k(x,y,t)u_{xx}- \Delta_y u+a(x,y,t)u_x= f(x,y,t)\) is studied in the domain \(x\in \mathbb{R}\), \(y\in \mathbb{R}^p\), \(t\in (0,T)\), with the initial condition \(u(x,y,0)=u_0 (x,y)\). The unique solvability is proved in Sobolev spaces with the weight \(\exp(-\alpha x)\), \(\alpha >0\).