[Paper Review] Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities

Dong-Hyun Kim, Hideaki Sunagawa|arXiv (Cornell University)|Jul 30, 2013

Advanced Mathematical Physics Problems13 references3 citations

TL;DR

This paper establishes improved $L^p$-decay estimates for small solutions to systems of cubic nonlinear Klein-Gordon equations in one space dimension with dissipative nonlinearities. Under a structural condition on the nonlinearity, it proves that solutions decay like $O((1+t)^{-(1/2-1/p)} / \sqrt{\log(2+t)})$ in $L^p$-norm for $2 \leq p \leq \infty$, gaining an additional logarithmic decay factor beyond the free evolution rate.

ABSTRACT

We consider the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution gains an additional logarithmic decay in comparison with the free evolution in the sense of $L^p$, $2\le p \le \infty$.

Motivation & Objective

• To analyze the large-time decay behavior of small-amplitude solutions to systems of cubic nonlinear Klein-Gordon equations in one space dimension.
• To identify structural conditions on the nonlinearity that lead to enhanced decay beyond the free evolution rate.
• To extend previous results on single-field nonlinear Klein-Gordon equations to the case of coupled systems with dissipative cubic nonlinearities.
• To establish sharp $L^p$-decay estimates, including logarithmic improvements, for both the solution and its derivatives.
• To address the open problem of obtaining pointwise asymptotic profiles in the presence of nonlinear dissipation in multi-component systems.

Proposed method

• Derives a system of ordinary differential equations in the self-similar variable $\tau = t/\omega_0(z)$, where $z = x/t$, by transforming the original PDE system.
• Introduces a complex-valued amplitude function $\beta(\tau, z)$ to represent the solution in the self-similar regime, enabling asymptotic analysis.
• Applies energy-type estimates using a modified inner product $\langle \cdot, A \cdot \rangle_{\mathbb{C}^N}$ to control the growth of $\beta$.
• Imposes a structural condition on the nonlinearity $F$ that ensures negative definite imaginary part of the nonlinear interaction, leading to damping.
• Uses logarithmic weightings in the energy functional to capture the additional decay, specifically $ (\log(\tau \omega_0(z)))^2 $, to derive decay bounds.
• Combines finite propagation speed with pointwise $L^\infty$ estimates to extend the decay result to $L^p$-norms for $2 \leq p \leq \infty$.

Experimental results

Research questions

• RQ1 Can small solutions to systems of cubic nonlinear Klein-Gordon equations in one space dimension exhibit improved decay rates due to dissipative nonlinearities?
• RQ2 What structural conditions on the nonlinearity ensure that the solution decays faster than the free evolution in $L^p$-norm?
• RQ3 Does the presence of nonlinear dissipation (e.g., $-(\partial_t u)^3$) lead to a logarithmic improvement in decay rates compared to the standard $t^{-(1/2-1/p)}$ rate?
• RQ4 Can the asymptotic profile of such solutions be characterized in terms of self-similar variables and logarithmic corrections?
• RQ5 How does the decay behavior change when the nonlinearity is not of the standard cubic type but includes dissipative terms?

Key findings

• Under a suitable structural condition on the nonlinearity, small solutions to the system of cubic Klein-Gordon equations in one space dimension gain an additional logarithmic decay factor in $L^p$-norm for $2 \leq p \leq \infty$.
• The solution decays as $\|u(t,\cdot)\|_{L^p} \leq C (1+t)^{-(1/2-1/p)} / \sqrt{\log(2+t)}$, which improves upon the free evolution rate $t^{-(1/2-1/p)}$.
• The logarithmic decay is a consequence of the dissipative nature of the nonlinearity, which induces a damping effect in the asymptotic profile via the imaginary part of the nonlinear interaction.
• The method establishes $L^\infty$-type pointwise estimates that are then extended to $L^p$-norms using finite propagation speed and scaling arguments.
• For first-order derivatives, the same decay rate $O((1+t)^{-(1/2-1/p)} / \sqrt{\log(2+t)})$ is obtained, confirming the robustness of the logarithmic improvement.
• The result confirms that nonlinear dissipation leads to a long-range effect distinct from the standard cubic nonlinearity, even in multi-component systems.