Boundedness for solutions of weighted forward–backward parabolic equations without assuming higher regularity. (July 2022)

• Record Type: Journal Article
• Authors: Paronetto, Fabio
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: 35B65 -- 35M10 -- 35B50 -- 35B45 -- 35J62 -- 35K65 -- 35J70; Parabolic equations -- Elliptic equations -- Elliptic–parabolic equations -- Forward–backward parabolic equations -- Mixed type equations -- De Giorgi classes -- Local and global boundedness -- Nonlinear equations -- Weights; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112863
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the solutions of two evolution equations of the type μ ( x, t ) u t + A u = 0 and ( μ ( x, t ) u ) t + A u = 0, where μ ∈ L 1 may be positive, null and negative and A a suitable monotone operator. The simple example is A = − Δ p with p ⩾ 2 . For these functions we prove an unusual local boundedness result using an approximation via the solutions of suitable equations, specifically ɛ B u + μ ( x, t ) u t + A u = 0 and ɛ B u + ( μ ( x, t ) u ) t + A u = 0 . For A = − Δ p, B u = − ( | u t | p − 2 u t ) t .