First boundary value problem for a direct and inverse ultraparabolic equation (Q1814382)

The equation (1) \(D_ \lambda u=u_{xx}\hbox{sgn} x\), where \(D_ \lambda=\sum^ n_{i=1}\lambda_ i(t)\), \(t=(t_ 1,t_ 2,\ldots,t_ n)\in E_ n\), is studied; the functions \(\lambda_ i(t):E_ n\to\mathbb{R}\) are of class \(C^ 1\), bounded, so that \(\sum^ n_{i=1}\lambda^ 2_ i\geq \alpha>0\). The equation (1) is considered in a domain \(Q_ T=G_ T\times(| x|<\infty)\) in which the set \(G_ T\subset E_ n\) is defined by the intermediate of the solution \(\tau(z,z_ 0,t)=(\tau_ 1(z,z_ 0,t),\ldots,\tau_ n(z,z_ 0,t))\) of the Cauchy problem \(d\lambda_ i/dz=\lambda_ i(\tau)\), \(\tau_ i(z_ 0,z_ 0,t)=t_ i\), \(i=1,\ldots,n\). The equation (1), with some initial conditions, is transformed into an Abel integral equation. For this integral equation some necessary and sufficient conditions for the existence of a unique solution in a Sobolev space \(H^{(q)}(Q_ T)\) are proved. An evaluation of the norm of the solution is also given.