Year: 2013
Author: V. A. Galaktionov, I. V. Kamotski
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 4 : pp. 305–362
Abstract
The Cauchy problem for a linear 2mth-order Schrödinger equation ut=−i(−Δ)mu,in RN×R+,u|t=0=u0;m≥1 is an integer, is studied, for initial data u0 in the weighted space L2ρ∗(RN), with ρ∗(x)=e|x|α and α=2m2m−1∈(1,2]. The following five problems are studied: (I) A sharp asymptotic behaviour of solutions as t→+∞ is governed by a discrete spectrum and a countable set Φ of the eigenfunctions of the linear rescaled operator B=−i(−Δ)m+12my⋅∇+N2mI,with the spectrum σ(B)=λβ=−|β|2m,|β|≥0. (II) Finite-time blow-up local structures of nodal sets of solutions as t?^- and a formation of "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" operator B∗=−i(−Δ)m−12my⋅∇,with the spectrum σ(B∗)=σ(B). Applications of these spectral results also include: (III) a unique continuation theorem, and (IV) boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schrödinger equations in the focusing ("+") and defocusing ("-") cases ut=−i(−Δ)mu±i|u|p−1u,in RN×R+,where P>1, as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" ut=−i(−Δ)mu(|u|nu),in RN×R+,where n>0. For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n→0+ and to use spectral theory of the pair B,B∗.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v26.n4.3
Journal of Partial Differential Equations, Vol. 26 (2013), Iss. 4 : pp. 305–362
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 58
Keywords: Higher-order Schrödinger operators