On a forward-backward parabolic equation

Boundary value problems for the equation $$operatorname{sgn} (x) cdot u_y - u_{xx} + ku = f(x,y)$$ (where k is a positive constant and ? is a given function) are investigated. The domain of the solutions will be the whole upper half-plane y>0, or the half-plane y>0 cut along the positive y-axis. We are interested in square integrable solutions u, with square integrable generalized derivatives u_y and u_xx. Existence theorems are proved, with an integral equations technique. Thus a theory is developed of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces.