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Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

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The first author was supported by the National Natural Science Foundation of China (11801114), the Heilongjiang Postdoctoral Foundation (LBH-Z15036), the China Scholarship Council (201706685064), the Fundamental Research Funds for the Central Universities. The second author was supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation (2013M540270), the Fundamental Research Funds for the Central Universities
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  • By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level.

    Mathematics Subject Classification: Primary: 35L05, 35L20, 35B44.

    Citation:

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  • Table 1.  Obtained results and open problems for problem (1)-(3)

    Global existence Asymptotic behavior Blow up
    Subcritical initial energy $E(0)<d$ Reference [4] Reference [4] Reference [4]
    Critical initial energy $E(0)=d$ Reference [4] Reference [4] Reference [4]
    Supercritical initial energy $E(0)>d$ Open problem Open problem $\omega=0$, $\mu>0$
    $\omega=0$, $\mu=0$
    Reference [4]
    $\omega>0$, $\mu>0$
    $\omega>0$, $\mu=0$
    $\omega=0$, $\mu>0$
    $\omega=0$, $\mu=0$
    Present paper
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