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

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

${u}_{tt}-{\Delta }u+h\left({u}_{t}\right)=g\left(u\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }×\right]0,\infty \left[,$

where ${\Omega }$ is a bounded domain of ${ℝ}^{2}$ 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.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35L20 Second order hyperbolic equations, boundary value problems 35B40 Asymptotic behavior of solutions of PDE
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