Numerical solution of an ill-posed Cauchy problem for a quasilinear   parabolic equation using a Carleman weight function
  This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations. 
Michael V. Klibanov, Nikolaj A. Koshev, Jingzhi Li and Anatoly G.   Yagola
math-ph math.MP