Blow-up for semilinear wave equations with slowly decaying data in high dimensions
type: Item
instance of: scholarly article
title: Blow-up for semilinear wave equations with slowly decaying data in high dimensions
zbMATH Open document ID: 0848.35017
author: H. Takamura
published in: Differential and Integral Equations
publication date: 1995-02-01T00:00:00Z
review text: Nonlinear wave equations are considered (for \(t, r\geq 0\)) \[ (1)\quad u_{tt}- u_{rr}- [(n- 1)/r] u_r= F(u),\quad \text{and} \quad(2)\quad u_{tt}- u_{rr}- [(n- 1)/r] u_r= F(u_r), \] with the initial conditions \(u(r, 0)= 0\), \(u_r(r, 0)= \varepsilon \psi\). It is assumed that \(F(s)\geq \text{const} |s|^p\), \(p> 1\), and that \(\psi\geq \text{const}(1+ r)^{- \kappa}\), where \(0< \kappa< (p+ 1)/(p- 1)\) in case (1) and \(0< \kappa< 1/(p- 1)\) in case (2). It is shown that the life span of the solution (or the blow up time) is of order \(\varepsilon^{-(p- 1)/[p+ 1 - (p- 1) \kappa]}\). The result depends on careful estimates of the spherical wave representation of the solution.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 716376
zbMATH Keywords: semilinear wave equations
zbMATH Keywords: initial data with noncompact support
zbMATH Keywords: blow up time
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 716376