TY - JOUR

T1 - The defocusing energy-critical wave equation with a cubic convolution

AU - Miao, Changxing

AU - Zhang, Junyong

AU - Zheng, Jiqiang

N1 - Publisher Copyright:
Indiana University Mathematics Journal ©

PY - 2014

Y1 - 2014

N2 - In this paper, we study the theory of the global wellposedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity utt - Δu + (|x|-4∗|u|2)u = 0 in spatial dimension d ≥ 5. The main difficulties are the absence of the classical finite speed of propagation (i.e., the monotonic local energy estimate on the light cone), which is a fundamental property to show global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity utt - Δu +|u|4/(d2)u = 0. To compensate for this, we resort to the extended causality and use the strategy derived from concentration compactness ideas. Then, the proof of global well-posedness and scattering is reduced to show the nonexistence of three enemies: finite-time blowup, soliton-like solutions, and low-to-high cascade. We use the Morawetz estimate, the extended causality, and the potential energy concentration to preclude the above three enemies.

AB - In this paper, we study the theory of the global wellposedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity utt - Δu + (|x|-4∗|u|2)u = 0 in spatial dimension d ≥ 5. The main difficulties are the absence of the classical finite speed of propagation (i.e., the monotonic local energy estimate on the light cone), which is a fundamental property to show global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity utt - Δu +|u|4/(d2)u = 0. To compensate for this, we resort to the extended causality and use the strategy derived from concentration compactness ideas. Then, the proof of global well-posedness and scattering is reduced to show the nonexistence of three enemies: finite-time blowup, soliton-like solutions, and low-to-high cascade. We use the Morawetz estimate, the extended causality, and the potential energy concentration to preclude the above three enemies.

KW - Concentration compactness

KW - Extended causality

KW - Morawetz estimate

KW - Scattering

KW - Wave-Hartree equation

UR - http://www.scopus.com/inward/record.url?scp=84908286545&partnerID=8YFLogxK

U2 - 10.1512/iumj.2014.63.5271

DO - 10.1512/iumj.2014.63.5271

M3 - Article

AN - SCOPUS:84908286545

SN - 0022-2518

VL - 63

SP - 993

EP - 1015

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 4

ER -