Global Spherically Symmetric Solutions of Non-linear Wave Equations with Null Condition on Extremal Reissner–Nordström Spacetimes

Abstract

We study spherically symmetric solutions of semilinear wave equations in the case where the nonlinearity satisfies the null condition on extremal Reissner-Nordström black hole spacetimes. We show that solutions which arise from sufficiently small compactly supported smooth data prescribed on a Cauchy hypersfurace Σ0 crossing the future event horizon H + are globally well-posed in the domain of outer communications up to and including H + . Our method allows us to close all bootstrap estimates under very weak decay results (compatible with those known for the linear case). Moreover we establish a certain number of non-decay and blow-up results along the horizon H + which generalize known instability results for the linear case. Our results apply to spherically symmetric wave maps for a wide class of target spaces.

A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

“…We can write (2) ∂ r (χΦ (2) ), J 2 [χΦ (2) ] . = r p+1 [−D r + 2D + 2D r −1 ]χΦ∂ r (χΦ (2) ), J 3 [χΦ (2) ] . = r p+3 [D + 4D r −1 + 2D r −2 ]χφ∂ r (χΦ (2) ).…”

“…where the inequality becomes an equality if ψ = ψ =2 . Hence, the bulk term on the right-hand side of (4.16) gets absorbed (or exactly cancelled) by the / ∇(χΦ (2) ) term in J 0 . Similarly, we can absorb the boundary integral on the right-hand side of (4.16) by the flux integral of J V [χΦ (2) ] · L at I + , where an exact cancellation occurs again if ψ = ψ =2 .…”

“…In this section, we will obtain additional r p -weighted estimates for the quantity ∂ k r T Φ (2) , with k ≥ 0, using the r p -weighted estimates from the previous sections. As an intermediate step, we will obtain r p -weighted estimates for the commuted quantities ∂ k r Φ (2) .…”

“…Finally, we put the above estimates together and apply the divergence theorem on divJ V [χ∂ n+1 r Φ (2) ] to conclude that (4.19) also holds for k = n + 1.…”

The Trapping Effect on Degenerate Horizons

“…Ori [28] and Sela [32] have numerically investigated the relation of long time dynamics of scalar fields and the conservation laws. Rigorous non-linear results have appeared in [10,2]. Further applications and extensions have been presented in [18,34,15,8].…”