In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ into the Orlicz space $L^{ { \phi}_p}(\R^2)$ associated to the function $\phi_p$ defined by $\phi_p(s):={ \rm{ e}^{ s^2}}-\Sum_{ k=0}^{ p-1} \frac{ s^{ 2k}}{ k!}\cdot$ We also undertake the study of a nonlinear wave equation with exponential growth where the Orlicz norm $\|.\|_{ L^{ \phi_p}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study.
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