Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity
In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-10-01
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| Series: | Communications in Analysis and Mechanics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2023033?viewType=HTML |
| _version_ | 1797361215013388288 |
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| author | Yuxuan Chen |
| author_facet | Yuxuan Chen |
| author_sort | Yuxuan Chen |
| collection | DOAJ |
| description | In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution. |
| first_indexed | 2024-03-08T15:50:44Z |
| format | Article |
| id | doaj.art-65e15d17d2ac44b9b9a90c19ef45f16f |
| institution | Directory Open Access Journal |
| issn | 2836-3310 |
| language | English |
| last_indexed | 2024-03-08T15:50:44Z |
| publishDate | 2023-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj.art-65e15d17d2ac44b9b9a90c19ef45f16f2024-01-09T06:03:51ZengAIMS PressCommunications in Analysis and Mechanics2836-33102023-10-0115465869410.3934/cam.2023033Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearityYuxuan Chen0School of Mathematical Sciences, Heilongjiang University, Harbin 150080, ChinaIn this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.https://www.aimspress.com/article/doi/10.3934/cam.2023033?viewType=HTMLglobal existencefinite time blow upground state solutiondegenerate parabolic equation |
| spellingShingle | Yuxuan Chen Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity Communications in Analysis and Mechanics global existence finite time blow up ground state solution degenerate parabolic equation |
| title | Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity |
| title_full | Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity |
| title_fullStr | Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity |
| title_full_unstemmed | Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity |
| title_short | Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity |
| title_sort | global dynamical behavior of solutions for finite degenerate fourth order parabolic equations with mean curvature nonlinearity |
| topic | global existence finite time blow up ground state solution degenerate parabolic equation |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2023033?viewType=HTML |
| work_keys_str_mv | AT yuxuanchen globaldynamicalbehaviorofsolutionsforfinitedegeneratefourthorderparabolicequationswithmeancurvaturenonlinearity |