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Blow-up and Lifespan of Solutions for a Nonlinear Viscoelastic Kirchhoff Equation

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Abstract

The blow-up of solutions of a class of nonlinear viscoelastic Kirchhoff equation with suitable initial data and Dirichlet boundary conditions is discussed. By constructing a suitable auxiliary function to overcome the difficulty of gradient estimation and making use of differential inequality technique, we establish a finite time blow-up result when the initial data is at arbitrary energy level. Moreover, a lower bound of the lifespan is also derived by constructing a control function with both nonlocal term and memory kernel. Compared with the previous literature, our approach to estimate the lifespan does not require the initial energy to control some norms of the solution.

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Acknowledgements

The author would like to thank the anonymous reviewers for his/her careful reading of the paper, giving valuable comments and suggestions. He would also like to thank Professor Qiuyi Dai for his continuous encouragement.

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  1. College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421002, Hunan, China

    Zhifeng Yang

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  1. Zhifeng Yang
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Correspondence to Zhifeng Yang.

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This work was supported by the National Science Foundation of China under 11671128, by the Science Research Project of Hengyang Normal University under 16D01, and by the Science Research Project of Education Department of Hunan Province under 17A029.

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Yang, Z. Blow-up and Lifespan of Solutions for a Nonlinear Viscoelastic Kirchhoff Equation. Results Math 75, 84 (2020). https://doi.org/10.1007/s00025-020-01223-2

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  • DOI: https://doi.org/10.1007/s00025-020-01223-2

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