Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
This work advances the understanding of multistability in diblock copolymer models, which is important for materials science and dynamical systems theory.
The authors provide a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, showing that solutions near the homogeneous state can be trapped by higher-energy stable states rather than converging to the global energy minimizer, a phenomenon absent in the Cahn-Hilliard equation.
We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer.