[Paper Review] Decay estimates for nonradiative solutions of the energy-critical focusing wave equation

Thomas Duyckaerts, Carlos E. Kenig+1 more|arXiv (Cornell University)|Dec 16, 2019

Advanced Mathematical Physics ProblemsMathematics25 references18 citations

TL;DR

This paper establishes sharp decay estimates for radial, nonradiative solutions of the energy-critical focusing wave equation in odd space dimensions $N \geq 3$. By leveraging a characterization of nonradiative solutions to the linear wave equation in odd dimensions, the authors prove that such solutions must exhibit a prescribed asymptotic behavior at spatial infinity, which is essential for proving the soliton resolution conjecture in the radial, odd-dimensional case. The key result is the classification of nonradiative solutions as being either zero or constant multiples of the ground state $W$, extending previous results from $N=3$ to all odd $N$.

ABSTRACT

Consider the energy-critical focusing wave equation in space dimension $N\\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution whose energy in the exterior $\\{|x|>|t|\\}$ of the wave cone vanishes asymptotically as $t\ o +\\infty$ and $t\ o -\\infty$. In a previous work (Cambridge Journal of Mathematics 2013, arXiv:1204.0031), we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, $0$ and $\\pm W$. This was crucial in the proof of soliton resolution in 3 space dimension. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behaviour as $r\ o \\infty$. We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag and the second author (Advances in Mathematics, 2015, arXiv:1409.3643) . We also study the propagation of the support of nonzero radial solutions with compactly supported initial data, and prove that these solutions cannot be nonradiative.

Motivation & Objective

• Establish the asymptotic behavior of radial, nonradiative solutions to the energy-critical focusing wave equation in odd space dimensions $N \geq 3$.
• Extend the classification of nonradiative solutions beyond $N=3$ to all odd dimensions, building on prior results in three dimensions.
• Provide decay estimates for the initial data of nonradiative solutions, showing they must decay with a specific rate at spatial infinity.
• Use these decay estimates to support the proof of the soliton resolution conjecture for radial, bounded energy solutions in odd dimensions.
• Prove that solutions with compactly supported initial data cannot be nonradiative, by establishing a uniform lower bound on the exterior energy.

Proposed method

• Utilize the characterization of nonradiative solutions to the linear wave equation in odd dimensions, established by Lawrie, Liu, Schlag, and Kenig in [15], as a foundational tool.
• Apply profile decomposition and exterior energy estimates to analyze the long-time behavior of solutions in the energy space $\mathcal{H}_{\text{rad}}$.
• Derive decay estimates for the initial data of nonradiative solutions by analyzing the asymptotic expansion of the solution in terms of spherical harmonics and radial modes.
• Use finite speed of propagation and radial Sobolev inequalities to control the growth of the nonlinearity $|u|^{\frac{4}{N-2}}u$ in the exterior region.
• Establish a lower bound on the exterior energy of solutions with compactly supported initial data by contradiction, showing it cannot vanish asymptotically.
• Employ a recursive inequality argument with geometric growth sequences (Appendix A) to control the growth of coefficients in the asymptotic expansion, proving uniform decay rates.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of radial, nonradiative solutions to the energy-critical focusing wave equation in odd space dimensions $N \geq 3$?
• RQ2 How do the initial data of nonradiative solutions decay at spatial infinity in odd dimensions?
• RQ3 Can solutions with compactly supported initial data be nonradiative, and what constraints does this impose on their exterior energy?
• RQ4 Does the classification of nonradiative solutions as $0$ or $\pm W$ extend beyond $N=3$ to all odd dimensions?
• RQ5 What role do decay estimates for nonradiative solutions play in the proof of the soliton resolution conjecture for radial, bounded energy solutions?

Key findings

• In odd space dimensions $N \geq 3$, radial nonradiative solutions of the energy-critical focusing wave equation must have initial data that decay as $r^{-k_{0}+\frac{1}{2}}$ at infinity, where $k_0$ is an integer determined by the solution's frequency content.
• The asymptotic decay rate $k_0$ is invariant in time, and the leading coefficient $\ell(T)$ satisfies $\ell(T) = \ell(0)$, implying a precise conservation law for the leading term.
• Nonradiative solutions in odd dimensions are classified as either identically zero or constant multiples of the ground state $W$, extending the $N=3$ result to all odd $N$.
• Any radial solution with compactly supported initial data cannot be nonradiative, as it satisfies a uniform positive lower bound on the exterior energy for all time.
• The exterior energy of such solutions remains bounded away from zero for all $t$, with $\int_{R+|t|}^{\infty} |\nabla_{t,x}u(t,x)|^2 r^{N-1} dr \geq \frac{1}{8} \int_R^\infty |\nabla_{t,x}u(0,x)|^2 r^{N-1} dr > 0$ for $R < \rho_0$.
• Using recursive estimates with geometric growth, the authors prove that the coefficients in the asymptotic expansion of nonradiative solutions grow at most like $C r^n$ or $C q^n$, depending on the parameters, ensuring uniform control.