Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations

TL;DR

This work analyzes a general quasilinear Schrödinger equation on $\mathbb{R}^N$ with nonlinearities encoded by $F(|u|^2)$ and a quasilinear term via $h(|u|^2)$. It develops a comprehensive framework: (i) global existence criteria (Theorem 1), (ii) finite-time blowup conditions (Theorem 2), and (iii) a sharp focusing threshold (Theorem 3) based on virial-type functionals and a ground-state energy level. The authors further derive a pseudo-conformal conservation law (Theorem 4) and establish detailed asymptotic behavior for global solutions, including decay rates and lower bounds on blow-up rates (Theorem 5). The results extend classical semilinear NLS theory to a broad class of quasilinear models, illuminating how the interplay between the quasilinear term and the nonlinearity shapes long-time dynamics and stability of solutions.

Abstract

In this paper, we study the Cauchy problem of the quasilinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+F(|u|^2)u \quad {\rm for} \ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, with various choices for models from mathematical physics. We examine the interplay between the quasilinear effect of $h$ and nonlinear effect of $F$ for the global existence and blowup phenomena. We provide sufficient conditions on the blowup in finite time and global existence of the solution. In some cases, we can deduce the watershed from these conditions. In the focusing case, we construct the sharp threshold for the blowup in finite time and global existence of the solution and lower bound for blowup rate of the blowup solution.