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Dans ce travail, on s’intéresse à l’existence globale de solutions classiques et au sens de Shatah-Struwe de l’équation des ondes critique à coefficients variables en dimension $d$ d’espace $${\square }_{A}u+{\left|u\right|}^{4/(d-2)}u={\partial }_t^{2}u-\mathrm{div}(A\left(x\right)·{\nabla }_{x}u)+{\left|u\right|}^{4/(d-2)}u=0,{ℝ}_t\times {ℝ}_x^d,$$ où $A$ est une fonction régulière à valeurs dans les matrices $d\times d$ définies positives, valant l’identité en dehors d’un compact fixe.

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty }$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function....