Stabilization to a singular steady state for the Frank-Kamenetskii equation in a critical dimension

Abstract

We study the phenomenon of the infinite-time stabilization of classical global solutions of nonlinear reaction–diffusion equations to an unbounded (singular) stationary state and we present a new case of such an asymptotic singularity pattern formation. We concentrate on the most famous parabolic model, namely the semi-linear Frank-Kamenetskii equation from combustion theory, where B is a ball in ℝ^N. Our goal is to show that a new asymptotic problem arises precisely in dimension 10, not being available in other dimensions (which were studied earlier). For N = 10, we fix the ball B = {|x| < 4} and take bounded initial data u₀ below the singular stationary solution U_s(x) = ln(16/|x|²), which is unbounded at the origin x = 0.

We establish a sharp estimate on the rate of convergence u(x, t) → U_s(x) as t → ∞ on compact subsets bounded away from x = 0 and also at the singularity point. We show that u(0, t) = α₀t + O(ln t) → ∞, where the positive constant α₀ is given by the first eigenvalue of the associated linear differential operator. We present formal asymptotic results showing that a detailed asymptotic analysis depends on a quite involved balance between various linear and nonlinear terms. Moreover, similar critical asymptotic behaviour is shown to exist in various related nonlinear second- and higher-order parabolic equation.