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

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


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