Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

Abstract

We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂x⁢x)s⁢u⁢(x)+∂u⁡F⁢(x,u⁢(x))=0{(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in ℝ{\mathbb{R}}. The smooth function F⁢(x,u){F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at +1{+1} and -1 for any x∈ℝ{x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.

Keywords

• fractional laplacian
• periodic solutions
• non-autonomous
• variational method
• 35j61
• 35b10