Infinite speed of support propagation for the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models Journal Article


Author(s): Fischer, Julian
Article Title: Infinite speed of support propagation for the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models
Affiliation
Abstract: We show that weak solutions of the Derrida-Lebowitz-Speer-Spohn (DLSS) equation display infinite speed of support propagation. We apply our method to the case of the quantum drift-diffusion equation which augments the DLSS equation with a drift term and possibly a second-order diffusion term. The proof is accomplished using weighted entropy estimates, Hardy's inequality and a family of singular weight functions to derive a differential inequality; the differential inequality shows exponential growth of the weighted entropy, with the growth constant blowing up very fast as the singularity of the weight becomes sharper. To the best of our knowledge, this is the first example of a nonnegativity-preserving higher-order parabolic equation displaying infinite speed of support propagation.
Keywords: Higher-order parabolic equation; Qualitative behaviour; Derrida-Lebowitz-Speer-Spohn equation; DLSS equation; Infinite speed of propagation
Journal Title: Nonlinear Differential Equations and Applications
Volume: 21
Issue 1
ISSN: 1420-9004
Publisher: Birkhäuser  
Date Published: 2014-01-01
Start Page: 27
End Page: 50
DOI: 10.1007/s00030-013-0235-0
Open access: no