Asymptotic behavior of solutions to Eells-Sampson equations near stable harmonic maps
The i.v. problem (ES) \(\partial f/\partial t=\Delta f\) on \(M\times [0,\infty)\), \(f(0)=f_ 0\) on M is considered. The following theorem is proved: Theorem. Let (M,g) and (N,h) be closed Riemannian manifolds, and F: \(M\to N\) be a stable harmonic map. There exists a positive number \(\epsilon\) such that if \(\| F-f_ 0\|_{H^ m(M,N)}<\epsilon\) for some \(m>\dim M+2\), then (ES) with initial value \(f_ 0\in H^ m(M,N)\) has a unique global solution in \(C^{1/2}([0,\infty);H^{m- 1}(M,N))\cap C^ 0([0,\infty);H^ m(M,N))\), and converge to F in \(H^ m(M,N)\) with exponential order as \(t\to \infty\).
scientific article; zbMATH DE number 4128100