2016-02-22Large time behavior, Lyapunov functionals and rearrangement theory for a nonlocal differential equationThanh Nam Nguyen (NIMS)
We consider an initial-boundary value problem for a nonlocal evolution equation of bistable type and study possible sharp transition layers at a very early stage that the solution might develop. It turns out that such transition layers can be investigated via the structure of the w-limit set of the corresponding nonlocal ordinary differential equation. We prove that for a large class of initial functions, the w-limit set of the nonlocal ordinary differential equation only contains one element. Furthermore, that element is a step function taking at most two values. The proof bases on the rearrangement theory and the existence of infinitely many Lyapunov functionals.