Steady state solutions of a bi-stable quasi-linear equation with saturating flux

Abstract

We consider the bistable equation proposed by Rosenau to replace the Allen-Cahn equation in the case of large gradients. We discuss the bifurcation problem for stationary solutions of this equation on an interval as the diffusion coefficient and the length of the interval are varied, concentrating on classical solutions.

Non-linear effects on canonical MEMS models

“…This seems to be similar to the behaviour observed in some onedimensional prescribed mean curvature equations (cf. [2,13]). Currently, the authors are looking at the one-dimensional solution set of equations (2.10) and (2.11) to see if it sheds any light on this two-dimensional radially symmetric case.…”

Subharmonic solutions of the prescribed curvature equation

“…Indeed, elementary phase-plane analysis and energy arguments, as in [14,15,16,17], show that any solution u, for which…”

“…The coexistence of classical and non-classical solutions of (1), according to the terminology introduced in [18,19,15,17,20], is determined by the specific structure of the curvature operator u / 1 + u 2 , which behaves like the 2-Laplacian u near zero and like the 1-Laplacian sgn(u ) at infinity. These considerations lead us to introduce the following concept of periodic solution for equation (1) that will be considered throughout this paper.…”

Optimal profiles in a phase-transition model with a saturating flux

“…In a series of papers (see, e.g., [37,12,32]) it was pointed out that, in some realistic diffusion processes, characterized for small gradients by linear gradient-flux relations, the flux response to an increase of gradients is expected to slow down and ultimately to approach saturation at large gradients. Accordingly, it was proposed in these contexts to penalize interfaces by a gradient term which is still quadratic for small values of the norm of the gradient but asymptotically linear.…”