A Mollification Method for a Noncharacteristic Cauchy Problem for a Parabolic Equation

is considered. With w gLp R., pgw1, `x, it is proved that a solution of NCP exists if and only if w is infinitely differentiable and 5w n. 5 Lp R.Fc 2n.!s2n, ;ngN, for certain constants c and s. NCP is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. The following mollification method is suggested for solving NCP in a stable way: If w gLp R. is given inexactly by wegLp R. then we mollify we by convolutions with the Dirichlet kernel and the de la Vall´ee Poussin kernel. The exact solution of NCP is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the H¨older type. By the method we can work with the data in Lp R., pgw1, `x and obtain several sharp stability estimates in Lp- and L`-norms of the H¨older type for the solution of the problem. The method can easily be implemented numerically using the fast Fourier transform. A stable marching difference scheme based on this method is suggested. Several numerical examples are given, which show that the method is effective.