The critical nonlinear wave equation in two space dimensions (Q363234)

The author establishes the well-posedness of the nonlinear wave equation \( u_{tt}-\Delta u+ue^{u^{2}}=0\) in \(\mathbb{R}\times \mathbb{R}^{2}\) with Cauchy data \((u,u_{t})_{t=0}=(u_{0},u_{1})\) for general \(u_{0},u_{1}\in C^{\infty }(\mathbb{R}^{2})\). The main result of the paper proves the existence of a unique smooth and global in time solution of this wave equation with critical nonlinearity. For the proof, the author argues indirectly assuming that a local solution cannot be smoothly extended. He indeed follows some of the arguments presented in his previous paper [Math. Ann. 350, No. 3, 707--719 (2011; Zbl 1227.35216)]. He also uses some energy estimates quoted from the paper by \textit{S. Ibrahim} et al. in [Commun. Pure Appl. Math. 59, No. 11, 1639--1658 (2006; Zbl 1117.35049)], where a similar problem is studied, but here assuming that the Cauchy data are such that their energy \(\int_{\mathbb{R}^{2}}(\frac{1}{2} \left| u_{1}\right| ^{2}+\left| \nabla u_{0}\right| ^{2}+e^{u_{0}^{2}})dx\leq 2\pi\). The author finally improves the Moser-Trudinger estimates, as the above nonlinear wave equation is closely linked to the critical Sobolev embedding in the 2D case.