@Article{JPDE-26-305,
author = {Galaktionov , V. A. and Kamotski , I. V.},
title = {Refined Scattering and Hermitian Spectral Theory for Linear Higher-order Schrödinger Equations},
journal = {Journal of Partial Differential Equations},
year = {2013},
volume = {26},
number = {4},
pages = {305--362},
abstract = {<p>The Cauchy problem for a linear 2mth-order Schrödinger equation $$u_t=-i(-Δ)^mu, in\ R^N×R_+, u|_{t=0}=u_0; m≥ 1$$ is an integer, is studied, for initial data u0 in the weighted space $L^2_{ρ^*}(R^N)$, with $ρ^*(x)=e^{|x|^α}$ and $α=\frac{2m}{2m-1}∈(1,2]$. 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 $$B=-i(-Δ)^m+\frac{1}{2m}y\cdot∇+\frac{N}{2m}I, with\ the\ spectrum\ σ(B)={λ_β=-\frac{|β|}{2m}, |β|≥ 0}.$$ (II) Finite-time blow-up local structures of nodal sets of solutions as t?^- and a formation of &quot;multiple zeros&quot; are described by the eigenfunctions, being generalized Hermite polynomials, of the &quot;adjoint&quot; operator $$B^*=-i(-Δ)^m-\frac{1}{2m}y\cdot∇, with\ the\ spectrum\ σ(B^*)=σ(B).$$ 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 (&quot;+&quot;) and defocusing (&quot;-&quot;) cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in\ R^N×R_+, where\ P&gt;1,$$ as well as for: (V) the quasilinear Schrödinger equation of a &quot;porous medium type&quot; $$u_t=-i(-Δ)^mu (|u|^nu), in\ R^N×R_+, where\ n&gt;0.$$ 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 ${B,B^*}$.</p>},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v26.n4.3},
url = {http://global-sci.org/intro/article_detail/jpde/5167.html}
}