Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-order PDEs (Journal of Partial Differential Equations, 2011)
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 "Hermitian" spectral theory for a pair {B,B^∗} of non self-adjoint rescaled operators  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'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 > 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.
Keywords: Odd-order linear and semilinear PDEs; fundamental solution; Hermitian spectral theory; polynomial eigenfunctions; self-similarity; very singular solutions; bifurcations; branching