Stabilization of ill-posed Cauchy problems for parabolic equations (Q1110728)

We study the noncharacteristic Cauchy problem \(u_ t-(a(x)u_ x)_ x=0\), \(x\in (0,\ell)\), \(t\in (0,T]\), \(u(0,t)=\phi (t)\), \(u_ x(0,t)=0\), \(0\leq t\leq T\), assuming only \(L^{\infty}\) for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.