## Differential and Integral Equations

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

### Degenerate Volterra equations in Banach spaces

Angelo Favini and Hiroki Tanabe

#### 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