Pointwise decay for semilinear wave equations in $\mathbb{R}^{!+3}$

Abstract

In this paper, we study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in R 3+1 . We prove that the solution decays as quickly as linear waves for p > 1+ √ 17 2, covering part of the subconformal case, while for the range 2 < p ≤ 1+ √ 17 2 , the solution still decays with rate at least t − 1 3 . As a consequence, the solution scatters in energy space when p > 2.3542.. As a consequence, the solution scatters in energy space for p > 2.7005.The aim of this paper are two folds: firstly we obtain pointwise decay estimate for the solution with data in some weighted energy space which is weaker than the conformal energy space required in previous works. We prove that the solution decays as quickly as linear waves (with the same initial data) for all p > 1+ √ 17 2, covering additional part of the subconformal range. Secondly for even smaller p with lower bound 2, we show that the solution decays at least t − 1 3 . This decay estimate immediately leads to the

On the defocusing semilinear wave equations in three space dimension with small power

“…Combined with the well known integrated local energy estimates (see for example [12]), a pigeon-hole argument leads to the improved time decay of the potential energy. This enables the second author in [18] to show that the solution decays at least t − 1 3 for 2 < p < 5 in three space dimension. The lower bound p > 2 arises due to the fact that the pigeon-hole argument works only for γ > 1.…”

“…Extensions could be found for example in [1], [6], [9], [13], [17]. The latest work [18] of the second author shows that the solution scatters to linear wave in energy space for 2.3542 < p < 5 in R 1+3 . However the precise asymptotics of the solutions remains unclear for small power p. One of the difficulties is that the equation degenerates to linear Klein-Gordon equation when p approaches to the end point 1.…”

“…Again, the essential idea is to introduce some new weighted vector fields as multipliers, which are partially inspired by our previous work [15] in space dimension two. This allows us to obtain potential energy decay for all 1 < p < 5 and time decay of the solutions for all p > √ 2, and hence filling the gap left in [18].…”

“…Remark 1.1. The theorem implies that the solution decays along out going null curves (|t| − |x| is constant) for all 1 < p ≤ 5 (see [18] for the case when 2 < p < 5 and [8] for the energy critical case). In other words, the solution vanishes on the future (and past) null infinity and blowing up can only occur at time infinity.…”

Pointwise decay for semilinear wave equations on Kerr spacetimes

“…In particular, it is well-known that for small initial data there is a unique global solution if p > 1 + √ 2, see [14], [12], [34]. In terms of pointwise decay of solutions, there are a number of results, see for example [28], [33], [37]. In the case of compactly supported smooth data, the optimal decay rate is…”