1999 Absorption evolution families and exponential stability of non-autonomous diffusion equations
Frank Räbiger, Roland Schnaubelt
Differential Integral Equations 12(1): 41-65 (1999). DOI: 10.57262/die/1367266993

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.

Citation

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Frank Räbiger. Roland Schnaubelt. "Absorption evolution families and exponential stability of non-autonomous diffusion equations." Differential Integral Equations 12 (1) 41 - 65, 1999. https://doi.org/10.57262/die/1367266993

Information

Published: 1999
First available in Project Euclid: 29 April 2013

zbMATH: 1015.34041
MathSciNet: MR1668533
Digital Object Identifier: 10.57262/die/1367266993

Subjects:
Primary: 47D06
Secondary: 35K15 , 47A55

Rights: Copyright © 1999 Khayyam Publishing, Inc.

Vol.12 • No. 1 • 1999
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