Scattering of solutions to NLW by Inward Energy Decay

Abstract

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ${\partial }_t^{2}u-\Delta u=-|u{|}^{p-1}u$ in the 3-dimensional space ($3\le p<5$) whose initial data are radial and come with a finite energy. In this work we prove scattering in the positive time direction by only assuming the inward part of the energy decays at a certain rate, as long as the total energy is finite, regardless of the decay rate or size of the outward energy. More precisely, we assume the initial data comes with a finite energy and \[ \int_{{\mathbb R}^3} \max\{1,|x|^κ\}\ (\ |\nabla u_0(x)\cdot \frac{x}{|x|} + \frac{u_0(x)}{|x|} + u_1(x)\ |^2 + \frac{2}{p+1}|u_0(x)|^{p+1}\ ) dx < \infty. \] Here $\kappa \ge {\kappa }_0(p)=\frac{5-p}{p+1}$ is a constant. If $\kappa >{\kappa }_0(p)$, we can also prove $\Vert u{\Vert }_{L^p{L}^{2p}}({\mathbb{R}}^{+}\times {\mathbb{R}}^3)<+\infty$ and give an explicit rate of $u$'s convergence to a free wave.

24 pages, 4 figures

References in corpus (1)

• Scattering for the radial 3d cubic wave equation

Cited by in corpus (3)

• Inward/outward Energy Theory of Wave Equation in Higher Dimensions
• Inward/outward Energy Theory of Non-radial Solutions to 3D Semi-linear Wave Equation
• Scattering theory for subcritical wave equation with inverse square potential