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Analytical Solution of Nonlinear Parabolic Equations with Non-Stationary Exponent of Nonlinearity

Authors: Granik I.S., Gribov A.F. Published: 01.08.2018
Published in issue: #4(79)/2018  
DOI: 10.18698/1812-3368-2018-4-4-13

 
Category: Mathematics | Chapter: Mathematical Physics  
Keywords: quasi-linear parabolic equation, non-stationary non-linearity, spatial localization

We study the evolution of thermal perturbation in a non-linear medium whose thermal conduction coefficient explicitly depends on time and is a power function of temperature, with its exponent also being time-dependent if there is volume absorption of heat in this medium. The presence of lowest terms in the quasi-linear parabolic equation governing these transport processes affects the properties of the heat transport process in the medium under consideration. The analysis of physical properties of the studied process is conducted and its specific characteristics, i. e. the spatial localization mode, and its variations, i. e. stable and metastable localization, are examined. The effect of an instantaneous heat source on propagation of thermal perturbation in an isotropic medium is considered. The study introduces the conditions for existence of a closed-form solution of the Cauchy problem describing the process, as well as the location of the heat wave front that separates the perturbed and unperturbed zones at an arbitrary moment in time. The conditions for the spatial localization of these perturbations are indicated, i. e. the boundaries are predicted beyond which thermal perturbations from this source do not penetrate

The work was supported by the Russian Foundation for Basic Research (project no. 18-07-00269)

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