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Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

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Abstract

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L2 and Lp+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

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Acknowledgement

We are grateful to the anonymous referees for a number of valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Qian Lei

  2. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Han Yang

  3. School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China

    Qian Lei

Authors
  1. Qian Lei
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  2. Han Yang
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Corresponding author

Correspondence to Qian Lei.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11501395, 71572156).

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Lei, Q., Yang, H. Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients. Chin. Ann. Math. Ser. B 39, 643–664 (2018). https://doi.org/10.1007/s11401-018-0087-3

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  • DOI: https://doi.org/10.1007/s11401-018-0087-3

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