Global existence for rough solutions of a fourth-order nonlinear wave equation
type: Item
instance of: scholarly article
title: Global existence for rough solutions of a fourth-order nonlinear wave equation
zbMATH Open document ID: 1200.35191
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-07-20T00:00:00Z
review text: Consider the Cauchy problem of the defocusing cubic forth-order wave equation \[ \partial^2_t u+\Delta^2 u+u+|u|^2u=0, \quad u|_{t=0}=u_0, \partial_t u|_{t=0}=u_1, \qquad (t,x)\in \mathbb R\times \mathbb R^n\;. \] It is easy to check that, for regular solutions, we have the conserved energy quantity \[ E(t)=\int_{\mathbb R^n} \left(|\partial_t u|^2+|\Delta u|^2+|u|^2+\tfrac{1}{2}|u|^4 \right)\,dx\simeq \|u\|_{H^2}^2+\|\partial_t u\|_{L^2}^2+\|u\|_{L^4}^4, \] which, as usual, can be exploited to prove the global well-posedness for the problem in \(H^2\times L^2\), when \(n<8\) (see \textit{B. Pausader} [J. Differ. Equations 241, No. 2, 237--278 (2007; Zbl 1145.35090)] for \(n\geq 5\) with general \(p\)). In the case \(n\leq 6\) (\(H^2\subset L^6\)), it follows from the classical energy argument without using the Strichartz type estimates. In this paper, the author studied the global well-posedness in \(H^s\times H^{s-2}\). The main result states that when \(3\leq n< 8\) and \(\min(\frac{n-2}{2},\frac{n}{4})<s<2\), the problem is globally well-posed in \(H^s\times H^{s-2}\). The proof follows from \textit{J. Bourgain}'s Fourier truncation idea [Int. Math. Res. Not. 1998, No. 5, 253--283 (1998; Zbl 0917.35126)], together with the Strichartz estimates for the corresponding linear equation in [\textit{B. Pausader}, loc. cit.].
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5757905
zbMATH Keywords: low regularity
zbMATH Keywords: Strichartz type estimates
zbMATH Keywords: defocusing cubic forth-order wave equation
zbMATH Keywords: global well-posedness
author: Junyong Zhang
reviewed by: Chengbo Wang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.04.003
OpenAlex ID: W1987541728
cites work: On global well-posedness for defocusing cubic wave equation
cites work: Q4393968
cites work: Q4255122
cites work: Analyticity of the scattering operator for semilinear dispersive equations
cites work: Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation.
cites work: On global solutions to a defocusing semi-linear wave equation.
cites work: Global well-posedness for semi-linear wave equations
cites work: On the optimal local regularity for the Yang-Mills equations in ℝ⁴⁺¹
cites work: On existence and scattering with minimal regularity for semilinear wave equations
cites work: \(H^{s}\)-global well-posedness for semilinear wave equations
cites work: Q4821882
cites work: Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations
cites work: Analyticity of the nonlinear scattering operator
cites work: Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on \(\mathbb R^3\)
cites work: Introduction to scattering for radial 3D NLKG below energy norm
cites work: Low regularity semi&mdash;linear wave equations1
cites work: Q5475508
DOI: 10.1016/J.JMAA.2010.04.003
description: scientific article; zbMATH DE number 5757905