Numerical modeling of the solution of the system semilinear heat
conduction problem with absorption at a critical parameter
Mukimov Askar
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471332
Keywords:
nonlinear heat equation, variable density, asymptotics of solutions, critical value of the parameter, upper
solution, lower solution, principle of comparison of solutions.
Abstract:
In this paper we study the asymptotic behavior (for
t
→
) of solutions of the system semilinear heat
conduction problem with absorption at a critical parameter. The asymptotics were established using the
method of standart equations. The proofs were carried out using the method of comparison of solutions and
the maximum principle. For numerical computations as an initial approximation we used founded the long
time asymptotic of the solution. Numerical experiments and visualization were carried for one and two
dimensional case.
1 INTRODUCTION
It is very important to establish the value of
numerical parameters which the character of the
asymptotic behavior of the solution will change. Such
values of numeric parameters are called critical or
critical values of the Fujita type. He first established
this for the semi-linear heat equation [1]. At the
critical parameters we can observe new effects such
as infinite energy, localization and others.
As is well known for the numerical
computation of a nonlinear problem, the choice of the
initial approximation is essential, which preserves the
properties of the final speed of propagation, spatial
localization, bounded and blow-up solutions, which
guarantees convergence with a given accuracy to the
solution of the problem with minimum number of
iterations.
In the domain
{( , ) :
0,
}
N
Q
t x t
x
R
=
the
following system semi linear heat conduction
equation
1
1
1
2
( )
0,
( )
0
u
L u
u v
t
v
L v
v u
t
−
+ −
=
− + −
=
(1)
0
0
(0, )
( )
0,
(0, )
( )
0,
N
u
x
u x
v
x
v x
x
R
=
=
(2)
t and x are, respectively, the temporal and spatial
coordinates where
1
2
,
1
2
2
1
/
N
i
i
x
=
=
.
Problem (1)-(2) is the basis for modeling various
processes of heat diffusion, magnetic hydrodynamics,
gas and liquid filtration, oil and gas with nonlinear
absorption part.
In an unlimited heat сconducting environment
with nonlinear absorption, the heat diffusion system
in which is described by equation (1)-(2), some initial
temperature perturbation
0
0
( )
0,
( )
0
u x
v x
is
introduced from the outside, of a rather arbitrary
form. As the results below show for critical values of
1
2
,
the asymptotic of the solution of the problem
(1)-(2) will be different.
Under some suitable assumptions, the existence,
uniqueness and regularity of a weak solution to the
Cauchy problem (1)-(2) and their variants have been
extensively investigated by many authors (see [3–5]
and the references therein).
Case
1
2
1
/ 2,
1, 2
1
i
N
i
+
=
=
−
Is called a critical case. Herero Escobedo
considered the problem Cauchy to the case
system with source
1
1
1
2
( )
0,
( )
0
u
v
L u
u
v
L v
v u
t
t
−
+ +
=
−
+ +
=
and proved that condition blow up solution is
1
2
1
/ 2,
1, 2
1
i
N
i
+
=
−
A lot of works studied properties of solutions
of problem with critical value of parameter
and
were established asymptotic behavior for
t
→
(see
[1],[2],[19]-[27]). The long time asymptotic of
solutions has been established for the critical value of
parameter
for following problem
(
)
*
2
p
u
u
u
u
t
−
=
−
0
(0, )
( ),
N
u
x
u x x
R
=
,
where critical parameter
*
1
/
p
p N
= − +
in [19].
The author in [20] established asymptotic solution for
double nonlinearity problem
(
)
*
1
m
u
u
u
u
t
−
=
−
0
(0, )
( ),
N
u
x
u x x
R
=
*
2 /
m
N
= +
.
In [21] authors were established the long
time asymptotic of solutions for the critical value of
parameter
2
1
N
= +
for problem (1)-(2) in case one
part of system.
They considered following semi-linear parabolic
equation
0,
0
t
u
u u
t
− +
=
(3)
0
(0, )
( )
0
u
x
u x
=
(4)
2
2
1
2
/
,
1
N
i
i
x
N
=
=
= +
The solution of problem (3)-(4) is “infinity” energy.
The initial solution is
2
0
( )
{exp(
)},
u x
o
x
x
=
−
→ +
0
They proved that for problem (3)-(4) the long time
asymptotic of the solutions is the following
approximate self-similar solution
2
*
1
2
( , )
(
) ln(
)
(
)
N
x
u t x
T
t
T
t
g
T
t
−
=
+
+
+
(5)
For the function
*
g
the following estimate of solution
2
2
2
2
*
exp
( )
exp
,
4
4
x
A
g
H
T
t
−
−
=
+
,
were obtained. Where A, H are constants.
For
2
1
N
+
, the approximate self-similar solution
is different from (5), which means that for critical
values the asymptotic of the solutions will change for
t
→
.
In [22] was considered following Cauchy
problem for nonlinear heat equation with absorption
1
(u
) u
t
u
+
=
−
(6)
in area
(0, )
N
Q
R
=
0
( , 0)
( )
u x
u x
=
for
N
x
R
(7)
Authors for critical value of parameter established the
the following long time asymptotic of the solution for
the critical exponent
*
1 2 /
N
=
= + +
.
u(x, t)
((
)ln(T t))
( ;a)
k
T
t
F
−
=
+
+
(8)
/
/ 2
(T t)
ln(T t)
k N
k
x
−
=
+
+
2
2
1/
0
1/
0
/ (N
2), F( ;a)
C (a
)
,
C
/ 2 (
1)
,
1,
0
k
N
k
N
T
a
+
=
+
=
−
=
+
where the value of the numerical parameter a
determined from the law of energy conservation
(x, t) dx
( ; a) d
M
w
F
M
=
=
/
1
N k
M
C a
=
where
/ 2
1
0
(N/ 2,1 1/ ) / Г(N/ 2)
N
C
C B
=
+
В аnd G is beta and gamma of Euler function.
They proved that solution (8) is the long time
asymptotic of the solution to problem (6)-(7) by
constructing lower and upper solutions. The
following lower and upper solution with variable a
((
)log(T t))
( ;a )
u(x, t)
((
)log(T t))
( ;a )
k
k
T
t
F
T
t
F
−
−
−
−
+
+
+
+
0
a
a
−
+
In [34] authors find asymptotic for the following
double nonlinear heat equation
(
)
*
2
1
p
m
k
u
u
u
u
u
t
−
−
=
−
with critical parameter
*
(
2)
p
k p
m
N
=
− + +
(N-
size of the dimension)
In [35] established large time asymptotic solution for
double nonlinear problem with critical exponent and
variable density.
In[24] established conditions of norm of
solution ( )
u t
with critical exponent
*
1
K
N
q
Nv
+
=
+
for
following Cauchy problem
*
1
1
(
)
q
m
v
p
t
u
div u
Du
Du
Du
u
−
−
−
= −
+
in
(0, )
N
R
T
0
( , 0)
( )
0
u x
u x
=
Here are some of them
*
1
*
1
( )
( )
[ln ]
A
vq
if q
q then u t
t
if q
q then u t
t
−
−
−
=
where
*
(
1)
q
q
A
Nv
H
−
=
+
,
(
1)(
1)
(
2)
H
vq
q m
=
+
− −
+ −
,
0,
1,
2
0
m
=
=
+ −
.
In [25] the long time asymptotic of the
solution of polygarmonic equation were established
for the following problem with critical parameter
1 2
/
c
p
m N
= +
1
(
)
p
m
t
u
u
u
−
= − − −
in area
N
R
R
+
0
( , 0)
( )
u x
u x
=
Following asymptotic
/ 2
/ (2
)
0
1/ 2
( , )
(ln )
(
)
(1)
N
m
N
m Q
m
x
u x t
C t
t
f
o
t
−
−
+
=
+
Mu et al. [26] studied the secondary critical
exponent for the following p-Laplacian equation with
slow decay initial values:
(
)
2
, ( , )
(0, )
p
q
N
t
u
div
u
u
u
x t
R
T
−
=
+
(9)
0
( ,0)
( ),
N
u x
u x x
R
=
(10)
where p > 2, q > 1, and showed that, for
*
1 ( /
)
c
q
q
p
p N
= − +
, there exists a secondary
critical exponent
*
( / (
1
))
c
a
p
q
p
=
+ −
such that the
solution ( , )
u x t
of (9)-(10) blows up in finite time for
the initial data
0
( )
u x
which behaves like
x
−
at
x
=
, if
*
(0,
)
c
a
a
, and there exists a global
solution for the initial data
0
( )
u x
, which behaves like
x
−
at
x
=
, if
*
( , )
c
a
a N
.
J.-S. Guo and Y. Y.Guo [27] obtained the secondary
critical exponent for the following porous medium
type equation in high dimensions:
,( , )
(0, )
m
p
N
t
u
u
u
x t
R
T
= +
(11)
0
( ,0)
( ),
N
u x
u x x
R
=
(12)
where p > 1, m > 1 or max{0, 1 – (2/N)} <m< 1,
0
( )
u x
is nonnegative bounded and continuous, and
proved that for
*
(2 /
)
m
p
p
m
N
= +
, there exists a
secondary critical exponent
*
2 / (
)
a
p m
=
−
such that
the solution ( , )
u x t
of (11)- (12) blows up in finite
time for the initial data
0
( )
u x
.
The authors of [28] considered the fast
diffusion equation for large t, when the degree of
absorption is critical for the following equation
0
m
q
t
u
u
u
− +
=
in the domain
(0, ) xR
N
0
(0, x)
u (x), x
R
N
u
=
*
2
:
q
q
m
N
=
= +
They established a clear lower bound, which
eliminates convergence to zero.
In this paper, we study the asymptotic
behavior (for
t
→
) of solutions of the system
semilinear heat conduction problem with absorption
at a critical parameter (1)-(2) and and we calculate a
computational experiment for the one-dimensional
and two-dimensional case.
2 Asymptotic of the solution
case
1
2
1
/ 2,
1, 2
1
i
N
i
+
=
−
Consider the system ordinary differential equation
1
2
,
du
dv
v
u
dt
dt
= −
= −
With a solution
1
2
1
2
1
2
1
2
1
2
( )
(
) , ( )
(
)
1
1
,
1
1
u t
A T
t
v t
B T
t
=
+
=
+
+
+
= −
= −
−
−
Where A, B satisfy to the system algebraic
equations
1 2
2 1
1
2
,
A
B
B
A
=
=
defined above the functions
1
2
1/ 2
1
1
( , )
(
)
( ),
( , )
(
)
( ),
(
)
u t x
T
t
f
v t x
T
t
f
x T
t
−
+
+
=
+
=
+
=
+
Consider the self-similar solution of the system
1
1
1
2
1
1
1
1
1
1
2
1 2
1
1
2
2
2
2
1
1 2
1
(
) ( / 2)
0
1
1
(
) ( / 2)
0
1
N
N
N
N
df
df
d
f
B f
d
d
d
df
df
d
f
A f
d
d
d
−
−
−
−
+
+
+
−
=
−
+
+
+
−
=
−
It is clear that the functions
2
2
1
2
( )
exp(
/ 4),
( )
exp(
/ 4)
f
A
f
B
=
−
=
−
Introduce the functions
1
2
1/ 2
1
1
1
2
1
2
1
2
1
2
( , )
(
)
( ),
( , )
(
)
( ) ,
(
)
1
1
,
1
1
u t x
T
t
f
v t x
T
t
f
x T
t
−
+
+
=
+
=
+
=
+
+
+
= −
= −
−
−
2
2
1
2
( )
exp(
/ 4),
( )
exp(
/ 4)
f
A
f
B
=
−
=
−
Theorem
1
Let
1
2
1
/ 2,
1, 2
1
i
N
i
+
=
−
.
(0, )
(0, ), (0, )
(0, ),
N
u
x
u
x v
x
v
x x
R
+
+
Then for any A, B>0 to the solution of the
problem (1), (2).
1
2
(
( , )),
(
( , ))
0
L u t x
L v t x
+
+
in Q
Really consider it is easy to calculate that
1
1
1
2
1
2
1
1
1
1
1
1
1
2
1
2
1
1
2
2
2
2
2
1
1
2
(
)
(
( , ))
(
)
1
( / 2)
0
1
(
)
(
( , ))
(
)
1
( / 2)
0
1
N
N
N
N
df
d
T
t
L u t x
d
d
df
f
B f
d
df
d
T
t
L v t x
d
d
df
f
A f
d
−
−
−
+
−
−
−
+
+
=
+
+
+
−
=
−
+
=
+
+
+
+
−
=
−
Since obvious
1
1
1
1
1
1
1
2
2
2
(
) ( / 2)
(
/ 2)
(
) ( / 2)
(
/ 2)
N
N
N
N
df
df
d
N
f
d
d
d
df
df
d
N
f
d
d
d
−
−
−
−
+
=
+
=
Then
1
1
1
2
1
2
1
1
1
2
1
2
2
2
2
1
1
2
1
(
)
( ( , )) [ (
/ 2)
]
1
1
(
)
( ( , )) [ (
/ 2)
]
1
T t
L u t x
N
f
B f
T t
L v t x
N
f
A f
−
+
−
+
+
+
= −
+
−
−
+
+
= −
+
−
−
According condition of the theorem 1 we hav
1
2
(
( , ))
0,
(
( , ))
0
L u t x
L v t x
+
+
In Q.
case
1
2
1
/ 2,
1, 2
1
i
N
i
+
=
=
−
The asymptotic of solutions of the system
has the following form
( , )
( ) ( )
u t x
u t f
=
( , )
( ) ( )
v t x
v t f
=
Where
1
1 2
1
1
( )
( ln )
u t
t
t
+
−
−
=
2
1 2
1
1
( )
( ln )
v t
t
t
+
−
−
=
2
2
4
( )
,
4
x
t
x
f
e
t
−
=
= −
The asymptotics were established using the
method of standart equations. The proofs were carried
out using the method of comparison of solutions and
the maximum principle and satisfy the following
conditions
( , )
( ) ( )
( , )
( , )
( ) ( )
u t x
u t f
u t x
u t x
u t f
+
−
=
=
Where
( , ),
( , )
u t x u t x
+
−
upper and lower
solutions.
3.
NUMERICAL COMPUTATION
From problem (1)-(2) we have following one
dimensional system of semi linear heat equations in
the domain
{( , ) :
[0, ],
[ , ]}
Q
t x
t
T
x
a b
=
1
2
2
2
2
2
u
u
v
t
x
v
v
u
t
x
=
−
=
−
(13)
with initial
0
0
(0, )
( )
0,
[ , ],
(0, )
( )
0,
[ , ],
u
x
u x
x
a b
v
x
v x
x
a b
=
=
and boundary conditions
1
2
1
2
( , )
( )
0,
[0, ],
( , )
( )
0,
[0, ],
( , )
( )
0,
[0, ],
( , )
( )
0,
[0, ].
u t a
t
t
T
u t b
t
t
T
v t a
t
t
T
v t b
t
t
T
=
=
=
=
,
Here
1
2
,
are positive constants,
0
( )
u x
and
0
( )
v x
initial distribution, respectively for the first
and second components,
1
( )
t
- value of the first
components on the left margin,
2
( )
t
- value of the
first component on the right end,
1
( )
t
and
2
( )
t
,
respectively for the second components.
For problem (13) we construct the spatial
grid x with steps h
,
0,
0,1,..., ,
h
i
x
ih
h
i
n
hn
b
=
=
=
=
And temporary grid with
,
0,
0,1,..., ,
j
t
j
j
m
m
T
=
=
=
=
replace problem (3.1.1) implicit two-layer difference
scheme and obtain the difference task with error
2
(
)
h
+
( )
( )
( )
1
2
1
1
1
1
1
1
2
1
1
1
1
1
1
2
0
0
0
1
2
2
(
)
,
2
(
)
1, 2,...,
1; j
0,1,...,
1
,
0,1,...,
,
1, 2,...,
,
1, 2,..
j
j
j
j
j
j
i
i
i
i
i
i
j
j
j
j
j
j
i
i
i
i
i
i
i
i
j
j
j
n
j
y
y
y
y
y
Y
h
Y
Y
Y
Y
Y
y
h
i
n
m
y
u
x
i
n
y
t
j
m
y
t
j
+
+
+
+
+
−
+
+
+
+
+
−
−
−
+
=
−
−
−
+
=
−
=
−
=
−
=
=
=
=
=
=
( )
( )
( )
0
0
0
1
2
.,
,
0,1,...,
,
1, 2,...,
,
1, 2,...,
i
i
j
j
j
n
j
m
Y
v
x
i
n
Y
t
j
m
Y
t
j
m
=
=
=
=
=
=
(14)
From different scheme (14) we will find tridiagonal
matrix coefficients A, B, C, F, A1, B1, C1, F1 and
solve following system linear equations by method
Thomas[29]
1
1
1
1
1
1
1
1
1
1
1
1
1
1
,
,
1, 2,..,
1,
0,1, 2,..,
1
j
j
j
j
j
j
j
i
i
i
i
i
i
i
j
j
j
j
j
j
j
i i
i i
i i
i
A y
C y
B y
F
A Y
C Y
B Y
F
i
n
j
m
+
+
+
−
+
+
+
+
−
+
−
+
= −
−
+
= −
=
−
=
−
With boundary conditions
0
1 1
1
y
y
=
+
2
1
2
N
N
y
y
−
=
+
and
0
1 1
1
Y
Y
=
+
2
1
2
N
N
Y
Y
−
=
+
Where
( )
1
2
2
,
,
1,
j
j
j
j
j
j
j
j
i
i
i
i
i
i
i
i
A
B
C
A
B
F
y
Y
h
h
=
=
=
+
+
=
−
( )
2
1
1
1
1
,
,
,
j
j
j
j
j
j
j
j
j
j
i
i
i
i
i
i
i
i
i
i
A
A B
B C
A
B F
Y
y
=
=
=
+
=
−
As an initial approximation, we should take:
(
)
2
1
1 2
1
4
1
ln
x
t
u
t
t
e
−
+
−
−
=
,
(
)
2
2
1 2
1
4
1
ln
x
t
v
t
t
e
−
+
−
−
=
.
The values of
1
2
,
must satisfy the following
expression
1
1
2
1
1
2
N
+
=
−
where for one-dimensional case N=1.
Problem (1)-(2) has no analytical solution.
Therefore, we will discuss result of the numerical
experiments. To find a solution of problem at some
point we are using numerical methods (see [29]-[31]).
The resulting asymptotic of the solutions were used
as an initial approximation for numerical
computation.
2.2
VISUALIZATION
Figure 1:
1
2
4,
2.75
=
=
Figure 2:
1
2
10,
2.3
=
=
Figure 3:
1
2
2.7,
3.1
=
=
Figure 4:
1
2
20,
2.3
=
=
CONCLUSIONS
Based on the obtained estimates of the
solutions, it is established that the proposed nonlinear
mathematical model with double nonlinearity
correctly reflects the physics of the process.
It is shown that the numerical analysis of the
results based on the obtained estimates of solutions
gives an exhaustive picture of the process in two-
component systems with the preservation of
localization properties.
The proposed method of selecting the initial
approximation proved to be effective and makes it
possible to numerically detect the processes of the
final propagation velocity and spatial localization.
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