DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Jacek Cyranka, Rutgers University

**Title: ***
Computer proof of heteroclinic connection in one-dimensional Ohta Kawasaki diblock copolymer model
*

**Abstract:**
We present a computer proof of a heteroclinic connection in the one dimensional Ohta--Kawasaki PDE model of diblock copolymers (Cahn-Hillard type) with Neumann boundary conditions.
For some pairs of parameters (sigma, epsilon) the equation exhibit rich attractor structure. We establish existence of a heteroclinic
connection between the homogeneous state u = 0 representing a perfect mixture of copolymers and a local
energy minimizers. This is a phenomenon not present in 1d Cahn-Hillard
PDE,
in which generic solutions are attracted by the global minimizer.
The computer assisted proof is conceptually simple, and combines several techniques from some of the authors' and P. Zgliczynski works. The most challenging part of the proof was to rigorously propagate in time an interval bounds consisting a piece of the unstable manifold up to a time when the bounds are within the basin of attraction of a stable fixed point. We will present the structure of the proof, and how the crucial steps were realized, and what algorithms were used.

**Time:** Friday, December 2, 2016, 1:30-2:20 p.m.

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491