Global existence and lifespan of smooth solutions of biwave maps

Abstract

We show that if f:R1+m→N is a biwave map from a Minkowski spacetime into a Riemannian manifold N with initial data such that the spatial dimension m > 6 and the Riemannian curvature of N satisfies certain conditions, then there exists a small constant ϵ0 such that the biwave map f has a unique global smooth solution for ϵ ∈ [0, ϵ0) (where the parameter ϵ indicates the size of the initial data). In case 1 ≤ m ≤ 6, we obtain the lifespan of the smooth solution of the above biwave map. Furthermore, if f:M→N is a biwave map on a Friedmann–Lemaître–Robertson–Walker spacetime under certain circumstances, we obtain the lifespan of the smooth solution of the biwave map for spatial dimension m ≥ 1.