Volume 26, Issue 4
Refined Scattering and Hermitian Spectral Theory for Linear Higher-order Schrödinger Equations

J. Part. Diff. Eq., 26 (2013), pp. 305-362.

Published online: 2013-12

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• Abstract

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 "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" 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 ("+") and defocusing ("-") cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in\ R^N×R_+, where\ P>1,$$ as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" $$u_t=-i(-Δ)^mu (|u|^nu), in\ R^N×R_+, where\ n>0.$$ 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 ${B,B^*}$.

• Keywords

Higher-order Schrödinger operators rescaled blow-up variables discrete real spectrum asymptotic behavior nodal sets of solutions unique continuation boundary characteristic point regularity quasilinear Schrödinger equations nonlinear eigen

35K55, 35K40

vag@maths.bath.ac.uk (V. A. Galaktionov)

i.kamotski@ucl.ac.uk (I. V. Kamotski)

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• RIS
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@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 = {

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 "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" 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 ("+") and defocusing ("-") cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in\ R^N×R_+, where\ P>1,$$ as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" $$u_t=-i(-Δ)^mu (|u|^nu), in\ R^N×R_+, where\ n>0.$$ 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 ${B,B^*}$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v26.n4.3}, url = {http://global-sci.org/intro/article_detail/jpde/5167.html} }
TY - JOUR T1 - Refined Scattering and Hermitian Spectral Theory for Linear Higher-order Schrödinger Equations AU - Galaktionov , V. A. AU - Kamotski , I. V. JO - Journal of Partial Differential Equations VL - 4 SP - 305 EP - 362 PY - 2013 DA - 2013/12 SN - 26 DO - http://doi.org/10.4208/jpde.v26.n4.3 UR - https://global-sci.org/intro/article_detail/jpde/5167.html KW - Higher-order Schrödinger operators KW - rescaled blow-up variables KW - discrete real spectrum KW - asymptotic behavior KW - nodal sets of solutions KW - unique continuation KW - boundary characteristic point regularity KW - quasilinear Schrödinger equations KW - nonlinear eigen AB -

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 "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" 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 ("+") and defocusing ("-") cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in\ R^N×R_+, where\ P>1,$$ as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" $$u_t=-i(-Δ)^mu (|u|^nu), in\ R^N×R_+, where\ n>0.$$ 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 ${B,B^*}$.

V. A. Galaktionov & I. V. Kamotski. (2019). Refined Scattering and Hermitian Spectral Theory for Linear Higher-order Schrödinger Equations. Journal of Partial Differential Equations. 26 (4). 305-362. doi:10.4208/jpde.v26.n4.3
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