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Global Existence and Energy Decay of Solutions to a Coupled Wave and Petrovsky System with Nonlinear Dissipations and Source Terms

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Abstract

In this paper, we consider the nonlinearly damped nonlinear coupled wave and Petrovsky system

$$\begin{aligned}&u'' + \Delta ^{2} u+ av + g_{1}(u')=f_{1}(u,v), \\&v'' - \Delta v+ au + g_{2}(v')=f_{2}(u,v). \end{aligned}$$

We prove the global existence of solutions by means of the stable set method in \(H_{0}^{2}(\Omega )\times H^1_0(\Omega )\) combined with the Faedo–Galerkin procedure. Furthermore, we establish a more general energy decay of solutions when the nonlinear dissipative terms \(g_{1},g_{2}\) do not necessarily have a polynomial growth near the origin.

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Acknowledgements

The authors would like to thank the reviewers for their useful remarks and suggestions. Baowei Feng was supported by the National Natural Science Foundation of China (No. 11701465) and by the Fundamental Research Funds for the Central Universities (No. JBK1902026).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact Sciences, University Mustapha Stambouli of Mascara, P. O. Box 305, Route de Mamounia, Mascara, 29000, Algeria

    Mounir Bahlil

  2. Laboratory of Analysis and Control of PDEs, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbès, 22000, Algeria

    Mounir Bahlil

  3. Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Baowei Feng

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  1. Mounir Bahlil
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  2. Baowei Feng
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Correspondence to Mounir Bahlil.

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Bahlil, M., Feng, B. Global Existence and Energy Decay of Solutions to a Coupled Wave and Petrovsky System with Nonlinear Dissipations and Source Terms. Mediterr. J. Math. 17, 60 (2020). https://doi.org/10.1007/s00009-020-1497-5

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  • DOI: https://doi.org/10.1007/s00009-020-1497-5

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