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Robustness of Fredholm properties of parabolic evolution equations under boundary perturbations
Author(s) -
Maniar Lahcen,
Schnaubelt Roland
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn001
Subject(s) - fredholm theory , fredholm integral equation , mathematics , mathematical analysis , extrapolation , boundary value problem , norm (philosophy) , integral equation , boundary (topology) , initial value problem , law , political science
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.