[Paper Review] Scattering of solutions to NLW by Inward Energy Decay

Ruipeng Shen|arXiv (Cornell University)|Sep 4, 2019

Advanced Mathematical Physics Problems9 references4 citations

TL;DR

This paper establishes scattering for radial, energy-subcritical defocusing nonlinear wave equations in 3D by requiring only inward energy decay at a specific rate, without assumptions on outward energy. Using a novel weighted energy estimate and a one-dimensional comparison argument, it proves global scattering in positive time and provides an explicit convergence rate for solutions to free waves under mild decay conditions on initial data.

ABSTRACT

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - Δu = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) 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 \[ \int_{{\mathbb R}^3} \max\{1,|x|^κ\}\ (\ | abla u_0(x)\cdot \frac{x}{|x|} + \frac{u_0(x)}{|x|} + u_1(x)\ |^2 + \frac{2}{p+1}|u_0(x)|^{p+1}\ ) dx < \infty. \] Here $κ\geq κ_0(p) = \frac{5-p}{p+1}$ is a constant. If $κ>κ_0(p)$, we can also prove $\|u\|_{L^p L^{2p}}({\mathbb R}^+ imes {\mathbb R}^3)< +\infty$ and give an explicit rate of $u$'s convergence to a free wave.

Motivation & Objective

• To establish scattering for radial solutions to the 3D defocusing nonlinear wave equation with finite energy, under minimal assumptions beyond energy finiteness.
• To remove the need for strong decay or size assumptions on outward energy by focusing on inward energy decay.
• To prove global existence and scattering in the positive time direction using only a weighted inward energy decay condition.
• To derive an explicit convergence rate for solutions to free waves under stronger decay assumptions on initial data.

Proposed method

• Introduces a weighted energy functional involving radial derivatives and the radial component of the solution, with weight $ |x|^{\kappa} $, $ \kappa > \kappa_0(p) = \frac{5-p}{p+1} $, to control inward energy decay.
• Applies a one-dimensional reduction via the radial wave equation in $ r $, transforming the problem into a 1D wave equation with a nonlinear term $ -\frac{|w|^{p-1}w}{r^{p-1}} $, where $ w = ru $.
• Uses d'Alembert's formula to express solutions in terms of initial data and nonlinear integrals over characteristic triangles.
• Employs a bootstrap argument with a smallness condition on initial data in weighted norms to control nonlinear growth.
• Applies a comparison lemma (Lemma 5.1) to bound nonlinear integrals using the size of the solution in the domain of dependence.
• Combines energy conservation and Strichartz-type estimates to control $ L^p L^{2p} $ norms and prove scattering in $ \dot{H}^1 \times L^2 $.

Experimental results

Research questions

• RQ1 Can scattering be proven for 3D defocusing NLW with finite energy and only inward energy decay, without assumptions on outward energy?
• RQ2 What is the minimal decay rate of initial data required to ensure scattering in the positive time direction?
• RQ3 Can an explicit convergence rate for the solution to a free wave be derived under stronger decay assumptions?
• RQ4 How does the solution behave in the exterior region $ |x| > t + R $ under the inward energy decay condition?
• RQ5 What is the sharp threshold for the weight exponent $ \kappa $ that enables the scattering result?

Key findings

• Scattering in $ \dot{H}^1 \times L^2 $ is proven for radial solutions to the 3D defocusing NLW with $ p \in [3,5) $, under the sole assumption that $ \int_{\mathbb{R}^3} (1 + |x|^\kappa) \left( |\nabla u_0|^2 + |u_1|^2 + |u_0|^{p+1} \right) dx < \infty $ for $ \kappa > \kappa_0(p) = \frac{5-p}{p+1} $.
• For $ \kappa > \kappa_0(p) $, the solution satisfies $ \|u\|_{L^p L^{2p}(\mathbb{R}^+ \times \mathbb{R}^3)} < \infty $, indicating strong integrability in space-time.
• An explicit convergence rate to a free wave is obtained: $ \|u(t) - v(t)\|_{\dot{H}^1 \times L^2} \to 0 $ as $ t \to \infty $, under the same condition.
• The solution satisfies $ |w(r,t)| \leq 3c r^\beta $ for $ r \geq t + R $, where $ w = ru $, $ \beta = \frac{2}{p-1} $, and $ c $ is small, implying $ u(r,t) \simeq c r^{-\frac{2}{p-1}} $ in the exterior region.
• An example is constructed where initial data decay like $ |x|^{-\frac{2}{p-1}} $, satisfying the weighted energy condition for $ \kappa \in (\frac{5-p}{p+1}, \frac{5-p}{p-1}) $, yet $ \|u\|_{L^{2(p-1)} L^{2(p-1)}} = \infty $, showing the sharpness of the $ L^p L^{2p} $ norm condition.
• The result holds without any restriction on the size or decay of outward energy, demonstrating that inward energy decay alone suffices for scattering.