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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

The system (1) in the domain, where

The process of the reaction-diffusion with double nonlinearity in the case of one equation has been investigated by many authors (see [

was established in [

and

where

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 [

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

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

Here, we obtain

Which are the solutions of following equations

Substituting (3), the system (1) is reduced to the following system of equations

where the functions

It is easy to establish that the system (4) has approximately self-similar solution of kind

where

It is easy to prove that as

for

In this case for the functions

where

In the case

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 [

Theorem 1. (A global solvability). Assume

Then for sufficiently small

where the functions

Proof. For proving theorem 1 we use a comparison principle. As a comparison solution we take the functions

It is easy to check that

If

Then we have

In order to apply a comparison principle we note that

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

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

property.

Critical case. The case

Theorem 2. Let

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

This inequality due to the comparison principle completes the proof of the theorem.

Value

corresponds to Fujita type critical exponent proved earlier by Escobedo, Herrero [

Now we study asymptotic of the weak compact supported solutions (c.s.s.) of the system (10) when

where

The existence of a self-similar weak c.s. solution for the problems (10), (15) in the case

We seek solution of the system (10) in the form

where

Theorem 3. Assume that

(c.s.s)

where the coefficients

Proof. It is easy to check that

and

We will show that the functions

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

The proof of the theorem is complete.

Theorem 4. Let

Here

1) if

2) if

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

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