Stability for backward problems in time for degenerate parabolic equations

Abstract

For solution $u(x,t)$ to degenearte parabolic equations in a bounded domain $\Omega$ with homogenous boundary condition, we consider backward problems in time: determine $u(\cdot ,t_0)$ in $\Omega$ by $u(\cdot ,T)$, where $t$ is the time variable and $0\le t_0<T$. Our main results are conditional stability under boundedness assumptions on $u(\cdot ,0)$. The proof is based on a weighted $L^2$-estimate of $u$ whose weight depends only on $t$, which is an inequality of Carleman's type. Moreover our method is applied to semilinear degenerate parabolic equations.