Finite time blow-up for damped wave equations with space-time dependent potential and nonlinear memory
type: Item
instance of: scholarly article
title: Finite time blow-up for damped wave equations with space-time dependent potential and nonlinear memory
zbMATH Open document ID: 1415.35201
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2019-02-28T00:00:00Z
review text: The article refers to the Cauchy problem for the nonlinear damped wave equation \[ \begin{aligned} u_{tt}-\Delta u+a(x)b(t)u_t={J^{\delta}}_{0|t}(|u|^p)(t),\\ u(0,x)=u_o(x),\quad u_t(0,x)=u_1(x),\quad t0, x\in \mathbb{R}^n, \end{aligned} \] where $\delta=1-\gamma$ and ${J^{\delta}}_{0|t}$ is the Riemann-Liouville integral operator. The coefficient of the damping term is given by $a(x)b(t):=a_o(1+{|x|^2})^{-{\alpha}/2}(1+t)^{-\beta}$ and initial data are assumed to belong to the energy space. In cases $n=1,2$ and $p\in(1,\infty)$ or $n\geq 3$ and $1p\leq n/(n-2)$, the existence of a unique maximal mild solution is proved. In both cases, under some additional assumptions on coefficients and data one proves that the mild solution of the stated problem blows up in a finite time. For the proof, the well known test function method is applied.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 7031110
zbMATH Keywords: local existence
zbMATH Keywords: subcritical potential
zbMATH Keywords: Riemann-Liouville integral operator
zbMATH Keywords: test function method
author: Ihab Dannawi
author: Mokhtar Kirane
author: Ahmad Z. Fino
MaRDI profile type: Publication
full work available at URL: s00030-018-0533-7
OpenAlex ID: W2885682372
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DOI: 10.1007/S00030-018-0533-7
Wikidata QID: Q129383785
description: scientific article; zbMATH DE number 7031110