Stability for backward problems in time for degenerate parabolic equations

TL;DR

This paper establishes conditional stability results for backward in time problems of degenerate parabolic equations, using Carleman estimates, and extends the approach to semilinear cases.

Contribution

It introduces a new weighted $L^2$-estimate approach for stability analysis in degenerate parabolic backward problems, including semilinear equations.

Findings

Conditional stability under boundedness assumptions.

Weighted $L^2$-estimate based on Carleman-type inequality.

Extension to semilinear 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.