On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior

Andrea   Giorgini

doi:10.3934/cpaa.2016.15.219

Article Contents

2016, Volume 15, Issue 1: 219-241. Doi: 10.3934/cpaa.2016.15.219

This issue Previous Article Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities Next Article Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species

On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior

Andrea Giorgini^1,

1.
Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, I-20133 Milano

Received: February 2015

Revised: April 2015

Published: January 2016

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Abstract

We propose a mathematical analysis of the Swift-Hohenberg equation arising from the phase field theory to model the transition from an unstable to a (meta)stable state. We also consider a recent generalization of the original equation, obtained by introducing an inertial term, to predict fast degrees of freedom in the system. We formulate and prove well-posedness results of the concerned models. Afterwards, we analyse the long-time behavior in terms of global and exponential attractors. Finally, by reading the inertial term as a singular perturbation of the Swift-Hohenberg equation, we construct a family of exponential attractors which is Hölder continuous with respect to the perturbative parameter of the system.

Keywords:

Swift-Hohenberg equation,
well-posedness,
global attractor,
exponential attractors,
robust family of exponential attractors.

Mathematics Subject Classification: Primary: 35B30, 35G31; Secondary: 82C26.

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References

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