Stabilization of ill-posed Cauchy problems for parabolic equations

Abstract

In this paper we study the noncharacteristic Cauchy problem, ut−(a(x)ux)x=0, x∈(0, l), t∈.(0, T], u(0, t)=ϕ(t), ux(0,t)=0, 0≦t≦T, assuming only L∞ for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Holder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.