Regularization of a non-characteristic Cauchy problem for a parabolic equation in multiple dimensions

In this paper we consider the non-characteristic Cauchy problem where with appropriate coefficient functions a, b and c. Assuming that the Cauchy data are given inexactly by a function satisfying ||-||Hr for some r0 and that f(y,t): = u(l,y,t) exists and belongs to Hs(n-1 × ) for some s, it is desired to calculate f from the improper data . This problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by projection on elements of Meyers multiresolution approximation {Vj}j. Within every space Vj the solution of the above problem depends continuously on the data, and we can find a mollification parameter J depending on the noise level in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. At the end of this paper numerical examples for our method will be given.