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Description of the lack of compactness in Orlicz spaces and applications. (English) Zbl 1363.46024

Summary: In this paper, we investigate the lack of compactness of the Sobolev embedding of \(H^1(\mathbb{R}^2)\) into the Orlicz space \(L^{{\phi}_p}(\mathbb{R}^2)\) associated to the function \(\phi_p\) defined by \(\phi_p(s):={\text{e}^{s^2}}-\sum_{k=0}^{p-1} \frac{s^{2k}}{k!}\). 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.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35P25 Scattering theory for PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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