(untitled) (Journal of Partial Differential Equations, 2020)
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=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)=λ_β=-|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 u_t=-i(-Δ)^mu ± i|u|^p-1u, 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^*.