The Neumann boundary problem for a nonlocal Cahn-Hilliard equation
The authors consider the following nonlocal Cahn-Hilliard equation: \[ \frac{\partial u}{\partial t} = \Delta\left(\int_\Omega J(x-y)dyu(x) -\int_\Omega J(x-y)u(y)dy + f(u)\right), \quad \text{in } \Omega, \;t>0, \] \[ \frac {\partial\left(\int_\Omega J(x-y)dyu(x)-\int_\Omega J(x-y)u(y)dy + f(u)\right) }{\partial n} = 0, \quad\text{on } \partial\Omega, \;t>0,\tag{1} \] \[ u(x,0) = u_0 (x), \] where \(0\leq J(x)=J(-x), T>0, \) \(\Omega \subset \mathbb{R}^n\) is bounded, and \(f\) is bistable. The authors prove the existence and uniqueness of solution to problem (1) as well as the continuous dependence on the initial data. Furthermore, they prove the existence of absorbing sets and apply their results to other nonlocal problems.
scientific article; zbMATH DE number 2173045
scientific article