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Journal of the London Mathematical Society Advance Access originally published online on March 3, 2008
Journal of the London Mathematical Society 2008 77(3):558-580; doi:10.1112/jlms/jdn001
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© 2008 London Mathematical Society

Robustness of Fredholm properties of parabolic evolution equations under boundary perturbations

Lahcen Maniar

Cadi Ayyad University
Faculty of Sciences
Semlalia
BP 2390, Marrakesh
Morocco
maniar@ucam.ac.ma

Roland Schnaubelt

Institut für Analysis
Fakultät für Mathematik
Universität Karls-ruhe
76128 Karlsruhe
Germany
schnaubelt@math.uni-karlsruhe.de

We study perturbations at the boundary of linear nonautonomous parabolic boundary value problems. Our approach relies on a transformation of the given inhomogeneous boundary value problem to an evolution equation in larger, time-varying extrapolation spaces. We establish the well-posedness of this equation and Duhamel's formulas relating the evolution families solving the perturbed and the unperturbed problem. By means of these formulas, we can show that the perturbed evolution equation inherits the exponential dichotomy and Fredholm properties of the unperturbed equation if the perturbations are small in norm or compact. This result leads to a Fredholm alternative for the given perturbed boundary value problem.

2000 Mathematics Subject Classification 35K20, 35K90, 47A53, 47A55, 47D06.

This work was supported by the DFG grant 445 MAR-113/10/0-2. Lahcen Maniar gratefully acknowledges the support of the Alexander-von-Humboldt foundation.

Received September 6, 2007; published online March 3, 2008.


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