American Institute of Mathematical Sciences

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December  2006, 16(4): 783-842. doi: 10.3934/dcds.2006.16.783

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

G. Bellettini 1, , Giorgio Fusco 2, and  Nicola Guglielmi 3,

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, Italy
3. 

Dipartimento di Matematica Pura e Applicata, Università de L'Aquila, I-67100 L'Aquila

Received  January 2006 Revised  June 2006 Published  September 2006

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 $\varepsilon\to 0^+$.
Keywords: singular perturbations, nonconvex functionals, Forward-backward parabolic equations, fourth order regularization, microstructures, stiff problems..
Mathematics Subject Classification: Primary: 35B25; Secondary: 47j06, 35K9.
Citation: G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783
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