A remark on decay rates of solutions for a system of quadratic nonlinear Schr\"odinger equations in 2D

Abstract

We consider the initial value problem for a three-component system of quadratic nonlinear Schr\"odinger equations with mass resonance in two space dimensions. Under a suitable condition on the coefficients of the nonlinearity, we will show that the solution decays strictly faster than $O(t^{-1})$ as $t \to +\infty$ in $L^{\infty}$ by providing with an enhanced decay estimate of order $O((t \log t)^{-1})$. Differently from the previous works, our approach does not rely on the explicit form of the asymptotic profile of the solution at all.