Estimates on front propagation for nonlinear higher-order parabolic equations: An algorithmic approach Journal Article


Author(s): Fischer, Julian
Article Title: Estimates on front propagation for nonlinear higher-order parabolic equations: An algorithmic approach
Affiliation
Abstract: We present an algorithm for the derivation of lower bounds on support propagation for a certain class of nonlinear parabolic equations. We proceed by combining the ideas in some recent papers by the author with the algorithmic construction of entropies due to Jüngel and Matthes, reducing the problem to a quantifier elimination problem. Due to its complexity, the quantifier elimination problem cannot be solved by present exact algorithms. However, by tackling the quantifier elimination problem numerically, in the case of the thin-film equation we are able to improve recent results by the author in the regime of strong slippage n ∈ (1, 2). For certain second-order doubly nonlinear parabolic equations, we are able to extend the known lower bounds on free boundary propagation to the case of irregular oscillatory initial data. Finally, we apply our method to a sixth-order quantum drift-diffusion equation, resulting in an upper bound on the time which it takes for the support to reach every point in the domain.
Keywords: Degenerate parabolic equation; Thin-film equation; Computer-assisted proof; Doubly nonlinear equation; Free boundaries; Quantum drift-diffusion equation; Waiting time phenomenon
Journal Title: Interfaces and Free Boundaries
Volume: 17
Issue 1
ISSN: 1463-9971
Publisher: European Mathematical Society Publishing House  
Date Published: 2015-01-01
Start Page: 1
End Page: 20
Sponsor: This research was supported by the Lithuanian-Swiss cooperation program under the project agreement No. CH-SMM-01/0.
DOI: 10.4171/IFB/331
Open access: no
IST Austria Authors
  1. Julian Fischer
    13 Fischer
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