Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation
type: Item
instance of: scholarly article
title: Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation
zbMATH Open document ID: 1151.35052
author: Shun-Tang Wu
author: L. Y. Tsai
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2006-08-14T00:00:00Z
review text: The authors study the following initial boundary value problem for nonlinear wave equation of Kirchhoff type with a dissipative term: \[ \begin{gathered} u_{tt}- M(\|\nabla u\|^2)\Delta u+ g(u_t)= f(u),\quad (t,x)\in\Omega\times [0,\infty),\\ u(x,0)= u_0(x),\;u_t(0,x)= u_1(x),\quad x\in\Omega,\\ u(x,t)= 0,\quad (t,x)\in(0,\infty)\times \partial\Omega,\end{gathered} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with boundary \(\partial\Omega\) so that divergence theorem can be applied, \(\|\cdot\|\) denotes the usual \(L^2\) norm in \(\Omega\), \(M\) is a nonnegative locally Lipschitz function, \(f\) is a nonlinear function typified by \(f(u)= |u|^{p-2}u\) with \(p> 2\) and \(g(u_t)\) is the strong dissipative term \(-\Delta u_t\) or the linear dissipative term \(u_t\) or nonlinear dissipative term \(|u_t|^{m-2} u_t\) with \(m> 2\). Then they prove the existence of a local solution for \(g(u_t)= u_t\) and \(g(u_t)= -\Delta u_t\) with \(p\leq(2N- 4)/(N- 4)\) for \(N> 4\) and \(p<\infty\) for \(N\leq 4\), and for \(|u_t|^{m-2} u_t\) with \(p\leq (2N- 6)/(N- 4)\) for \(N> 4\) and \(p<\infty\) for \(N\leq 4\). Moreover, the authors prove the existence of a blow-up solution for \[ {sf(s)\over 2\int^s_0 f(r)\,dr}\geq 1+\delta\geq {sM(s)\over \int^s_0 M(r)\,dr} \] with \(s> 0\) and \(\delta> 0\) under some suitable assumptions to the initial energy.
reviewed by: Fumihiko Hirosawa
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 5045903
zbMATH Keywords: Kirchhoff equations
zbMATH Keywords: dissipation
zbMATH Keywords: blow-up
zbMATH Keywords: life span
zbMATH Keywords: nonlinear wave equations
MaRDI profile type: Publication
full work available at URL: j.na.2004.11.023
OpenAlex ID: W1982241160
cites work: REMARKS ON BLOW-UP AND NONEXISTENCE THEOREMS FOR NONLINEAR EVOLUTION EQUATIONS
cites work: Existence of a solution of the wave equation with nonlinear damping and source terms
cites work: Blow-up theorems for nonlinear wave equations
cites work: On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms
cites work: On solutions of nonlinear wave equations
cites work: Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Pu tt = -Au + ℱ(u)
cites work: Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations
cites work: Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients
cites work: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation
cites work: Global nonexistence theorems for quasilinear evolution equations with dissipation
cites work: Existence and nonexistence of global solutions of some system of semilinear wave equations
cites work: Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation
cites work: Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation
cites work: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings
cites work: On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation
cites work: On continuity of functions with values in various Banach spaces
cites work: Global nonexistence theorems for a class of evolution equations with dissipation
DOI: 10.1016/J.NA.2004.11.023
description: scientific article