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 |
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De Gruyter
2020-08-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2020-2075 |
| _version_ | 1817997004453707776 |
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| author | Feng Zhenping Du Zhuoran |
| author_facet | Feng Zhenping Du Zhuoran |
| author_sort | Feng Zhenping |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T02:31:50Z |
| format | Article |
| id | doaj.art-20b4303507f143898a524e5b719e543a |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-14T02:31:50Z |
| publishDate | 2020-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-20b4303507f143898a524e5b719e543a2022-12-22T02:17:39ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752020-08-0120372573710.1515/ans-2020-2075Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional LaplacianFeng Zhenping0Du Zhuoran1School of Mathematics, Hunan University, Changsha410082, P. R. ChinaSchool of Mathematics, Hunan University, Changsha410082, P. R. ChinaWe 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.https://doi.org/10.1515/ans-2020-2075fractional laplacianperiodic solutionsnon-autonomousvariational method35j61 35b10 |
| spellingShingle | Feng Zhenping Du Zhuoran Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian Advanced Nonlinear Studies fractional laplacian periodic solutions non-autonomous variational method 35j61 35b10 |
| title | Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian |
| title_full | Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian |
| title_fullStr | Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian |
| title_full_unstemmed | Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian |
| title_short | Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian |
| title_sort | periodic solutions of non autonomous allen cahn equations involving fractional laplacian |
| topic | fractional laplacian periodic solutions non-autonomous variational method 35j61 35b10 |
| url | https://doi.org/10.1515/ans-2020-2075 |
| work_keys_str_mv | AT fengzhenping periodicsolutionsofnonautonomousallencahnequationsinvolvingfractionallaplacian AT duzhuoran periodicsolutionsofnonautonomousallencahnequationsinvolvingfractionallaplacian |