Almost global existence for quadratic Schrödinger equations in two dimensions

Xuan Liu

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arXiv · 2610.34237

Almost global existence for quadratic Schrödinger equations in two dimensions

Xuan Liu

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Abstract

We study the two-dimensional quadratic Schrödinger equation $(i\partial_t+Δ)u=λ_1u^2+λ_2\overline u^{\,2}$ with initial data of size $\varepsilon$ in a weighted $H^2$ space. We prove almost global existence, with a lifespan of at least $\exp(c/\varepsilon)$ and decay at the optimal linear dispersive rate $t^{-1}$. Our proof is based on the space--time resonance method introduced by Germain, Masmoudi, and Shatah. The main new ingredient is a Besov-type profile norm $\mathcal Z$, which allows us to close the bootstrap argument without controlling the global second spatial momentof the free profile, thereby avoiding the substantial time-dependent losses arising from direct second-moment estimates.

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Xuan Liu. 2026-10-25. Almost global existence for quadratic Schrödinger equations in two dimensions. https://arxiv.org/abs/2610.34237

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