Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian
We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂xx)su(x)+∂uF(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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2020-08-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2020-2075 |
| Summary: | We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂xx)su(x)+∂uF(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. |
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| ISSN: | 1536-1365 2169-0375 |