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Bounds for Blow-up Time to a Viscoelastic Hyperbolic Equation of Kirchhoff Type with Variable Sources

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Abstract

The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition:

$$ |u_{t}|^{\rho }u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int _{0}^{t}g(t- \tau )\Delta u(\tau )d\tau +|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u. $$

Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the modular and the norm. The authors construct suitable function spaces to discuss the upper bound for blow-up time with positive initial energy by means of a differential inequality technique. In addition, lower bounds for blow-up time in different range of exponent are obtained. These improve and generalize some recent results.

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Acknowledgements

Authors wish to express their gratitude to Professor Wenjie Gao for his support and constant encouragement.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, Jilin Province, 130012, China

    Menglan Liao, Bin Guo & Xiangyu Zhu

  2. Department of Mathematics, College of Science, Yanbian University, Yanji, Jilin Province, 133002, China

    Xiangyu Zhu

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  1. Menglan Liao
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  2. Bin Guo
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  3. Xiangyu Zhu
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Correspondence to Bin Guo.

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The project is supported by NSFC (11301211), by the Scientific and Technological Project of Jilin Province’s Education Department in Thirteenth-five-year (JJKH20180111KJ).

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Liao, M., Guo, B. & Zhu, X. Bounds for Blow-up Time to a Viscoelastic Hyperbolic Equation of Kirchhoff Type with Variable Sources. Acta Appl Math 170, 755–772 (2020). https://doi.org/10.1007/s10440-020-00357-3

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  • DOI: https://doi.org/10.1007/s10440-020-00357-3

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