A remark on forward–backward parabolic equations

ABSTRACT

In this note we deal with a problem regarding forward–backward parabolic equations like $r(x)u_t-u_{\mathit{xx}}=0$ with r which changes sign. It is natural to have the continuity of the function $t\mapsto \int u^2(x,t)r(x)\mathrm{d}x$. Here we show that, under very mild assumptions, one also gets the continuity of the function $t\mapsto \int u^2(x,t)|r(x)|\mathrm{d}x$ 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.

KEYWORDS:

• Forward–backward parabolic equations
• abstract mixed type equations

AMS SUBJECT CLASSIFICATIONS:

• 35K90
• 35M10

Notes

No potential conflict of interest was reported by the author.