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On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

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Abstract

This paper is concerned with the study of the nonlinear damped wave equation

$${u_{tt} - \Delta u+ h(u_t)= g(u) \quad \quad {\rm in}\,\Omega \times ] 0,\infty [,}$$

where Ω is a bounded domain of \({\mathbb{R}^2}\) having a smooth boundary ∂Ω = Γ. Assuming that g is a function which admits an exponential growth at the infinity and, in addition, that h is a monotonic continuous increasing function with polynomial growth at the infinity, we prove both: global existence as well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, optimal and uniform decay rates of the energy are proved for global solutions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Federal University of Campina Grande, 58109-970, Campina Grande, PB, Brazil

    Claudianor O. Alves

  2. Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

    Marcelo M. Cavalcanti

Authors
  1. Claudianor O. Alves
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  2. Marcelo M. Cavalcanti
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Correspondence to Claudianor O. Alves.

Additional information

The author is Supported by CNPq 300959/2005-2, CNPq/Universal 472281/2006-2 and CNPq/Casadinho 620025/2006-9.

Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0.

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Alves, C.O., Cavalcanti, M.M. On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. Calc. Var. 34, 377–411 (2009). https://doi.org/10.1007/s00526-008-0188-z

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  • DOI: https://doi.org/10.1007/s00526-008-0188-z

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