Formation of singularities in solutions of a wave equation
type: Item
instance of: scholarly article
title: Formation of singularities in solutions of a wave equation
zbMATH Open document ID: 0942.35116
DOI: 10.1016/S0893-9659(99)00027-0
published in: Applied Mathematics Letters
publication date: 2000-08-15T00:00:00Z
review text: The author studies the following Cauchy problem: \[ \begin{aligned} w_{tt}(x,t) &=a( x)(\varphi(w_x(x,t))w_{xx}, x\in \mathbb R, t>0,\\ w(x,0)&=u_0(x), w_t(x,0)=v_0(x,t), x\in \mathbb R, \end{aligned} \] where \(a>0, \varphi >0\). Suppose that \(a\in W^{1,\infty}(\mathbb R)\), \(\varphi \in C^2(\mathbb R)\), \(a(x)\geq\alpha\), \(\varphi(\xi)\geq \alpha'\), \(x,\xi\in \mathbb R\), \(\alpha>0\). If \(u_0,v_0 \in H^2(\mathbb R)\), the Cauchy problem has a unique local solution such that \(u,u_t\in C([0,T); H^2(\mathbb R)) \cap C^1([0,T);H^1(\mathbb R))\). Moreover, the author proves that if \(a''\in L^{\infty}(\mathbb R)\), \(\varphi'(0)<0\), then there exist the initial data \(u_0,v_0\) such that the solution blows up in finite time.
reviewed by: Andrei Perjan
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1369342
zbMATH Keywords: nonlinear wave equation
zbMATH Keywords: blow up
author: Salim A. Messaoudi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1369342