On the decay estimate for small solutions to nonlinear Klein-Gordon equations with dissipative structure

TL;DR

The paper analyzes the cubic nonlinear Klein–Gordon equation in one space dimension with small initial data and identifies dissipative nonlinearities that yield faster decay than the free solution. By expressing the nonlinearity through the complex function $K_F(z)$ and the polynomial $P_F(y)$, the authors classify nonlinearities into regimes $(A_0)$, $(B_j)$, and $(C)$, and prove corresponding $L^p$-decay estimates for $u$ and its derivatives. The core method combines a reduction to hyperbolic coordinates, a profile equation for a complex amplitude, and Sunagawa-type asymptotics to derive leading-order behavior, with the decay rates explicitly depending on the dissipative structure (logarithmic corrections appear in several cases). The results provide a unified framework for understanding how dissipative structure controls long-time decay in 1D NLKG and include concrete examples illustrating each regime. The work advances the theory of nonlinear dispersive decay by linking dissipativity, asymptotic profiles, and precise $L^p$-decay rates, with potential implications for related equations and higher-dimensional analogues.

Abstract

We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the $L^p$-decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity has the suitable dissipative structure.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• Proposition 3.1
• Lemma 3.1