[Paper Review] Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions

Satoshi Masaki, Jun‐ichi Segata|arXiv (Cornell University)|Dec 1, 2016

Advanced Mathematical Physics Problems22 references3 citations

TL;DR

This paper establishes modified scattering for the quadratic nonlinear Klein-Gordon equation in two space dimensions by constructing a solution that converges to a given asymptotic profile with a logarithmic phase correction. Using Fourier series expansion of the nonlinearity and a refined approximate solution, the authors prove global existence and optimal decay rates, showing that the solution approaches the profile in $ L^2 $ at rate $ t^{-d} $ for $ 1/2 < d < 1 $, despite the nonlinearity's lack of smoothness.

ABSTRACT

In this paper, we consider the long time behavior of solution to the quadratic gauge invariant nonlinear Klein-Gordon equation (NLKG) in two space dimensions. For a given asymptotic profile, we construct a solution to (NLKG) which converges to given asymptotic profile as t goes infinity. Here the asymptotic profile is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

Motivation & Objective

• To address the final state problem for the quadratic nonlinear Klein-Gordon equation in two dimensions, where standard scattering fails due to non-smooth nonlinearity.
• To construct a solution that asymptotically approaches a given profile $ u_{\text{ap}} $, which includes a logarithmic phase correction.
• To overcome the challenge of the critical nonlinearity $ |u|u $, which lies outside the scope of prior results due to lack of smoothness.
• To establish global existence and optimal $ L^2 $-decay for the solution under small initial data.

Proposed method

• Construct an approximate solution $ u_{\text{ap}} $ as the leading term of the linear solution with a logarithmic phase correction.
• Decompose the nonlinearity using Fourier series expansion to handle the lack of smoothness in $ |u|u $.
• Define a correction term $ v_{\text{ap}} $ to absorb residual nonlinear terms and improve approximation accuracy.
• Use weighted norms in frequency space, particularly involving $ \hat{\phi}_0, \hat{\phi}_1 $, and their derivatives, to control growth.
• Apply energy estimates and $ L^2 $-based bounds to show that the residual error decays as $ t^{-2} (\log t)^2 $.
• Prove that the full solution $ u $ to the nonlinear equation converges to $ u_{\text{ap}} $ in $ L^2 $ at rate $ t^{-d} $ for $ 1/2 < d < 1 $.

Experimental results

Research questions

• RQ1 Can modified scattering be established for the quadratic nonlinear Klein-Gordon equation in two dimensions when the nonlinearity $ |u|u $ lacks smoothness?
• RQ2 What is the correct asymptotic profile that captures the long-time behavior of solutions, and how should it differ from the linear solution?
• RQ3 How can one construct a solution that converges to a given profile in $ L^2 $ despite the nonlinearity being at the borderline of short- and long-range behavior?
• RQ4 What role does the logarithmic phase correction play in stabilizing the asymptotic profile and enabling the construction of a global solution?

Key findings

• The solution $ u $ to the quadratic nonlinear Klein-Gordon equation in two dimensions converges to the asymptotic profile $ u_{\text{ap}} $ in $ L^2 $ as $ t \to \infty $, with decay rate $ \|u - u_{\text{ap}}\|_{L^2} \leq C t^{-d} $ for $ 1/2 < d < 1 $.
• The asymptotic profile $ u_{\text{ap}} $ includes a logarithmic phase correction, which is essential for capturing the correct long-time dynamics.
• The residual error between the approximate solution and the true nonlinear equation decays as $ t^{-2} (\log t)^2 $, ensuring sufficient accuracy for the construction.
• The method successfully handles the non-smooth nonlinearity $ |u|u $ by using Fourier series expansion to decompose and control the nonlinear interaction.
• The construction relies on a weighted norm $ \|(.\,.\|_Y $ involving frequency derivatives of initial data, ensuring control over high-frequency contributions.
• The result confirms that modified scattering occurs for the critical nonlinearity $ |u|u $ in 2D, extending previous results beyond the smoothness threshold.