The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation

Irena Pawłow; Wojciech M. Zajączkowski

doi:10.3934/cpaa.2014.13.859

Article Contents

2014, Volume 13, Issue 2: 859-880. Doi: 10.3934/cpaa.2014.13.859

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The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation

Irena Pawłow¹ and
Wojciech M. Zajączkowski²

1.
System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw
2.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw

Received: November 30, 2012

Revised: May 31, 2013

Published: February 2014

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Abstract

We consider again the sixth order Cahn-Hilliard type equation with a nonlinear diffusion, addressed in our previous paper in Commun. Pure Appl. Anal. 10 (2011), 1823--1847. Such PDE arises as a model of oil-water-surfactant mixtures. Applying the approach based on the Bäcklund transformation and the Leray-Schauder fixed point theorem we generalize the existence result of the above mentioned paper by imposing weaker assumptions on the data. Here we prove the global unique solvability of the problem in the Sobolev space $H^{6,1}(\Omega\times(0,T))$ under the assumption that the initial datum is in $H^3(\Omega)$ whereas previously $H^6(\Omega)$-regularity was required. Moreover, we admit a broarder class of nonlinear terms in the free energy potential.

Keywords:

Sixth orfer Cahn-Hilliard type equation with nonlinear diffusion,
Bäcklund transformation,
global existence.

Mathematics Subject Classification: Primary: 35K35, 35K60; Secondary: 35Q72, 35L205.

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References

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