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Existence and Blow-up Study of a Quasilinear Wave Equation with Damping and Source Terms of Variable Exponents-type Acting on the Boundary

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Abstract

In this work, we are concerned with a quasilinear wave equation with nonlinear damping and source terms of variable exponents-type acting in a part of the boundary. Under suitable conditions on the exponents and the initial data, we study the blow-up properties. Firstly, by using Faedo-Galerkin method and Banach-Fixed-Point Theorem, we establish the existence of a weak solution, under suitable assumptions on the variable exponents and the initial data. Secondly, we show a finite time blow-up with lower and upper bound as well. Next, an infinite time blow-up is proved under some conditions in the exponents and the initial data.

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Acknowledgements

The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. Also, the authors would like to thank the referees for their value comments and remarks that improved our manuscript.

Funding

This work is funded by KFUPM, Grant No. INCB2404.

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Authors and Affiliations

  1. Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Mohammad Kafini, Mohammad M. Al-Gharabli & Adel M. Al-Mahdi

  2. The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Mohammad Kafini, Mohammad M. Al-Gharabli & Adel M. Al-Mahdi

Authors
  1. Mohammad Kafini
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  2. Mohammad M. Al-Gharabli
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  3. Adel M. Al-Mahdi
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Contributions

Kafini wrote the first draft of the main manuscript text and Al-Gharabli and Al-Mahdi reviewed the manuscript.

Corresponding author

Correspondence to Adel M. Al-Mahdi.

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The authors declare no competing interests.

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Kafini, M., Al-Gharabli, M.M. & Al-Mahdi, A.M. Existence and Blow-up Study of a Quasilinear Wave Equation with Damping and Source Terms of Variable Exponents-type Acting on the Boundary. J Dyn Control Syst 30, 11 (2024). https://doi.org/10.1007/s10883-024-09695-z

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  • DOI: https://doi.org/10.1007/s10883-024-09695-z

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