Well-posedness and stability for a class of fourth-order nonlinear parabolic equations
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917577600532480 |
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| author | Li, Xinye Melcher, Christof |
| author_facet | Li, Xinye Melcher, Christof |
| contents | In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-Δ)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(ξ) = \pm \lvert ξ\rvert^2 ξ$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08398 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Well-posedness and stability for a class of fourth-order nonlinear parabolic equations Li, Xinye Melcher, Christof Analysis of PDEs 35K25, 35K55, 35B35 In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-Δ)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(ξ) = \pm \lvert ξ\rvert^2 ξ$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively. |
| title | Well-posedness and stability for a class of fourth-order nonlinear parabolic equations |
| topic | Analysis of PDEs 35K25, 35K55, 35B35 |
| url | https://arxiv.org/abs/2308.08398 |