Identification of the memory kernel in the strongly damped wave equation by a flux condition
type: Item
instance of: scholarly article
title: Identification of the memory kernel in the strongly damped wave equation by a flux condition
zbMATH Open document ID: 1191.45008
author: Fabrizio Colombo
author: Davide Guidetti
published in: Communications on Pure and Applied Analysis
publication date: 2009-03-20T00:00:00Z
review text: The authors consider an abstract parabolic integrodifferential problem which is motivated by an initial and boundary value problem for a semilinear strongly damped wave equation with memory effects. The problem is formulated as follows. Let \(X\) be a Banach space and let \(A\) be a sectorial operator in \(X\), with domain \(D(A)\). Further, let~\(B\) be a linear operator of domain \(D(A)\) and let \(f\) be a nonlinear perturbation. Then, given \(p\in(1,\infty)\) and \(T>0\), and knowing function \(G\in W^{1,p}(0,T;X)\) and initial value \(U_0\in D(A)\), one looks for \(\tau\in(0,T]\) and \(U\in W^{2,p}(0,\tau;X)\cap W^{1,p}(0,\tau;D(A))\), \(h\in L^p(0,\tau)\), satisfying the system \[ U'(t)=AU(t)+h*BU(t)+f(U(t))+G(t),\quad t\in(0,\tau),\quad U(0)=U_0, \] where \[ (h*f)(t):=\int_0^t h(t-s)f(s)\,ds. \] The supplementary information, necessary to find \(h\) together with~\(U\), is given by the condition \[ \Phi_0(U(t))+\Phi_1(h*U(t))=g(t),\quad t\in(0,\tau), \] where \(g\in W^{1,p}(0,T)\) and~\(\Phi_0\), \(\Phi_1\) are functionals defined in~\(D(A)\). Moreover, \(\Phi_1\neq 0\). Applying common techniques from analytic semigroup theory, the authors formulate and prove first the local in time existence and uniqueness theorem. Then, assuming that the nonlinearity \(f\) has bounded Fréchet-derivative, they proceed with the global in time existence and uniqueness result.
Mathematics Subject Classification ID: 45Q05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45N05
zbMATH DE Number: 5530379
zbMATH Keywords: abstract parabolic integrodifferential equation
zbMATH Keywords: identification of memory kernel
zbMATH Keywords: local in time existence
zbMATH Keywords: global in time existence
zbMATH Keywords: uniqueness
zbMATH Keywords: strongly damped wave equation with memory
reviewed by: Oleh Omel'chenko
MaRDI profile type: Publication
full work available at URL: cpaa.2009.8.601
OpenAlex ID: W2073116165
DOI: 10.3934/CPAA.2009.8.601
description: scientific article; zbMATH DE number 5530379