Linear and Nonlinear Tails Ii: Exact Decay Rates in Spherical Symmetry

Abstract

We derive the exact late-time asymptotics for small spherically symmetric solutions of nonlinear wave equations with a potential. The dominant tail is shown to result from the competition between linear and nonlinear effects.

Linear and Nonlinear Tails I: General Results and Perturbation Theory

“…Further, we develop perturbation theory for these equations and by its means argue that presented estimates give optimal decay rates at late times. A rigorous proof of this fact will appear in a following publication [3] (part II).…”

“…This will be studied in more detail in a forthcoming publication [3] dealing with spherical symmetry where the lowest perturbation orders can be calculated almost explicitly.…”

“…p in the norm L ∞ 1,p is optimal) then theorem 1 gives an optimal estimate for u ∈ L ∞ 1,p with the same decay rate p. Here, we consider only the asymptotics in direction of timelike infinity (the case of spatial infinity can be treated similarly). Assume, we are able to show (by some explicit calculation, like in [3]…”

“…It is expected that it will be true for u 2 ∼ = εd 2 (x)t −q , what means, that already u 2 would have the same rate of decay as u (see [3] for such results in spherical symmetry).…”

“…It is the same as that of the given perturbation order because all higher terms in the perturbation series, summed up, are too small to be able to modify the asymptotics. The issue of optimal decay estimates and precise asymptotics compared with numerical results will be addressed in a forthcoming publication [3] which will be focused on spherical symmetry where the perturbation equations can be solved almost exactly.…”