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Classification of Initial Energy to a Pseudo-parabolic Equation with p(x)-Laplacian

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Abstract

This paper deals with a pseudo-parabolic equation involving p(x)-Laplacian and variable nonlinear sources. Firstly, we use the Faedo-Galerkin method to give the existence and uniqueness of weak solution in the Sobolev space with variable exponents. Secondly, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely by using the initial energy, characterized with the mountain pass level, and Nehari energy. In the supercritical case, we construct suitable auxiliary functions to determine the quantitative conditions on the initial data for the existence of blow-up or global solutions. The results in this paper are compatible with the problems with constant exponents. Moreover, we give broader ranges of exponents of source and diffusion terms in the discussion of blow-up or global solutions.

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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 Shandong Provincial Natural Science Foundation of China (ZR2021MA003, ZR2020MA020).

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xizheng Sun

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

    Bingchen Liu

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  1. Xizheng Sun
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  2. Bingchen Liu
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Correspondence to Bingchen Liu.

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Sun, X., Liu, B. Classification of Initial Energy to a Pseudo-parabolic Equation with p(x)-Laplacian. J Dyn Control Syst 29, 873–899 (2023). https://doi.org/10.1007/s10883-022-09629-7

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  • DOI: https://doi.org/10.1007/s10883-022-09629-7

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