A remark on forward–backward parabolic equations

In this note we deal with a problem regarding forward–backward parabolic equations like r(x)ut-uxx=0 with r which changes sign. It is natural to have the continuity of the function t↦∫u2(x,t)r(x)dx . Here we show that, under very mild assumptions, one also gets the continuity of the function t↦∫u2(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.