In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitative properties of solutions of such reaction-diffusion systems. We also study the finite speed of propagation (FSP) properties of solutions, an asymptotic behavior of the compactly supported solutions and free boundary asymptotic solutions in quick diffusive and critical cases.
Let’s consider properties of the Cauchy problem for the following system of nonlinear reaction-diffusion equations in the domain
where are given positive numbers, and, ,. System (1) describes different physical process in two componential inhomogeneous nonlinear environments. For example, the processes of the reaction-diffusion, heat conductivity, polytrophic filtration of liquids and gas with a source power which is equal to Cases, when, were considered in [1] -[7] .
The system (1) in the domain, where is degenerated, and in the domain of degeneration it may not have the classical solution. Therefore, we study the weak solutions of system (1) which also have physical sense: and satisfy some integral identity in the sense of distribution [1] . For the solution of system (1) there are phenomena of the finite speed of a propagation (FSP). That is, there are functions that satisfy and at and. In the case of, a solution of problems (1), (2) is called space localization of a disturbance. The surfaces and are called a free boundary or a front, respectively.
The process of the reaction-diffusion with double nonlinearity in the case of one equation has been investigated by many authors (see [8] -[15] and the references therein). FSP and blow-up property for equations with variable density
was established in [8] [9] . An asymptotic of self-similar solutions was studied in [15] . Martynenko and Tedeev [10] [11] studied the Cauchy problem for the following two equations with variable coefficients:
and
where or
They showed that under some restrictions to the parameters and initial data, any nontrivial solution to the Cauchy problem blows up in finite time. Moreover, the authors established a sharp universal estimate of the solution near the blow-up point.
It is well know that qualitative properties of solutions of the equation similar to (1) have not been investigated thoroughly. There are some results in [1] -[6] corresponding to the case.
In the present work, the qualitative properties of solutions of system (1) are studied based on the self-similar and approximately self-similar approach. We establish one way of construction of the critical exponent and property finite speed of perturbation (FSP) for system (1). An asymptotic property of compactly supported solutions (c.s.s.) of the considered problem and the behavior of the free boundary for the case are obtained. We prove the existence of solution with finite property. An asymptotic of a self-similar solution for the fast diffusion case and a critical case are also studied.
2. Approximate Self-Similar and Self-Similar Equations
Below we provide a method of nonlinear splitting for construction of self-similar and approximately self-similar equation. For construction of the self-similar and approximately self-similar solutions of system (1) we search the solutions in the form
Here, we obtain as
Which are the solutions of following equations
Substituting (3), the system (1) is reduced to the following system of equations
where the functions are chosen as following
It is easy to establish that the system (4) has approximately self-similar solution of kind
where and the functions satisfies the approximately self-similar system equations
It is easy to prove that as
for, where -Hardy’s body [2] , are constants. In this case, it is easy to show that system (1) becomes a self-similar for a sufficient large t. Therefore it is possible to consider the system (7) as an asymptotically self-similar system of equation corresponding to system (1). In particular case, when approximately self-similar systems (7) will be as self-similar if
In this case for the functions we have the following self-similar system of equation in “radial” form
where
In the case or in (10), the properties of the different solutions as computing aspects of the system Equation (10) were studied by many authors [8] -[15] . In singular, one equation case, when the existence of positive solutions of the Equation (10) was studied in [14] .
3. Slowly Diffusion Case: <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/3-1720275x62.png" xlink:type="simple"/></inline-formula>3.1. A Global Solvability of Solutions
where
In the case,
where
Fujita type critical exponent for the system (1) is numerical parameters for which the following equality holds:
This result consists of the result of Escobedo, Herero [15] for the case when in (1).
Theorem 1. (A global solvability). Assume,
Then for sufficiently small the followings holds
where the functions defined as above, are constants.
Proof. For proving theorem 1 we use a comparison principle. As a comparison solution we take the functions where
It is easy to check that
If
Then we have
In order to apply a comparison principle we note that in
Since
Therefore,
Then according to the hypotheses of Theorem 1 and comparison principle we have
if
The proof of the theorem is complete.
We notice that if
then
It means that
if
3.2. Property of Finite Speed of a Perturbation
Corollary 1. Suppose that the hypotheses of Theorem 1 holds. Then a solution of the problems (1), (2) has FSP property.
Indeed, for a weak solution of the problems (1), (2) we have
It follows that
where It means that the solution of the problems (1), (2) have FSP
property.
Critical case. The case will be called a critical case.
Theorem 2. Let Then for sufficiently small
the problems (1), (2) have global solution and the following inequalities in Q hold
here
Proof. Proof of the theorem is based on the comparison principle. We take for comparison the functions
where
It is easy to check that
From the hypothesis of Theorem 2 and last expressions we have
if the constants such that
This inequality due to the comparison principle completes the proof of the theorem.
Value for which
corresponds to Fujita type critical exponent proved earlier by Escobedo, Herrero [15] for the case p = 2.
4. Asymptotic of the Self-Similar Solutions
Now we study asymptotic of the weak compact supported solutions (c.s.s.) of the system (10) when Consider this system equation with boundary condition
where.
The existence of a self-similar weak c.s. solution for the problems (10), (15) in the case was studied in [6] where the authors obtained conditions for existence of the c.s. solution.
We seek solution of the system (10) in the form
where
Theorem 3. Assume that Then the weak compactly support solutions
(c.s.s) of the system (10) as has asymptotic
where the coefficients satisfied to system of the algebraic equations
Proof. It is easy to check that
and
We will show that the functions should be main member of asymptotic of solution of the system (10). For this goal we search the solution of system (10) in the form
By using expression (10) it is easy to cheek that
Therefore according transformation (16) the system (10) reduced to the system
where
Analysis of solution of last system shows that as where constants are the solutions of the algebraic system equations
Theorem 4. Let Then regular (quenching) solution of the system (10) as has asymptotic
Here
1) if then the coefficients are the roots of the nonlinear system of the algebraic equations
2) if then the coefficients are the roots of the nonlinear system of the algebraic equations
Proof. We will seek a solution of system (10) in following form
Since
By substituting (21) into (10) we get
where
Analyzing of solutions system (22) when we conclude that the solutions of this system where constants are solutions of the algebraic system (19), (20).
Cite this paper
MersaidAripov,Shakhlo A.Sadullaeva, (2015) Qualitative Properties of Solutions of a Doubly Nonlinear Reaction-Diffusion System with a Source. Journal of Applied Mathematics and Physics,03,1090-1099. doi: 10.4236/jamp.2015.39135
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