Qualitative Properties of Solutions of a Doubly Nonlinear Reaction-Diffusion System with a Source

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.


Introduction
Let's consider properties of the Cauchy problem for the following system of nonlinear reaction-diffusion equations in the domain ∈ , +∞ . 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 1 2 v u β β , . Cases, when 1 2 2 0 k l p m m =, = , = = , were considered in [1]- [7].
The system (1) in the domain, where 0 u v = = 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: 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 ( ) ( ) 1 2 l t l t , In the case of ( ) ( ) x l t = 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 [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: 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 2 p = .
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

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 Which are the solutions of following equations Substituting (3), the system (1) is reduced to the following system of equations are chosen as following It is easy to establish that the system (4) has approximately self-similar solution of kind where ( ) It is easy to prove that as t → ∞ for where H -Hardy's body [2], 1 2 c c , 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 ( ) const t γ = 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 In the case 2 p = or 1 m = 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 2 m p β = + − the existence of positive solutions of the Equation (10) was studied in [14].

A Global Solvability of Solutions
We prove properties of a global solvability of weak solutions of the system (1) using a comparison principle (see [1]). For this goal we construct a new system of equation using the standard equation method as in [3]: In the case, Then for sufficiently small In order to apply a comparison principle we note that ( ) 0 Then according to the hypotheses of Theorem 1 and comparison principle we have The proof of the theorem is complete. We notice that if ( ) It means that

Property of Finite Speed of a Perturbation
Proof of the theorem is based on the comparison principle. We take for comparison the functions

Asymptotic of the Self-Similar Solutions
Now we study asymptotic of the weak compact supported solutions (c.s.s.) of the system (10) where 0 d < < +∞ . The existence of a self-similar weak c.s. solution for the problems (10), (15) in the case 0 2 k p = , = was studied in [6] where the authors obtained conditions for existence of the c.s. solution.
We seek solution of the system (10)