@Article{JPDE-24-207,
author = {R. S. Fernandes  and V. A. Galaktionov },
title = {Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-order PDEs},
journal = {Journal of Partial Differential Equations},
year = {2011},
volume = {24},
number = {3},
pages = {207--263},
abstract = {<p>Asymptotic large- and short-time behavior of solutions of the linear dispersion equation u_t = u_{xxx} in R×R_+, and its (2k+1)th-order extensions are studied. Such a refined scattering is based on a &quot;Hermitian&quot; spectral theory for a pair {B,B^∗} of non self-adjoint rescaled operators &nbsp;B=D^3_y+\frac13yD_y+\frac13I, and the adjoint one B^∗=D^3_y-\frac13yD_y, with the discrete spectrum σ(B)=σ(B^∗)={λ_l=-l/3, l=0,1,2,...} and eigenfunctions for B, {ψ_l(y)=[(-1)^l/\sqrt{l!}]D^l_yAi(y), l ≥ 0}, where Ai(y) isAiry&#39;s classic function. Eigenfunctions of B^∗ are then generalized Hermite polynomials. Applications to very singular similarity solutions (VSSs) of the semilinear dispersion equation with absorption, u_S(x,t)=t^{-\frac{1}{p-1}}f(\frac{x}{t^{\frac13}}): u_t=u_{xxx}-|u|^{p-1}u in R×R_+, p &gt; 1, and to its higher-order counterparts are presented. The goal is, by using various techniques, to show that there exists a countable sequence of critical exponents {p_l=1+3/(l+1), l=0,1,2,...} such that, at each p= p_l , a p-branch of VSSs bifurcates from the corresponding eigenfunction ψ_l of the linear operator B above.</p>},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v24.n3.2},
url = {http://global-sci.org/intro/article_detail/jpde/5208.html}
}