Stabilization of ill-posed Cauchy problems for parabolic equations

In this paper we study the non characteristic Cauhy problem for the parabolic equation u_t-- (a(x)u_x)_x = 0, 0<x<l, O <_ t <- T, assuming a essentially bounded in (0,l). In the case of mild a priori bounds on u, we derive stability estimates on u of Hoelder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.