A. Jüngel, M. Winkler:

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

in: "ASC Report 6/2014", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 37.

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 L1 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∞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.

Degenerate parabolic equation, fourth-order parabolic equation, blow-up, weak solutions, entropy inequality, Bose-Einstein condensation, weighted spaces.

http://www.asc.tuwien.ac.at/preprint/2014/asc06x2014.pdf

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