Optimal regularity and stability analysis in the \(\alpha-\)norm for a class of partial functional differential equations with infinite delay
type: Item
instance of: scholarly article
title: Optimal regularity and stability analysis in the \(\alpha-\)norm for a class of partial functional differential equations with infinite delay
zbMATH Open document ID: 1219.34098
author: Aziz Ouhinou
author: Abdelhai El Azzouzi
published in: Discrete and Continuous Dynamical Systems
publication date: 2011-05-16T00:00:00Z
review text: The paper deals with the partial functional differential equation with infinite delay \[ \begin{cases}\frac{dx}{dt}(t)=-Ax(t)+F(x_t),&\text{for }t\geq 0,\\ x_0=\varphi\in {\mathcal B},&{}\end{cases}\tag{P} \] where \(-A\) generates an analytic semigroup \((T(t))_{t\geq 0}\) on a Banach space \(X\), and the phase space \(\mathcal B\) is the Banach space of functions mapping \((-\infty,0]\) to \(X\), which satisfies some fundamental axioms. The function \(F:{\mathcal B}_{\alpha}\to X\) is continuously differentiable with \(F'\) locally Lipschitz, where \({\mathcal B}_{\alpha}\) is the subspace of \(\mathcal B\) defined by \[ {\mathcal B}_{\alpha}=\{\varphi\in {\mathcal B};\;\varphi(\theta)\in D(A^{\alpha})\text{ for }\theta\leq 0\text{ and }A^{\alpha}\varphi\in{\mathcal B}\}, \] for \(0<\alpha<1\), \(A^{\alpha}\) denotes the fractional power of \(A\), the function \(A^{\alpha}\varphi\) is defined by \((A^{\alpha}\varphi)(\theta):=A^{\alpha}(\varphi(\theta))\) for \(\theta\leq 0\), and, for every \(t\geq 0\), the history function \(x_t\in {\mathcal B}_{\alpha}\) is defined by \(x_t(\theta):=x(t+\theta)\) for \(\theta\leq 0\). The authors establish necessary and sufficient conditions for the existence of strict solutions for problem (P); then they give a new characterisation of the infinitesimal generator of the solution semigroup and study the regularity of mild solutions. Spectral properties of the infinitesimal generator of the semigroup solution and the asymptotic behavior for the solution semigroup are also investigated. An example of a partial differential equation with infinite delay which illustrates the obtained results is finally presented.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 5888332
zbMATH Keywords: partial functional differential equation
zbMATH Keywords: infinite delay
zbMATH Keywords: analytic semigroup
zbMATH Keywords: fractional power of operators
zbMATH Keywords: mild and strict solutions
zbMATH Keywords: uniform fading memory space
zbMATH Keywords: essential spectrum
zbMATH Keywords: stability
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: dcds.2011.30.115
OpenAlex ID: W2040309286
DOI: 10.3934/DCDS.2011.30.115
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description: scientific article; zbMATH DE number 5888332