TY - JOUR

T1 - Global center stable manifold for the defocusing energy critical wave equation with potential

AU - Jia, Hao

AU - Liu, Baoping

AU - Schlag, Wilhelm

AU - Xu, Guixiang

N1 - Publisher Copyright:
© 2020 by Johns Hopkins University Press.

PY - 2020/10

Y1 - 2020/10

N2 - In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension 3. We prove that the set of initial data for which solutions scatter to an unstable excited state (φ, 0) forms a finite co-dimensional path connected C1 manifold in the energy space. This manifold is a global and unique center-stable manifold associated with (φ, 0). It is constructed in a first step locally around any solution scattering to φ, which might be very far away from φ in the Ḣ1 × L2 (R3) norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends a previous result of ours to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from Beceanu-Goldberg to construct a local center stable manifold near any solution that scatters to (φ, 0). This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle, which is used to show that solutions that start off but near the local manifolds associated with φ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state φ.

AB - In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension 3. We prove that the set of initial data for which solutions scatter to an unstable excited state (φ, 0) forms a finite co-dimensional path connected C1 manifold in the energy space. This manifold is a global and unique center-stable manifold associated with (φ, 0). It is constructed in a first step locally around any solution scattering to φ, which might be very far away from φ in the Ḣ1 × L2 (R3) norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends a previous result of ours to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from Beceanu-Goldberg to construct a local center stable manifold near any solution that scatters to (φ, 0). This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle, which is used to show that solutions that start off but near the local manifolds associated with φ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state φ.

UR - http://www.scopus.com/inward/record.url?scp=85093883602&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85093883602&partnerID=8YFLogxK

U2 - 10.1353/ajm.2020.0038

DO - 10.1353/ajm.2020.0038

M3 - Article

AN - SCOPUS:85093883602

SN - 0002-9327

VL - 142

SP - 1497

EP - 1557

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 5

ER -