Explosion for some semilinear wave equations
type: Item
instance of: scholarly article
title: Explosion for some semilinear wave equations
zbMATH Open document ID: 0679.35068
DOI: 10.1016/0022-0396(88)90016-2
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/27226
review text: The author studies the explosion of the solution to the semilinear wave equation with quadratic nonlinearity. First of all, he treats the one- dimensional case and reduces the semilinear wave equation as well as the Carleman model of transport theory to the following nonlinear (2\(\times 2)\) system \[ (1)\quad \partial_{\pm}u_{\pm}=f_{\pm}(u_+,u_- ),\quad \partial_{\pm}=\partial_ t\pm \partial_ x,\quad f_{\pm}=A_{\pm}u^ 2_{\pm}+B_{\pm}u_+u_-+C_{\pm}u^ 2_{\pm}, \] where \(A_{\pm}\), \(B_{\pm}\), \(C_{\pm}\) are constants. The first result asserts that, when \(A_+\neq 0\) or \(C_-\neq 0\), one can find initial data \(u^ 0_{\pm}\in L^{\infty}({\mathbb{R}})\), piecewise \(C^{\infty}\) and of compact support such that the Cauchy problem for (1) with initial data \(u_{\pm}(0,x)=\epsilon u^ 0_{\pm}(x)\) does not have a global solution for any \(\epsilon\neq 0\). The proof is based on comparison of \(L^{\infty}\) norm of \(u_+\), \(u_-\) with the non- negative solution of a suitable ordinary differential equation. The case of the nonlinear wave equation in higher dimensional space is considered too.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 4113165
zbMATH Keywords: explosion of solutions
zbMATH Keywords: wave equation
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: Carleman model
zbMATH Keywords: transport theory
author: Jeffrey Rauch
reviewed by: Vladimir Georgiev
MaRDI profile type: Publication
OpenAlex ID: W2062385938
cites work: Q3736173
cites work: Q3736177
cites work: Q3736172
cites work: Global existence for nonlinear wave equations
cites work: Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: creation and propagation
cites work: Discontinuous progressing waves for semilinear systems
description: scientific article; zbMATH DE number 4113165