Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schr\"{o}dinger equations

TL;DR

This paper investigates the conditions under which solutions to a class of quasilinear Schrödinger equations exist globally or blow up in finite time, analyzing the effects of nonlinearities and providing thresholds and bounds.

Contribution

It introduces new criteria for global existence and blowup, including sharp thresholds and blowup rate bounds, for a broad class of quasilinear Schrödinger equations.

Findings

Established sufficient conditions for finite-time blowup.

Derived criteria for global existence of solutions.

Constructed sharp thresholds for blowup and global solutions.

Abstract

In this paper, we study the Cauchy problem of the quasilinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2uh'(|u|^2)\Delta h(|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.