Existence and perturbation of principal eigenvalues for a periodic-parabolic problem

Existence and perturbation of principal eigenvalues for a periodic-parabolic problem

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We give a necessary and sufficient condition for the existence of a positive principal eigenvalue for a periodic-parabolic problem with indefinite weight function.The condition was originally established by Beltramo and Hess [extit{frenchspacing Comm.Part.Diff.Eq.

}, extbf{9} (1984), 919--941] in the framework of the Vaccum Flask Schauder theory of classical solutions.In the present paper, the problem is considered in the framework of variational evolution equations on arbitrary bounded domains, assuming that the coefficients of the operator and the weight function are only bounded and measurable.We also establish a general perturbation theorem for the principal eigenvalue, which in particular allows quite singular perturbations of the domain.Motivation for the problem comes from population dynamics 24" Double Wall Oven taking into account seasonal effects.