We study the initial–boundary value problem 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.

Bertsch, M., Smarrazzo, F., Tesei, A. (2016). Nonuniqueness of solutions for a class of forward-backward parabolic equations. NONLINEAR ANALYSIS, 137, 190-212 [10.1016/j.na.2015.12.028].

Nonuniqueness of solutions for a class of forward-backward parabolic equations

BERTSCH, MICHIEL;
2016-01-01

Abstract

We study the initial–boundary value problem 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.
2016
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Forward–backward parabolic equations; Radon measure-valued solutions; Nonuniqueness
Bertsch, M., Smarrazzo, F., Tesei, A. (2016). Nonuniqueness of solutions for a class of forward-backward parabolic equations. NONLINEAR ANALYSIS, 137, 190-212 [10.1016/j.na.2015.12.028].
Bertsch, M; Smarrazzo, F; Tesei, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/173541
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