## Differential and Integral Equations

- Differential Integral Equations
- Volume 12, Number 1 (1999), 41-65.

### Absorption evolution families and exponential stability of non-autonomous diffusion equations

Frank Räbiger and Roland Schnaubelt

#### Abstract

For a positive, strongly continuous evolution family $\mathcal{U}$ on $L^p(\Omega)$ and a
positive, measurable, time dependent potential $V(\cdot)$ we construct a corresponding
*absorption evolution family* $\mathcal {U}_V$ by a procedure introduced by J.
Voigt, [24, 25], in the autonomous case. We give sufficient conditions on $\mathcal{U}$
and $V$ such that $\mathcal {U}_V$ is strongly continuous and satisfies variation of
constants formulas. For an evolution family $\mathcal {U}$ on $L^1(\mathbb{R}^N)$
satisfying upper and lower Gaussian estimates exponential stability of $\mathcal {U}_V$ is
characterized by a condition on the size of $V$ extending recent results by W.Arendt and
C.J.K. Batty, [2, 3, 6], in the autonomous case and by D. Daners, M. Hieber, P. Koch
Medina, and S. Merino, [8, 12], in the time periodic case. An application to a second
order parabolic equation with real coefficients and singular potential is given.

#### Article information

**Source**

Differential Integral Equations, Volume 12, Number 1 (1999), 41-65.

**Dates**

First available in Project Euclid: 29 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367266993

**Mathematical Reviews number (MathSciNet)**

MR1668533

**Zentralblatt MATH identifier**

1015.34041

**Subjects**

Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Secondary: 35K15: Initial value problems for second-order parabolic equations 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]

#### Citation

Räbiger, Frank; Schnaubelt, Roland. Absorption evolution families and exponential stability of non-autonomous diffusion equations. Differential Integral Equations 12 (1999), no. 1, 41--65. https://projecteuclid.org/euclid.die/1367266993