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

Abstract

The article considers self-similar solutions of the nonlinear heat equation with a three-dimensional source that evolve in a blow-up setting. The self-similar problem is a boundary-value problem for a nonlinear equation of elliptical type that has a nonunique solution. We investigate the eigenfunction spectrum of the self-similar problem in two-and three-dimensional space. The problem is solved on a grid by Newton's iteration method. The implementation of Newton's method requires analysis of a linearized equation and construction of initial approximations. The eigenfunctions are continued in a parameter. Structures of various symmetry are obtained. New types of multidimensional structures are observed: these are multiply connected three-dimensional heat localization regions.

Bifurcation analysis of the spectrum of two-dimensional thermal structures evolving with blow-up

“…The grid is typically N = 100 × 100 or N = 200 × 200 nodes. The self-similar equation and the boundary conditions are approximated on the grid to second order of accuracy [29]. In [28] the finite element method has been used.…”

“…The eigenfunctions corresponding to different eigenvalues are linked by a similarity transformation [29], and therefore without loss of generality we set τ β = − ( ) − 1 1 and find the corresponding spectrum of the functions Θ( ) ξ ϕ τ , , . Previous studies (see, e.g., [12 -15]) have shown the existence of three types of self-similar solutions with blow-up depending on the values of the parameters β and σ.…”

“…These authors have constructed for the first time certain types of eigenfunctions in Cartesian and polar coordinates and have proposed a first attempt at their classification. New types of complex two-dimensional eigenfunctions have been identified in [29] and three-dimensional structures have been constructed for the first time. Different eigenfunctions describe structures of different complexity and have different symmetry and architecture.…”

“…22, pp. 50-75, 2005. three-dimensional structures have been found representing multiply connected combustion region, i.e., structures that enclose a region with zero temperature in their interior [23,29].…”

Stability and localization of unbounded solutions of a nonlinear heat equation in a plane

“…Eigenfunctions are positive on the entire plane and go to zero at infinity. For sufficiently large ξ the eigenfunctions Θ( , ) ξ ϕ are described by the following asymptotic behavior [13]:…”