Differential and Integral Equations

Non-autonomous Miyadera perturbations

Abdelaziz Rhandi, Frank Räbiger, Roland Schnaubelt, and Jürgen Voigt

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Abstract

We consider time dependent perturbations $B(t)$ of a non--autonomous Cauchy problem $\dot{v}(t)=A(t)v(t)$ $(CP)$ on a Banach space $X$. The existence of a mild solution $u$ of the perturbed problem is proved under Miyadera type conditions on $B(\cdot)$. In the parabolic case and $X=L^d(\Omega)$, $1 < d < \infty$, we show that $u$ is differentiable a.e. and satisfies $\dot{u}(t)=(A(t)+B(t))u(t)$ for a.e. $t$. Our approach uses perturbation results due to one of the authors, [29], and S. Monniaux and J. Prüß, [14], which are applied to the evolution semigroup induced by the evolution family related to $(CP)$. As an application we obtain solutions of a second order parabolic equation with singular lower order coefficients.

Article information

Source
Differential Integral Equations Volume 13, Number 1-3 (2000), 341-368.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356124303

Mathematical Reviews number (MathSciNet)
MR1811962

Zentralblatt MATH identifier
0980.34056

Subjects
Primary: 34G10: Linear equations [See also 47D06, 47D09]
Secondary: 35K10: Second-order parabolic equations 35K90: Abstract parabolic equations 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Citation

Räbiger, Frank; Schnaubelt, Roland; Rhandi, Abdelaziz; Voigt, Jürgen. Non-autonomous Miyadera perturbations. Differential Integral Equations 13 (2000), no. 1-3, 341--368. https://projecteuclid.org/euclid.die/1356124303.


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