# American Institute of Mathematical Sciences

January  2017, 37(1): 131-167. doi: 10.3934/dcds.2017006

## On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations

 Department of Mathematics, Florida International University, Miami, FL 33199, USA

Received  February 2016 Revised  September 2016 Published  November 2016

We consider a doubly nonlocal Cahn-Hilliard equation for the nonlocal phase-separation of a two-component material in a bounded domain in the case when mass transport exhibits non-Fickian behavior. Such equations are important for phase-segregation phenomena that exhibit non-standard (anomalous) behaviors. Recently, four different cases were proposed to handle this important equation and the two levels of nonlocality and interaction that are present in the equation. The so-called strong-to-weak interaction case (when one kernel is integrable in some sense while the other is not) was investigated recently for the doubly nonlocal parabolic equation with a regular polynomial potential. In this contribution, we address the so-called strong-to-strong interaction case when both kernels are strongly singular and non-integrable in a suitable sense. We establish well-posedness results along with some regularity and long-time results in terms of finite dimensional global attractors.

Citation: Ciprian G. Gal. On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 131-167. doi: 10.3934/dcds.2017006
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##### References:
$X\subset \mathbb{R}^{n}$ is a bounded domain with Lipschitz continuous boundary $\partial X.$ The general model covered is the mass-conserved one given by (1.5) with the following choices of operators $A,B$. A physically relevant choice that satisfies our assumptions is the double-well potential $F\left( s\right) =\theta s^{4}-\theta _{c}s^{2}$, $0<\theta <\theta _{c}.$
 Model Classical CHE Doubly nonlocal CHE, case (2) $\mathit{A}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{1}}, s_{1}\in \left( 1/2,1\right)$ $\mathit{B}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{2}}, s_{2}\in \left( 1/2,1\right)$
 Model Classical CHE Doubly nonlocal CHE, case (2) $\mathit{A}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{1}}, s_{1}\in \left( 1/2,1\right)$ $\mathit{B}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s_{2}}, s_{2}\in \left( 1/2,1\right)$
The information is the same as in Table 1.
 Model CHE: anamolous transport CHE: nonlocal strong energy $\mathit{A}$ $(-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$ $-\Delta _{X,N}$ $\mathit{B}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$
 Model CHE: anamolous transport CHE: nonlocal strong energy $\mathit{A}$ $(-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$ $-\Delta _{X,N}$ $\mathit{B}$ $-\Delta _{X,N}$ $(-\Delta )_{X,N}^{s}, s\in \left( 1/2,1\right)$
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