Investigating the Spectrum of Self-Similar Solutions of the Nonlinear Heat Equation

Abstract

Self-similar solutions of the nonlinear heat equation with a three-dimensional source and density that varies as a power function of the radius are considered in planar, cylindrical, and spherical geometries. The self-similar solutions evolve in a blow-up setting and constitute time-dependent dissipative structures. The eigenfunction spectrum of the self-similar problem is investigated for various values of the model parameters by computational methods involving continuation in a parameter and bifurcation analysis. It is shown that the spectral problem may have a nonunique solution. We establish the number of eigenfunctions and their existence domain in the parameter space. The evolution of the eigenfunctions with changes in the parameter is examined. The stability of the self-similar solutions is shown to depend on the parameter values, the eigenfunction index, and the eigenfunction parity. New structurally stable and metastable self-similar solutions are obtained. The metastable solutions follow the self-similar law almost during the entire blow-up time and preserve their complex structure as the temperature is increased by two orders of magnitude.