A remark on forward–backward parabolic equations. Issue 6 (26th April 2019) - Paronetto, Fabio
ABSTRACT: In this note we deal with a problem regarding forward–backward parabolic equations liker ( x ) u t - u xx = 0 with r which changes sign. It is natural to have the continuity of the functiont ↦ ∫ u 2 ( x, t ) r ( x ) d x . Here we show that, under very mild assumptions, one also gets the continuity of the functiont ↦ ∫ u 2 ( x, t ) | r ( x ) | d x 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.