Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-order PDEs

Asymptotic large- and short-time behavior of solutions of the linear dispersion equation μt = Uxxx in IR× IR＋, and its （2k＋l）th-order extensions are studied. Such a refined scattering is based on a ＂Hermitian＂ spectral theory for a pair {B,B＊} of non self-adjoint rescaled operators     

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrōdinger Equations

The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=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 Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same 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 Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜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^＊}.     

On the behaviour of blow-up interfaces for an inhomogeneous filtration equation

We study the asymptotic behaviour of blow-up interfaces of thesolutions to the one-dimensional nonlinear filtration equationin inhomogeneous media where m>1 isa constant and (x) = |x|^– (for |x| 1, with > 2) isa bounded, positive, smooth, and symmetric function. The initialdata are assumed to be smooth, bounded, compactly supported,symmetric, and monotone. It is known that due to the fast decayof the density (x) as |x| the support of the solution increasesunboundedly in a finite time T. We prove that as tT^– theinterface behaves like O((T – t)^–b), where the exponentb > 0 (which depends on m and only) is given by a uniqueself-similar solution of the second kind satisfying the equation|x|^– u_t = (u^m)_xx. The corresponding rescaled profilesalso converge. We establish the stability of the self-similarsolution of the second kind for the exponential density (x)=e^–|x|for |x| 1. We give a formal asymptotic analysis of the blow-upbehaviour for the non-self-similar density (x) = e^–|x|2.Several exact self-similar solutions and their correspondingasymptotics are constructed.