Regularization of the solution of the Cauchy problem: the quasi-reversibility method

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.