Lifespan of solutions to the damped wave equation with a critical nonlinearity
type: Item
instance of: scholarly article
title: Lifespan of solutions to the damped wave equation with a critical nonlinearity
zbMATH Open document ID: 1361.35114
author: Masahiro Ikeda
author: Takayoshi Ogawa
published in: Journal of Differential Equations
publication date: 2016-05-19T00:00:00Z
reviewed by: Satyanad Kichenassamy
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 6582628
zbMATH Keywords: blow-up time
zbMATH Keywords: small data
zbMATH Keywords: Fujita exponent
MaRDI profile type: Publication
full work available at URL: j.jde.2016.04.016
OpenAlex ID: W2345440008
cites work: A modified test function method for damped waves
cites work: Semi-linear wave equations with effective damping
cites work: Q5547841
cites work: Existence of a solution of the wave equation with nonlinear damping and source terms
cites work: On nonexistence of global solutions of some semilinear parabolic differential equations
cites work: Damped wave equation with super critical nonlinearities.
cites work: On the critical nonlinear damped wave equation with large initial data
cites work: Large time behavior and \(L^{p}\)-\(L^{q}\) estimate of solutions of 2-dimensional nonlinear damped wave equations
cites work: Small data blow-up of \(L^2\) or \(H^1\)-solution for the semilinear Schrödinger equation without gauge invariance
cites work: A note on the lifespan of solutions to the semilinear damped wave equation
cites work: Decay estimates of solutions for dissipative wave equations in \(\mathbb R^N\) with lower power nonlinearities
cites work: Global existence of solutions for semilinear damped wave equations in \(\mathbb R^N\) with noncompactly supported initial data
cites work: Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential
cites work: Fujita’s Exponent for a Semilinear Wave Equation with Linear Damping
cites work: Q4444800
cites work: Global Existence, Large Time Behavior and Life Span of Solutions of a Semilinear Parabolic Cauchy Problem
cites work: Breakdown of solutions to \(\square u+u_ t=| u| ^{1+\alpha}\)
cites work: Critical exponent for the semilinear wave equation with time-dependent damping
cites work: On the asymptotic behavior of solutions of semi-linear wave equations
cites work: \(L^p\)-\(L^q\) estimates for damped wave equations and their applications to semi-linear problem
cites work: \(L^p\)-\(L^q\) estimates of solutions to the damped wave equation in 3-dimensional space and their application
cites work: Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping
cites work: Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations
cites work: On the existence and nonexistence of solutions of some quasilinear hyperbolic equations
cites work: On the dependence of the critical exponent of the nonlinear heat equation on the initial function
cites work: Q3995410
cites work: Q5187767
cites work: The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
cites work: Critical exponent for a nonlinear wave equation with damping
cites work: Weighted \(L^2\)-estimates for dissipative wave equations with variable coefficients
cites work: Small data global existence for the semilinear wave equation with space-time dependent damping
cites work: Blow-up results for nonlinear parabolic equations on manifolds
cites work: A blow-up result for a nonlinear wave equation with damping: the critical case
cites work: A blow up result for a nonlinear wave equation with damping and vanishing initial energy in \(\mathbb R^N\)
cites work: Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations
DOI: 10.1016/J.JDE.2016.04.016
review text: The main result of this paper is an inequality of the form \(T(\varepsilon)\leq\exp(C\varepsilon^{-p})\) for the blow-up time for a mild solution of NEWLINE\[NEWLINE u_{tt}-\Delta u + u_{t}=|u|^p, NEWLINE\]NEWLINE where the spatial domain is all of \(\mathbb R^n\), with \(n\geq 1\), \(p=1+2/n\), and the data have the form \((\varepsilon u_0, \varepsilon u_1)\), with \((u_0,u_1)\in H^1\times L^2\), \(u_0+u_1\in L^1\), and \(\int_{\mathbb R^n}(u_0+u_1)dx>0\). The method is similar to that of \textit{M. Kirane} and \textit{M. Qafsaoui} [Adv. Nonlinear Stud. 2, No. 1, 41--49 (2002; Zbl 1056.35020)] and \textit{Q. S. Zhang} [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 2, 109--114 (2001; Zbl 1056.35123)].
description: scientific article; zbMATH DE number 6582628