Differential and Integral Equations

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

Frank Räbiger and Roland Schnaubelt

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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

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

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

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