On the asymptotic behavior of spherically symmetric wave maps
``Wave maps are maps from an Einsteinian manifold \(M\), i.e. a manifold with a metric \(g\) of index 1, into a Riemannian manifold \(N\), i.e. one with a positive definite metric \(h\). Formally, they are the critical points \(U: M\to N\) of the Lagrangian \[ {\mathcal L}[U] = {1\over 2} \int_ M h_{ab} \partial_ \mu U^ a \partial^ \mu U^ b. \] In a previous paper [the authors, Commun. Pure Appl. Math. 46, No. 7, 1041-1091 (1993; Zbl 0744.58071)] we proved the global regularity of the solution to the Cauchy problem for a spherically symmetric wave map in the case where \(M\) is the \((2+1)\)-dimensional Minkowski space-time and \(N\) satisfies the following two conditions. (C1) There exists an orthonormal frame of smooth vectorfields \(\Omega_ A\) on \(N\) whose structure functions \(e^ C_{AB}\) are bounded. (C2) For each \(p \in N\), let \(\Sigma(p,s)\) be the geodesic sphere of radius \(s\) centered at \(p\) and let \(k_{AB}\) be its second fundamental form. Then there exists constants \(c\) and \(C\) such that \[ s \lambda \geq c\quad\text{and}\quad s \Lambda \leq C(1+s), \] where \(\lambda\) and \(\Lambda\) are respectively the smallest and largest eigenvalues of \(k_{AB}\). The map \(U\) satisfies the Euler-Lagrange system that in local coordinates can be written as \[ \square U^ a + \Gamma^ a_{bc} \partial_ \mu U^ b \partial^ \mu U^ c = 0,\tag{1} \] where \(\square = -\partial^ 2_ t + \partial^ 2_ r + r^{-1} \partial_ r\), and \(\Gamma\)'s are the Christoffel symbols of the target \(N\). Here we obtain the asymptotic behavior of these solutions. In particular, it will be shown that the asymptotic behavior is like that of the linear wave equation.
scientific article; zbMATH DE number 495436