Long time solutions for wave maps with large data (Q2842368)

The \((2+1)\)-dimensional wave map problem with \(\mathbb{S}^2\) as target, i.e., existence of a map \(\varphi =(\varphi _1,\varphi _2,\varphi _3)\: \mathbb{R}^{2+1}\rightarrow \mathbb{S}^2\) satisfying the system NEWLINE\[NEWLINE \varphi _{tt} -\Delta \varphi =\left( -\sum _{j=1}^3| \partial _t\varphi _j| ^2+\sum_{j=1}^3 \sum_{i=1}^2| \partial _{x_i}\varphi _j| ^2\right)\varphi NEWLINE\]NEWLINE and the initial data \((\varphi ^{(0)},\varphi ^{(1)}),\) is discussed. The authors show that for all positive numbers \(T_0,E_0>0\) there exist initial data with energy at least \(E_0>0\) such that the life-span of the solution is \([0,T_0]\) at least.