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Blow up for initial-boundary value problem of wave equation with a nonlinear memory in 1-D

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Abstract

The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory:

$${u_{tt}} - {u_{xx}} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int_0^t {{{\left( {t - s} \right)}^{ - \gamma }}{{\left| {u\left( s \right)} \right|}^p}ds} .$$

The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.

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Acknowledgement

The authors are very grateful to the referees for the valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Lishui University, Lishui 323000, Zhejiang, China

    Ning-An Lai

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Jianli Liu

  3. College of Education, Lishui University, Lishui 323000, Zhejiang, China

    Jinglei Zhao

Authors
  1. Ning-An Lai
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  2. Jianli Liu
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  3. Jinglei Zhao
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Corresponding author

Correspondence to Jianli Liu.

Additional information

This work was supported by the National Natural Sicence Foundation of China (Nos. 11301489, 11401367,11501273), the Natural Science Foundation of Zhejiang Province (Nos. LQ13A010013, LY14A010010), and the Doctoral Fund of Ministry of Education of China (No. 20133108120002).

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Lai, NA., Liu, J. & Zhao, J. Blow up for initial-boundary value problem of wave equation with a nonlinear memory in 1-D. Chin. Ann. Math. Ser. B 38, 827–838 (2017). https://doi.org/10.1007/s11401-017-1098-1

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  • DOI: https://doi.org/10.1007/s11401-017-1098-1

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