A wave equation with structural damping and nonlinear memory
type: Item
instance of: scholarly article
title: A wave equation with structural damping and nonlinear memory
zbMATH Open document ID: 1320.35218
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2015-02-23T00:00:00Z
review text: The author obtains a critical exponent for a wave equation with structural damping and nonlinear memory \(u_{tt}-\Delta u+\mu (-\Delta)^{1/2} u_t=\int_0^t (t-s)^{-\gamma} |u(s)|^p ds\), posed for \(x\in \mathbb{R}^n\), where \(\mu>0\) and \(\gamma\in (0,1)\). Let \(\bar p(n,\gamma):=\max(1+\frac{3-\gamma}{n+\gamma-2}, \frac{1}{\gamma})\) with \(n\geq 2\). It is proved that when \(p>\bar p(n,\gamma)\), we have global existence for sufficiently small data. On the other hand, in the special case that \(\mu=2\), it is proved that there is no global solutions for suitable arbitrarily small data, when \(1<p<\bar p(n,\gamma)\) for any \(n\geq 2\) and \(1<p<\infty\) for \(n=1\). This shows that the critical power for the problem to admit global small solutions is \(\bar p(n,\gamma)\).
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6407191
zbMATH Keywords: global existence
author: M. D'Abbicco
reviewed by: Chengbo Wang
MaRDI profile type: Publication
full work available at URL: s00030-014-0265-2
OpenAlex ID: W2124655802
cites work: Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space
cites work: A maximum principle applied to quasi-geostrophic equations
cites work: Local and global existence of solutions to semilinear parabolic initial value problems
cites work: The influence of a nonlinear memory on the damped wave equation
cites work: Diffusion phenomena for the wave equation with structural damping in the \(L^p-L^q\) framework
cites work: An application of \(L^p-L^q\) decay estimates to the semi-linear wave equation with parabolic-like structural damping
cites work: Semi-linear wave equations with effective damping
cites work: Semilinear structural damped waves
cites work: Nonlinear perturbations of the kirchhoff equation
cites work: High-order energy decay for structural damped systems in the electromagnetical field
cites work: Critical exponent for damped wave equations with nonlinear memory
cites work: Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation
cites work: Asymptotic profiles for wave equations with strong damping
cites work: Energy decay estimates for wave equations with a fractional damping.
cites work: Global existence of solutions for semilinear damped wave equations in \(\mathbb R^N\) with noncompactly supported initial data
cites work: Wave equations with strong damping in Hilbert spaces
cites work: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations
cites work: Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations.
cites work: Rates of decay for structural damped models with decreasing in time coefficients
cites work: A decay estimate for a class of hyperbolic pseudo-differential equations
cites work: \(L^p\)-\(L^q\) estimates for damped wave equations and their applications to semi-linear problem
cites work: L 1 Estimates for Oscillating Integrals Related to Structural Damped Wave Models
cites work: Q4233638
cites work: Q3105452
cites work: Q4726489
cites work: Q4946797
cites work: Critical exponent for a nonlinear wave equation with damping
DOI: 10.1007/S00030-014-0265-2
description: scientific article