Description of the lack of compactness in Orlicz spaces and applications

In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^1({\mathbb{R}}^2)$ into the Orlicz space $L^{{\varphi }_p}({\mathbb{R}}^2)$ associated to the function ${\varphi }_p$ 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 $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_{L^{{\varphi }_p}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study.