On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. (English) Zbl 1172.35043

Alves, Claudianor O.; Cavalcanti, Marcelo M.

Calc. Var. Partial Differ. Equ. 34, No. 3, 377-411 (2009).

This paper is concerned with the study of the nonlinear damped wave equation
\[ {u_{tt} - \Delta u+ h(u_t)= g(u) \quad \text{in }\Omega \times ] 0,\infty [,} \]
where \(\Omega \) is a bounded domain of \({\mathbb{R}^2}\) having a smooth boundary \(\partial \Omega = \Gamma \). 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, the authors prove the global existence as well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, for global solutions, they also gave the optimal and uniform decay rates of the energy.

Reviewer: Cheng-Hsiung Hsu (Chung-Li)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35L20	Initial-boundary value problems for second-order hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

damped wave equation; Nehari manifold; Trudinger-Moser inequality; global existence; blow up

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