Regularization of an Ill-Posed Partial Differential Equation Problem

We consider the problem K ( x ) uxx = utt , 0 = g ( t ) E L2 and ux ( 0 , t ) = 0, where K ( x ) is continuous and does not come close to zero. This is an ill-posed problem in the sense that, if the solutions exists, it does not depend cotinuously on g. Considering the existence of a solution u ( x , . ) belonging to the Sobolev space H1 ( R ) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem approaching it by well-posed problems in the scalins spaces.