Advances in Differential Equations

A free boundary characterization of measure-valued solutions for forward-backward diffusion

Dirk Horstmann and Ben Schweizer

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We consider a nonlinear one-dimensional scalar equation of diffusion type in which, depending on the gradient of the solution, the diffusion coefficient may be positive or negative. We compare two concepts of Young measure solutions which are based on different methods to construct approximate solutions, the SP-solutions (singular perturbation) and the EM-solution (energy minimization). We show that the SP-solution can recover classical solutions where the EM-solution fails to do so, and that EM-solutions are more stable under perturbations of the initial values. We characterize the EM-solution with a free boundary problem and determine its long-time behavior.

Article information

Adv. Differential Equations, Volume 13, Number 3-4 (2008), 201-227.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions 35K65: Degenerate parabolic equations 35R35: Free boundary problems


Horstmann, Dirk; Schweizer, Ben. A free boundary characterization of measure-valued solutions for forward-backward diffusion. Adv. Differential Equations 13 (2008), no. 3-4, 201--227.

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