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Zbl 1105.35007
Bellettini, Giovanni; Fusco, Giorgio; Guglielmi, Nicola
A concept of solution and numerical experiments for forward-backward diffusion equations. (English)
[J] Discrete Contin. Dyn. Syst. 16, No. 4, 783-842 (2006). ISSN 1078-0947; ISSN 1553-5231/e

The authors study the gradient flow associated with the functional $F_{\phi} (u): = \frac12 \int_I \phi (u_x)\, dx$, where $\phi$ is non convex, and with its singular perturbation $F_{\phi}^x (u): = \frac12 \int_I (\varepsilon^2 (u_{xx})^2 + \phi (u_x))\,dx$. With the support of numerical simulations, various aspects of the global dynamics of solutions $u^{\varepsilon}$ of the singularly perturbed equation $u_t= -\varepsilon^2 u_{xxxx} +\frac12 \phi'' (u_x) u_{xx}$ for small values of $\varepsilon > 0$ are discussed. Their analysis leads to a reinterpretation of the unperturbed equation $u_{tt} =\frac12 (\phi' (u_x))_x$, and to a well defined notion of a solution. Examine the conjecture that this solution coincides with the limit of $u^{\varepsilon}$ as $\varepsilon \to 0^+$ is given.
[Qin Mengzhao (Beijing)]

MSC 2000:: *35B25 Singular perturbations (PDE)
35K55 Nonlinear parabolic equations
34E13 Multiple scale methods
49L25 Viscosity solutions
35A15 Variational methods (PDE)

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