A remark on forward–backward parabolic equations. Issue 6 (26th April 2019)

• Record Type: Journal Article
• Authors: Paronetto, Fabio
• Is Part Of: Applicable analysis. Volume 98:Issue 6(2019)
• Page Start: 1042
• Page End: 1051
• Publication Date: 2019-04-26
• Subjects: Forward–backward parabolic equations -- abstract mixed type equations; 35K90 -- 35M10; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2017.1408084
• Languages: English
• ISSNs: 0003-6811

Abstract

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.