ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 255–263.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
A.L. Kazakov, P.A. Kuznetsov, 2018, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2018, Vol. XXI, No. 2, pp. 56–65.
On the Analytic Solutions of a Special Boundary Value Problem
for a Nonlinear Heat Equation in Polar Coordinates
A. L. Kazakov
and P. A. Kuznetsov
Matrosov Institute for System Dynamics and Control Theory,
ul. Lermontova 134, Irkutsk, 664033 Russia
Irkutsk State University, ul. Karla Marksa 1, Irkutsk, 664033 Russia
Received July 25, 2017
Abstract—The paper addresses a nonlinear heat equation (the porous medium equation) in the case
of a power-law dependence of the heat conductivity coeﬃcient on temperature. The equation is used
for describing high-temperature processes, ﬁltration of gases and ﬂuids, groundwater inﬁltration,
migration of biological populations, etc. The heat waves (waves of ﬁltration) with a ﬁnite velocity
of propagation over a cold background form an important class of solutions to the equation under
consideration. A special boundary value problem having solutions of such type is studied. The
boundary condition of the problem is given on a suﬃciently smooth closed curve with variable
geometry. The new theorem of existence and uniqueness of the analytic solution is proved.
Keywords: nonlinear heat equation, power series, convergence, existence and uniqueness
Under study is the nonlinear parabolic equation of the form
= uΔu +
Here u = u(t, ¯x) is the sought function, time t and spatial coordinates ¯x =(x
) are the inde-
pendent variables, and the operators Δ and ∇ act on spatial variables. The nonlinear equation of heat
conduction is reduced to (0.1) in the case of a power-law dependence of the heat conductivity coeﬃcient
on temperature .
Equation (0.1) is of interest because of its numerous applications. In particular, it is used in the
description of high-temperature processes , ﬁltration of a polytropic gas in porous media , ground-
water movement , and migration of biological populations . A more detailed survey of applications
is given in .
An important class of solutions to the nonlinear heat equation is formed by the heat waves that
propagate with ﬁnite velocity along a cold background. Geometrically, a heat wave is represented by
two solutions of (0.1): the perturbed u ≥ 0 and the trivial u ≡ 0 functions continuously joined along
some suﬃciently smooth line, the wave front.
The ﬁrst publications on heat waves in the nonlinear case dated back to the middle of the last century
[2, 3]. Not trying to give a detailed survey of all available results, we note that the initial-boundary
value problems presupposing the existence of such solutions were studied in the scientiﬁc school of
A. F. Sidorov to which the authors of the paper belong. In addition to the methods for constructing
exact solutions (see, for example, ), another approach to the study of these problems is successfully
developed in this school which is based on the method of power series and its generalizations. For
example, in  (see also ), an approximate solution of the initial-boundary value problem for (0.1)