The diffusion phenomenon for damped wave equations with space-time dependent coefficients
type: Item
instance of: scholarly article
title: The diffusion phenomenon for damped wave equations with space-time dependent coefficients
zbMATH Open document ID: 1411.35187
author: Montgomery Taylor
published in: Discrete and Continuous Dynamical Systems
publication date: 2018-10-10T00:00:00Z
review text: The paper investigates the long-time behavior of solutions for IVP to damped wave equations of the form \(u_{tt}+u_t-\nabla\cdot (a(x,t)\nabla u)=0\) with initial data \(u_0(x)\), \(u_1(x)\), \(x\in \mathbb{R}^N\). Decay estimates in \({L_x}^2\) norm for the solution, its gradient and its derivative with respect to \(t\) are obtained and a large discussion on improved decay is done. Results on exponentially decay outside a ball of the derivatives of the solution are shown. Under some special assumptions one proves that the solution \(u(x, t)\) behaves like the solution to the diffusion problem \(v_t-\nabla\cdot (a(x,t)\nabla v)=0\), \(v(x,0)=u_0(x)+u_1(x)\). In the acknowledgments part, the author writes: ``I would like to ask Kat Ashdown, the most caring person I have ever known, to marry me''.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A08
zbMATH DE Number: 6951278
zbMATH Keywords: decay estimates
zbMATH Keywords: improved decay
zbMATH Keywords: weighted energy
MaRDI profile type: Publication
full work available at URL: dcds.2018257
OpenAlex ID: W2888047779
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DOI: 10.3934/DCDS.2018257
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description: scientific article; zbMATH DE number 6951278