Interfaces and Free Boundaries


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Volume 17, Issue 1, 2015, pp. 1–20
DOI: 10.4171/IFB/331

Published online: 2015-05-26

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

Julian Fischer[1]

(1) Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

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 $n \in (1,2)$. 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: Free boundaries, waiting time phenomenon, degenerate parabolic equation, thin-film equation, doubly nonlinear equation, quantum drift-diffusion equation, computer-assisted proof

Fischer Julian: Estimates on front propagation for nonlinear higher-order parabolic equations: An algorithmic approach. Interfaces Free Bound. 17 (2015), 1-20. doi: 10.4171/IFB/331