Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation ∂tu+(−Δ)2u=∇⋅F(∇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)=±|ξ|2ξ, we also e...

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Bibliographic Details
Published in:Journal of Differential Equations 2024-05, Vol.391, p.25-56
Main Authors: Li, Xinye, Melcher, Christof
Format: Article
Language:English
Subjects:
Bounded mean oscillation
Epitaxial growth
Fourth-order parabolic equation
Gradient nonlinearity
Stability
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Summary:In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation ∂tu+(−Δ)2u=∇⋅F(∇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)=±|ξ|2ξ, we also examine the large time behaviour and stability of global solutions for arbitrary and small initial data in VMO, respectively.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2024.01.038