A remark on forward-backward parabolic equations

In this note we deal with a problem regarding forward-backward parabolic equations like $r(x) u_t - u_{xx} = 0$ with $r$ which changes sign. It is natural to have the continuity of the function $t \mapsto \int u^2(x,t) r(x) \, dx$. Here we show that, under very mild assmptions, one also gets the continuity of the function $t \mapsto \int u^2(x,t) |r(x)| \, dx$ also in more general and abstract situations. \\ This result turns out to be useful when dealing with De Giorgi classes for these type of equations.