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 blowup time and preserve their complex structure as the temperature is increased by two orders of magnitude.

Two-Dimensional and Three-Dimensional Thermal Structures in a Medium with Nonlinear Thermal Conductivity

“…Previous studies (see, e.g., [14,17,25]) have shown that three types of self-similar blow-up solutions exist depending on the parameters β and σ. For β σ ≥ + 1 they describe localized combustion.…”

“…The eigenfunction spectrum has been studied in [12 -14] for the one-dimensional case and in [13 -18] for the radially symmetrical case and for a medium with distributed density. Bifurcation analysis has been applied in [17] to determine the number of eigenfunctions of the self-similar problem and to investigate in detail the properties of the spectrum as a function of the geometry of the region and the density distribution.…”

“…Among the eigenfunctions we distinguish structurally stable solutions and metastable solutions [12,17]. Structurally stable solutions are asymptotics of many solutions with arbitrary initial values.…”

“…Until quite recently it has been thought that only solutions with a single maximum at the center of symmetry can be structurally stable. It has been shown in [17,18], however, that the radially symmetrical self-similar solution with a single maximum and a zero region at the center (a structure in the shape of a spherical or cylindrical layer) is also a structurally stable solution. All other eigenfunctions of the self-similar problem are metastable solutions.…”

“…Metastable eigenfunctions with multiple maxima, unlike arbitrary temperature distributions, exist almost during the entire blow-up time and may preserve their structure as the temperature increases by two orders of magnitude. Near the blow-up point they either separate into simple structures that burn independently with their own blow-up instants, or degenerate into a single structure with a single maximum [17]. New applications of time-dependent dissipative structures in sociology [9,11], in particular in problems of coevolution of complex systems [6,20,21], and in quantum mechanics [15, 20 -22] have aroused the interest in the spectrum of multidimensional thermal structures, which are the solutions of the self-similar problem for the three-dimensional nonlinear heat equation.…”