Nonuniqueness of solutions for a class of forward–backward parabolic equations. (May 2016)

• Record Type: Journal Article
• Authors: Bertsch, Michiel; Smarrazzo, Flavia; Tesei, Alberto
• Is Part Of: Nonlinear analysis. Volume 137(2016)
• Page Start: 190
• Page End: 212
• Publication Date: 2016-05
• Subjects: primary 35D99 35K55 35R25 -- secondary 28A33 28A50; Forward–backward parabolic equations -- Radon measure-valued solutions -- Nonuniqueness; 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.2015.12.028
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the initial–boundary value problem { u t = [ φ ( u ) ] x x + ε [ ψ ( u ) ] t x x in Ω × ( 0, ∞ ) φ ( u ) + ε [ ψ ( u ) ] t = 0 in ∂ Ω × ( 0, ∞ ) u = u 0 ≥ 0 in Ω × { 0 } with measure-valued initial data. Here Ω is a bounded open interval, φ ( 0 ) = φ ( ∞ ) = 0, φ is increasing in ( 0, α ) and decreasing in ( α, ∞ ), and the regularising term ψ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Nonnegative Radon measure-valued solutions are known to exist and their construction is based on an approximation procedure. Until now nothing was known about their uniqueness. In this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions, whence uniqueness fails. In addition, we classify the steady state solutions.