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The Principal Spectrum for Linear Nonautonomous Parabolic PDEs of Second Order: Basic Properties

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Abstract:

We put forward a theory extending the notion of principal eigenfunction and principal eigenvalue to the case of linear nonautonomous parabolic PDEs of second orderut=∑ij=1N aij(tx) ∂2uxixj+∑i=1N ai(tx) ∂uxi+a0(tx) u,  xΩ,on a bounded domain Ω⊂ℝN, with Dirichlet or Robin boundary conditions. A canonically defined one-dimensional subbundle  (corresponding to the solutions that are globally defined and of the same sign) serves as an analog of principal eigenfunction. The principal spectrum is defined to be the dynamical (Sacker–Sell) spectrum of the linear skew-product flow on . Characterizations of principal spectrum in terms of (logarithmic) growth rates of positive solutions are given. Finally, monotonicity of the principal spectrum with respect to zero order terms is proved. Copyright 2000 Academic Press.

Document Type: Research Article

Affiliations: Institute of Mathematics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, Wrocław, PL-50-370, Poland

Publication date: December 1, 2000

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