Mosaic solutions and entropy for discrete coupled phase-transition equations

We consider arrays of coupled scalar differential equations organized on a spatial lattice. One component of this system is analogous to the Allen–Cahn partial differential equation which on the integer lattice has interactions of nearest neighbor type, and the other to the Cahn–Hilliard partial differential equation which has interactions of nearest and next nearest neighbor type. Our coupling functions are forms of the so-called double obstacle nonlinearity. The interaction strengths of both equations are not restricted in magnitude or in sign and need not be near a continuum limit. We prove existence and uniqueness results for the initial value problem and consider the existence and stability of a class of equilibrium solutions called mosaic solutions. These equilibrium solutions take only the values +1,−1, and 0 at each lattice point. Using the notion of a weakly forward invariant set we provide criteria for weak Lyapunov and weak asymptotic stability. Rigorous results are then obtained for the spatial entropy of these stable mosaic solutions and it is shown that the existence and stability results obtained on the integer lattice can be used to obtain similar results on an arbitrary lattice. Numerical results are presented that illustrate the importance of the analytical results.