Generalized monotonicity from global minimization in fourth-order ODEs

We consider solutions of the stationary Extended Fisher-Kolmogorov equation with general potential that are global minimizers of an associated variational problem. We present results that relate the global minimization property to a generalized concept of monotonicity of the solutions. This monotonicity can be described as the lack of intersections of the solution curve when projected onto the $(u,u')$--plane. Our method is based on applying a cut-and-paste argument in the space H^2( {mathbb{R ) to intersections in the $(u,u')$--plane. The statements and proofs are presented for the Extended Fisher-Kolmogorov equation, but the method can be directly extended to a wide class of fourth-order ordinary differential equations that derive from minimization problems

Non-existence and uniqueness results for the extended Fisher-Kolmogorov equation

We give a classification of all bounded solutions of the equation [ u'''' + p u'' + F'(u) = 0, qquad -infty < t< infty, ] in which $F$ is a general quartic polynomial and $p$ is restricted to various subsets of $(-infty,0]$. These results are obtained by combining an a priori estimate with geometric arguments in the $(u,u'')$-plane

Spatial localization for a general reaction-diffusion system

We use a local energy method to study the spatial localization of the supports of the solutions of a reaction-diffusion system with nonlinear diffusion and a general reaction term. We establish finite speed of propagation and the existence of waiting times under a set of weak assumptions on the structural form of the system. These assumptions allow for additive and multiplicative reaction terms and space-and time-dependence of the coefficients, as well as a divergence-free convection term

Diffusive gradients in the PTS system

It has recently been conjectured that metabolic pathways with membrane-bound enzymes can give rise to concentration gradients in the cytosolic pathway components. We investigate this issue using a theoretical model for the Phosphoenolpyruvate-dependent Phosphotransferase system in {it E. coli/, for which accurate measurements of the kinetic parameters are available. We show that significant spatial gradients indeed exist, and we discuss the potential implications of this finding

A continuum model of lipid bilayers

We study a one-dimensional continuum model for lipid bilayers. The system consists of water and lipid molecules; lipid molecules are represented by two ‘beads’, a head bead and a tail bead, connected by a rigid rod. We derive a simplified model for such a system, in which we only take into account the effects of entropy and hydrophilic/hydrophobic interactions. We show that for this simple model membrane-like structures exist for certain choices of the parameters, and numerical calculations suggest that they are stable

Cylindrical shell buckling: a characterization of localization and periodicity

A hypothesis for the prediction of the circumferential wavenumber of buckling ofthe thin axially-compressed cylindrical shell is presented, based on the addition of a length effect to the classical (Koiter circle) critical load result. Checks against physical and numerical experiments, both by direct comparison of wavenumbers and via a scaling law, provide strong evidence that the hypothesis is correct