This article is cited in 21 papers

Differentical equations, dynamical systems and optimal control

On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation

A. L. Kazakov

Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontova str., Irkutsk, 664033, Russia

Abstract: The paper deals with a nonlinear second order parabolic PDE, which is usually called “the nonlinear heat equation”. We construct and study a particular class of solutions having the form of a heat wave that propagates on a cold (zero) background with finite velocity. The equation degenerates on the front of a heat wave and its order decreases. This fact complicates the study. We prove a new existence and uniqueness theorem for a boundary-value problem with a given heat-wave front in the class of analytical functions. Also, we are looking for exact heat-wave type solutions. The construction of these solutions is reduced to integration of the nonlinear second order ODE with singularity.

Keywords: partial differential equations, nonlinear parabolic heat equation, existence and uniqueness theorem, exact solution.

UDC: 517.956.45, 517.911

MSC: 35A09,35A10,35A24

Received May 28, 2019, published August 7, 2019

DOI: 10.33048/semi.2019.16.073

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