Differential and Integral Equations

Degenerate Volterra equations in Banach spaces

Angelo Favini and Hiroki Tanabe

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper is concerned with degenerate Volterra equations $Mu(t)+\int_0^tk(t-s)Lu(s)\, ds=f(t)$ in Banach spaces, both in the hyperbolic case and the parabolic one. The key assumption is played by the representation of the underlying space $X$ as a direct sum $X=N(T)\oplus\overline{R(T)}$, where $T$ is the bounded linear operator $T=ML^{-1}$. Hyperbolicity means that the part $\tilde{T}$ of $T$ in $\overline{R(T)}$ is an abstract potential operator, i.e., $-\tilde{T}^{-1}$ generates a $C_0$-semigroup, and parabolicity means that $-\tilde{T}^{-1}$ generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel $k(\cdot)$ is degenerate or singular at $t=0$ using the results of Prüss [8] on analytic resolvents. Finally we consider the case where $\lambda$ is a pole for $(\lambda L+M)^{-1}$.

Article information

Source
Differential Integral Equations, Volume 14, Number 5 (2001), 613-640.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123260

Mathematical Reviews number (MathSciNet)
MR1824747

Zentralblatt MATH identifier
1013.45005

Subjects
Primary: 45N05: Abstract integral equations, integral equations in abstract spaces
Secondary: 35K65: Degenerate parabolic equations 35L80: Degenerate hyperbolic equations 45D05: Volterra integral equations [See also 34A12]

Citation

Favini, Angelo; Tanabe, Hiroki. Degenerate Volterra equations in Banach spaces. Differential Integral Equations 14 (2001), no. 5, 613--640. https://projecteuclid.org/euclid.die/1356123260


Export citation