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Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping

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  • Ding, Hang
  • Zhou, Jun

Abstract

This paper deals with a class of quasilinear wave equations with structural damping or strong damping. By virtue of the improved Faedo–Galerkin method and some technical efforts, we first establish the local well-posedness of weak solutions, especially the continuity of weak solutions with respect to time in the natural phase space. Then we investigate the global existence, asymptotic behavior and finite time blow-up of weak solutions with subcritical or critical initial energy. As for the supercritical initial energy case, we show that the weak solutions may blow up in finite time with arbitrarily high initial energy. Last but not least, the upper and lower bounds of the blow-up time for blow-up solutions are derived.

Suggested Citation

  • Ding, Hang & Zhou, Jun, 2022. "Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    • Handle: RePEc:eee:chsofr:v:163:y:2022:i:c:s0960077922007469
    DOI: 10.1016/j.chaos.2022.112553
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    References listed on IDEAS

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    1. Xiao, Ti-Jun & Ding, Hui-Sheng & Liang, Jin, 2008. "Global attractors for semilinear hyperbolic equations," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1040-1047.
    2. Fan Geng & Ruizhai Li & Xiaojun Zhang & Xiangyu Ge, 2016. "Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-11, March.
    3. Xiao, Haibin, 2009. "Global attractors for semilinear damped wave equations with critical exponent on Rn," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2546-2552.
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    1. Ding, Hang & Zhou, Jun, 2023. "A note on “Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping” [Chaos Solitons Fractals 163 (2022) 112553]," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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