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Classification of Singular Solutions in a Nonlinear Fourth-Order Parabolic Equation

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Abstract

This paper deals with the fourth-order parabolic equation \(u_{t}+{\varDelta }^{2}u=u^{p(x)}\log u\) in a bounded domain, subject to homogeneous Navier boundary conditions. For subcritical and critical initial energy cases, we combine the Galerkin’s method with the generalized potential well method to prove the existence of global solutions. By the concavity arguments, we obtain the results about blow-up solutions. For super critical initial energy case, we use some ordinary differential inequalities to study the extinction of solutions. Moreover, extinction rate, blow-up rate and time, and decay estimate of solutions are discussed.

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Acknowledgements

The authors would like to express their sincerely thanks to the Editor and the Reviewers for the constructive comments to improve this paper.

Funding

This paper is supported by the Shandong Provincial Natural Science Foundation of China (ZR2021MA003, ZR2020MA020).

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Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266580, People’s Republic of China

    Bingchen Liu & Min Zhang

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  1. Bingchen Liu
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  2. Min Zhang
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Correspondence to Bingchen Liu.

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Liu, B., Zhang, M. Classification of Singular Solutions in a Nonlinear Fourth-Order Parabolic Equation. J Dyn Control Syst 29, 455–474 (2023). https://doi.org/10.1007/s10883-022-09597-y

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  • DOI: https://doi.org/10.1007/s10883-022-09597-y

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