An apparently unnatural estimate about forward-backward parabolic equations

Abstract

In this note we come back to face a problem regarding forward-backward parabolic equations like $(r(x,t) u)_t - u_{xx} = 0$ and $r(x,t) u_t - u_{xx} = 0$ ($r$ is both positive and negative): the continuity of $t \mapsto \int u^2(x,t) |r(x,t)| \, dx$.