A concept of solution and numerical experiments for forward-backward diffusion equations

G. Bellettini; Giorgio Fusco; Nicola Guglielmi

doi:10.3934/dcds.2006.16.783

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2006, Volume 16, Issue 4: 783-842. Doi: 10.3934/dcds.2006.16.783

This issue Previous Article Dissipative and weakly--dissipative regimes in nearly--integrable mappings Next Article Boundedness of solutions in a class of Duffing equations with a bounded restore force

A concept of solution and numerical experiments for forward-backward diffusion equations

1.
Dipartimento di Matematica, Università di Roma 'Tor Vergata', 00133, Roma
2.
Dipartimento di Matematica Pura e Applicata, Università de L’Aquila, I-67100 L’Aquila
3.
Dipartimento di Matematica Pura e Applicata, Università de L'Aquila, I-67100 L'Aquila

Received: January 2006

Revised: June 2006

Published: December 2006

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Abstract

We study the gradient flow associated with the functional $F_\phi(u)$ := $\frac{1}{2}\int_{I} \phi(u_x)~dx$, where $\phi$ is non convex, and with its singular perturbation $F_\phi^\varepsilon(u)$:=$\frac{1}{2}\int_I (\varepsilon^2 (u_{x x})^2 + \phi(u_x))dx$. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2 u_{x x x x} + \frac{1}{2} \phi''(u_x)u_{x x}$ for small values of $\varepsilon>0$. Our analysis leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of $u^\varepsilon$ as $\varepsilon\to 0^+$.

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Mathematics Subject Classification: Primary: 35B25; Secondary: 47j06, 35K99.

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