Scattering for the radial 3d cubic wave equation

Abstract

Consider the Cauchy problem for the radial cubic wave equation in 1+3 dimensions with either the focusing or defocusing sign. This problem is critical in ${\dot{H}}^{\frac{1}{2}}\times {\dot{H}}^{-\frac{1}{2}}$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time-interval of existence, then the solution must in fact be global-in-time and scatter to free waves as $t\to \pm \infty$.

Minor revisions. To appear in Analysis and PDE. 29 pages

References in corpus (4)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}
• Exotic blowup solutions for the u^5 focusing wave equation in R^3
• Blowup behaviour for the nonlinear Klein--Gordon equation

Cited by in corpus (21)

• Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm
• Global well-posedness and scattering for the radial, defocusing, cubic wave equation with almost sharp initial data
• Global well-posedness and scattering for the radial, defocusing, cubic nonlinear wave equation
• Scattering theory for the radial ${\dot{H}}^{1/2}$-critical wave Equation with a cubic convolution
• Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations
• Scattering for defocusing energy subcritical nonlinear wave equations
• Inward/outward Energy Theory of Wave Equation in Higher Dimensions
• Scattering of solutions to NLW by Inward Energy Decay
• Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials
• Scattering for radial energy-subcritical wave equations
• Numerical simulations for the energy-supercritical nonlinear wave equation
• Inward/outward Energy Theory of Non-radial Solutions to 3D Semi-linear Wave Equation
• Scattering theory for subcritical wave equation with inverse square potential
• Scattering of solutions to the defocusing energy sub-critical semi-linear wave equation in 3D
• Global well-posedness and scattering for the radial, defocusing, cubic wave equation with initial data in a critical Besov space
• Morawetz Estimates Method for Scattering of Radial Energy Sub-critical Wave Equation
• Energy Distribution of Radial Solutions to Energy Subcritical Wave Equation with an Application on Scattering Theory
• Dynamics of nonlinear wave equations
• Exterior scattering of non-radial solutions to energy subcritical wave equations
• A sharp Lorentz-invariant Strichartz norm expansion for the cubic wave equation in ${\mathbb{R}}^{1+3}$
• Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space