Michal Kowalczyk

Kink dynamics in the ⁴ model: Asymptotic stability for odd perturbations in the energy space

We consider a classical equation known as the

Dynamics of an Interior Spike in the Gierer--Meinhardt System

\phi^4

Multi-bump ground states of the Gierer–Meinhardt system in R2

model in one space dimension. The kink, defined by

A Variational Principle for Molecular Motors

H(x)=\tanh(x/{\sqrt{2}})

Resonance and Interior Layers in an Inhomogeneous Phase Transition Model

, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

Nondegeneracy of the saddle solution of the Allen-Cahn equation

We study the dynamics of an interior spike of the Gierer--Meinhardt system. Under certain assumptions on the domain size, the diffusion coefficients, and the decay rates, we prove that the velocity of the center of the spike is proportional to the negative gradient of

Nonexistence of small, odd breathers for a class of nonlinear wave equations

R(\xi,\xi)

Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality

, where

Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences

R(x,\xi)

On De Giorgi's conjecture and beyond.

is the regular part of the Greens function of the Laplacian with the Neumann boundary condition. Hence, an interior spike moves towards local minima of