Periodic solutions for the Allen-Cahn equation

Rui Huang, Haochuan Huang, Shanming Ji, Jingxue Yin
2015 Advances in Difference Equations  
In this paper we consider the time-periodic Allen-Cahn equation subject to homogeneous boundary value condition and time-periodic condition. For the case of a smooth bounded domain with spatial dimension N ≤ 3, we prove the existence of classical nontrivial periodic solutions. For the case of a star shaped domain with N ≥ 4, we prove the nonexistence of nontrivial periodic solutions. For the case of an annulus domain with N ≥ 3, we prove the existence of nontrivial radial periodic solutions.
more » ... iodic solutions. Some numerical simulations are also presented to illustrate our results.
doi:10.1186/s13662-015-0631-3 fatcat:gmr5kb7vb5dzxb3amaovmsae2i