Opuscula Math. 40, no. 1 (2020), 111-130
https://doi.org/10.7494/OpMath.2020.40.1.111

 
Opuscula Mathematica

Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity

Wei Lian
Md Salik Ahmed
Runzhang Xu

Abstract. In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.

Keywords: global existence, blow-up, logarithmic and polynomial nonlinearity, potential well.

Mathematics Subject Classification: 35L71, 35L20, 35L05.

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  • Runzhang Xu (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-4703-9319
  • College of Automation, College of Mathematical Sciences, Harbin Engineering University, 150001, People's Republic of China
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2019-02-23.
  • Accepted: 2019-08-01.
  • Published online: 2020-02-17.
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Cite this article as:
Wei Lian, Md Salik Ahmed, Runzhang Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math. 40, no. 1 (2020), 111-130, https://doi.org/10.7494/OpMath.2020.40.1.111

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