Strongly damped semilinear equations
type: Item
instance of: scholarly article
title: Strongly damped semilinear equations
zbMATH Open document ID: 0841.34062
DOI: 10.1155/S1048953395000360
author: Dhirendra Bahuguna
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1996-07-25T00:00:00Z
full work available at URL: https://eudml.org/doc/47447
review text: If \(\Omega\subset \mathbb{R}^n\) is a bounded domain with sufficiently smooth boundary \(\partial\Omega\) and if the differential operator \(Lu= \sum_{|\alpha|\leq 2m} a_\alpha(x) D^\alpha u\) is strongly elliptic of order \(2m\) in \(\Omega\), the strongly damped wave equation \[ u_{tt}+ (aL+ b) u_t+ (cL+ d)u= f(t, u,u_t)\quad \text{in}\quad \Omega\times (t_0, T), \] \[ u(x, t_0)= u_0\quad\text{and} \quad u_t(x, t_0)= u_1,\;x\in \Omega, \] \[ D^\alpha u= 0\quad\text{for} \quad (x, t)\in \partial \Omega\times [t_0, T),\;|\alpha|\leq m- 1 \] is considered as a special case of the abstract second-order semilinear differential equation in a Banach space \(X\), \[ u''(t)+ (aA+ bI) u'(t)+ (cA+ dI)u= f(t, u(t), u'(t)),\;t> t_0, \] \[ u(t_0)= u_0,\;u'(t_0)= u_1 \] with \(A= L\) and assuming that \(-A\) generates an analytic semigroup \(T(t)\) in \(X\). Sufficient conditions on \(T\), \(A\) and \(f\) are given to assure the existence and uniqueness of local and global classical solutions.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35A07
zbMATH DE Number: 847757
zbMATH Keywords: strongly damped wave equation
zbMATH Keywords: abstract second-order semilinear differential equation
zbMATH Keywords: Banach space
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: global classical solutions
MaRDI profile type: Publication
OpenAlex ID: W2005728115
description: scientific article; zbMATH DE number 847757