A degenerate fourth-order parabolic equation modeling Bose-Einstein condensation. Part II: Finite-time blow-up

TL;DR

This paper studies a degenerate fourth-order parabolic equation modeling Bose-Einstein condensation, proving finite-time blow-up of solutions under certain initial conditions and showing the set of initial data causing blow-up is dense.

Contribution

It establishes finite-time blow-up results for a kinetic model approximation of Bose-Einstein condensation, including the density of initial data leading to blow-up.

Findings

Solutions blow up in finite time if initial data is large enough.

The set of initial conditions causing blow-up is dense among admissible data.

Finite-time blow-up is proven using entropy inequalities and approximation methods.

Abstract

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted $L^1$ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the $L^\infty$ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.