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

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 nonconvex, 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 numericalsimulations, various aspects of the global dynamics of solutions$u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2u_{x x x x} + \frac{1}{2}\phi''(u_x)u_{x x}$ for smallvalues of $\varepsilon>0$. Our analysis leads to a reinterpretationof the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$,and to a well defined notion of a solution. We also examine theconjecture that this solution coincides with the limit of $u^\varepsilon$as $\varepsilon\to 0^+$.