A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on $\mathbb {R}^2$

Shen, Ruipeng; Staffilani, Gigliola

doi:10.1090/s0002-9947-2015-06513-1

Cited by 7 publications

(14 citation statements)

References 60 publications

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“…The method of conformal conservation laws assumes that E 2 (u 0 , u 1 ) < +∞. The initial data in works [42,49] satisfy E 1+ε (u 0 , u 1 ) < +∞. The decay rate of initial data in the scattering results of this work is lower than these previously known results.…”

Section: Resultsmentioning

confidence: 68%

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Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

Li¹,

Shen²,

Wei³

2021

Preprint

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We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends to infinity. In addition, the inward/outward part of energy gradually vanishes as time tends to positive/negative infinity. These behaviours resemble those of free waves. We also prove some decay estimates of the solutions if the initial data decay at a certain rate as the spatial variable tends to infinity. As an application, we prove a couple of scattering results for solutions whose initial data are in a weighted energy space. Our assumption on decay rate of initial data is weaker than previous known scattering results.

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Section: Resultsmentioning

confidence: 68%

“…In a joint work [42] with Staffilani, the second author applies a transformation (introduced in Tataru [48]) between wave equation in Euclidean space and shifted wave equation in hyperbolic space and proves the scattering of solutions to the quintic equation p = 5, if the initial data are radial and satisfy…”

Section: Local Theorymentioning

confidence: 99%

Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

Li¹,

Shen²,

Wei³

2021

Preprint

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“…A key property we need in the proof is an integrated local energy decay estimate; see (10.11). As we will see, a single crucial multiplier argument, which goes back to [28,53], underlies all three proofs; see Lemma 10.1 below.…”

Section: Integrated Local Energy Decay Morawetz Estimates and Strichmentioning

confidence: 92%

Profile decompositions for wave equations on hyperbolic space with applications

Lawrie

Shahshahani

2015

Math. Ann.

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Abstract. The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gérard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential.Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive timeindependent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t → ±∞. This proof will serve as a blueprint for the arguments in a forthcoming work, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

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“…Sobolev Embedding As in Euclidean Spaces, we have the Sobolev embedding H 0,1 (H 3 ) ֒→ L 6 (H 3 ). (Please see [50] for more details.) This implies that the energy…”

Section: Set-up Of the Proofmentioning

confidence: 99%

A semi-linear energy critical wave equation with an application

Shen

2016

Journal of Differential Equations

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In this work we consider a semi-linear energy critical wave equation in R d (3 ≤ d ≤ 5)Here the function φ ∈ C(R d ; (0, 1]) converges to zero as |x| → ∞. We follow the same compactness-rigidity argument as Kenig and Merle applied in their paper [32] on the Cauchy problem of the equationu and obtain a similar result when φ satisfies some technical conditions. In the defocusing case we prove that the solution scatters for any initial data in the energy spaceḢ 1 × L 2 . While in the focusing case we can determine the global behaviour of the solutions, either scattering or finite-time blow-up, according to their initial data when the energy is smaller than a certain threshold.By the Sobolev embeddingḢ 1 (R d ) ֒→ L 2 * (R d ), the energy E φ (u 0 , u 1 ) is finite for any initial data (u 0 , u 1 ) ∈Ḣ 1 × L 2 (R d ).

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A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on $\mathbb {R}^2$

Cited by 7 publications

References 60 publications

Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

Energy Distribution of solutions to defocusing semi-linear wave equation in two dimensional space

Profile decompositions for wave equations on hyperbolic space with applications

A semi-linear energy critical wave equation with an application

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