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.

Mathematics Subject Classification (2000)

35L05 35L20 35A07 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsFederal University of Campina GrandeCampina GrandeBrazil
  2. 2.Department of MathematicsState University of MaringáMaringáBrazil

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