[Paper Review] Energy decay for small solutions to semilinear wave equations with weakly dissipative structure

Yoshinori Nishii, Hideaki Sunagawa+1 more|arXiv (Cornell University)|Feb 22, 2020

Advanced Mathematical Physics Problems27 references6 citations

TL;DR

This paper establishes energy decay for small data solutions to two-dimensional semilinear wave equations with weakly dissipative structure, proving that the energy decays like $\|u(t)\|_E \leq C\varepsilon / (1 + \varepsilon^2 \log(t+2))^\lambda$ under the Agemi condition (A) and the quadratic null condition, even when the cubic null condition fails. The result extends known decay rates under stronger structural assumptions to a broader class of nonlinearities via pointwise estimates and a modified Grönwall-type argument.

ABSTRACT

This article gives an energy decay result for small data solutions to a class of semilinear wave equations in two space dimensions possessing weakly dissipative structure relevant to the Agemi condition.

Motivation & Objective

• To establish energy decay for small data solutions to 2D semilinear wave equations with weakly dissipative structure.
• To determine whether energy decay occurs when the Agemi condition (A) holds but the cubic null condition and (A+) are violated.
• To extend known decay results beyond the cubic null condition by analyzing the interplay between the quadratic null condition and the Agemi condition.
• To derive sharp decay rates for the energy norm in terms of the vanishing order of the cubic nonlinearity's profile on the unit sphere.

Proposed method

• Derives a detailed pointwise estimate for the solution under the quadratic null condition and the Agemi condition (A), using commuting vector fields and weighted energy estimates.
• Introduces a transformation $U(t,x) = D(|x|^{1/2}u(t,x))$ to reduce the wave equation to a first-order ODE in $t$ for fixed $\sigma = r - t$ and $\omega = x/|x|$.
• Applies a modified Grönwall-type lemma (Lemma 6.3) to the resulting ODE $\partial_t V = -P(\omega)/(2t) V^3 + G(t)$, where $P(\omega) = F_c(\hat{\omega})$.
• Uses the structure of $P(\omega)$ to bound the solution $V(t; \sigma, \omega)$ and derive decay in $\log t$ via the vanishing order of $P(\omega)$ at its zeros.
• Combines pointwise estimates with weighted $L^2$ energy norms to control $\|u(t)\|_E$ and derive the final decay rate.
• Employs the radiation field and asymptotic analysis to handle the long-time behavior of the solution in the weakly dissipative regime.

Experimental results

Research questions

• RQ1 Does energy decay occur for small solutions to 2D semilinear wave equations when the Agemi condition (A) holds but the cubic null condition fails?
• RQ2 What is the optimal decay rate for the energy norm $\|u(t)\|_E$ under the Agemi condition (A) and the quadratic null condition?
• RQ3 How does the decay rate depend on the vanishing order of the cubic nonlinearity $F_c(\partial u)$ on the unit sphere?
• RQ4 Can the decay rate be quantified in terms of the profile $P(\omega) = F_c(\hat{\omega})$?

Key findings

• The energy norm decays as $\|u(t)\|_E \leq C\varepsilon / (1 + \varepsilon^2 \log(t+2))^\lambda$ for some $\lambda > 0$, under the Agemi condition (A) and the quadratic null condition, even when the cubic null condition is violated.
• The decay rate $\lambda$ is determined by the maximum vanishing order $2\nu$ of $P(\omega) = F_c(\hat{\omega})$ on the unit circle, with $\lambda = 1/(4\nu) - \delta$ for arbitrarily small $\delta > 0$.
• For $F_c(\partial u) = -(∂_1 u)^2 \partial_t u$, $P(\omega) = \omega_1^2$, which vanishes to order 2, yielding $\lambda = 1/4 - \delta$, so $\|u(t)\|_E = O((\log t)^{-1/4 + \delta})$.
• For $F_c(\partial u) = -(∂_1 u)^2(\partial_t u + \partial_2 u)$, $P(\omega) = \omega_1^2(1 - \omega_2)$, vanishing to order 4 at $\omega = (0,1)$ and 2 at $\omega = (0,-1)$, so $\nu = 2$, and $\|u(t)\|_E = O((\log t)^{-1/8 + \delta})$.
• For $F_c(\partial u) = -(∂_t u + \partial_2 u)^3$, $P(\omega) = (1 - \omega_2)^3$, vanishing to order 6 at $\omega = (0,1)$, so $\nu = 3$, and $\|u(t)\|_E = O((\log t)^{-1/12 + \delta})$.