Finite Speed of the Perturbation Distribution and Asymptotic Behavior of the Solutions of a Parabolic System not in Divergence Form

The property of a finite speed of a perturbation distribution to the Cauchy problem for a parabolic system not in divergence form based on comparison method and an asymptotic behavior of a self-similar solution for both slow and fast diffusion cases are established. It is shown that the coefficients of the main term of the asymptotic of solution satisfy some system of nonlinear algebraic equations. It is found the Zeldovich-Kompaneets-Barenblatt type solution to the parabolic system.

The equations and system of equations not in divergence form, comparing with the classical divergence form equations are more close to the actual circumstances in some cases. For example, for the biological species, the diffusion of divergence form implies that the species is able to move to all locations within its environment with equal probability, but if we consider this problem with the objective conditions, the population density will affects the rate of diffusion, so a kind of 'biased' diffusion equation will be more realistic, for the non-divergence form diffusion. The diffusion rate is regulated by population density that is increasing for large populations and decreasing for small populations.
It is observed by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][19][20][21] nonlinear equations of non-divergence form are a source of new nonlinear effects, such as finite speed of a perturbation of distribution, space localization, blow up, etc. In particular, nonlinear problems in the non-divergence form of the following form 0, x , t 0, are discussed by several authors, see Friedman and Mcleod [1], Gage [2], Passo and Luckhaus [3], Wang [4] et al. Let 1 λ be the first eigenvalue of ∆ in Ω with homogeneous Dirichlet boundary condition. In paper [1], Friedman and Mcleod proved the existence of global solutions if a an 2 1 p d λ > = in (3), and the solution has a blow-up property if a an 2 1 . d p λ < = In [2] Gage studied in more detail the solutions with the blow-up property for p 2 = and set the estimates for the blow-up solution. In [5] for the cases a an 2 1 p d λ > > in problem (3), Zimmer studied the existence of global solutions. In paper [6], Chen studied the solution of problem (1) with the blow-up property 58 Finite Speed of the Perturbation Distribution and Asymptotic Behavior of the Solutions of a Parabolic System not in Divergence Form in the domain Ω for a 1 λ < , 1 p 2 < < and large initial data. In [7], Wiegner considered problem (3) for the case p 1 > and proved that the solution of problem (3) has the blow-up property for a 1 λ < . In [4], Wang et al. studied problem (3) for p 1 > . They obtained uniqueness conditions for the solution and proved that all positive solutions of problem (3) exist globally if and only if a 1 λ ≥ . In paper [3], Passo and Luckhaus studied some properties of solutions of problem (3).
Later, some authors generalized the problem (3). In [8], Raimbekov investigated some properties of solutions of the Cauchy problem for a nonlinear degenerate parabolic equation in a non-divergence form with variable density ( ) He studied a self-similar solution of the Zeldovich-Kompaneets-Barenblatt type, established the asymptotics of self-similar solutions in the case of fast and slow diffusion.
In [9], Zhou et al. investigated positive solutions of a doubly degenerate parabolic equation in a non-divergent form with a gradient term ( ) with zero Dirichlet boundary conditions. They established the local existence of weak solutions of the problem, and then determined how the gradient term affects the behavior of the solutions. Nonlinear systems of equations in a non-divergence form were discussed in the papers of Wang, Xie et al.
In [10] Wang and Xie considered for the system In [11], Wang studied for the system > λ , and the initial datum ( ) u ,..., u 10 n0 satisfies to some assumptions, then the positive classical solution is unique and blows up in finite time.
Later, in [19][20][21], the authors studied the Cauchy problem for the system They studied the properties of self-similar solutions of the system in a two-component medium with variable density and source, in particular, solutions of Zeldovich-Kompaneets-Barenblatt type for cross-diffusion systems of a non-divergent type were constructed, and slow and fast diffusion cases were investigated.
The system (1) is degenerate. Therefore it does not have classical solution in the domain Q at 0, 0, 0, . In this case we consider weak solution with v t x and we introduce the following definitions of solution (see [8,10,11,13,14]).

Definition 1 (weak solution). A nonnegative functions
with respect to x , if the following conditions ( )

Definition 3 (vanishing-at-infinity solution). A solution
. → +∞ t In this paper we study asymptotic behavior of self-similar solutions for a degenerate parabolic system not in divergence form (1) for slow and the fast diffusion cases. And based on comparison method the property FSPD of the Cauchy problem for a cross-diffusion parabolic system not in divergence form is established. On the basis of this asymptotic of solutions, suitable initial approximations are offered in the iterative process for the slow and fast diffusion cases, depending on the values of the numeric parameters.

The Self-similar System of Equations
Here we provide a method of nonlinear splitting [14] for construction of self-similar system of equations. We look for It is easy to check that for unknown functions , w ψ we obtain the following system of equations and in the case Put in (5) and (6) where ξ is the self-similar variable.
And in the case ( ) It is easy to check that following functions In accordance with the statement of the original problem we will consider nontrivial, nonnegative solutions of the system equations (8)-(9) satisfying the following conditions: Applying the method of comparison of solutions [13] and method of standard equations [14] for solving the problem (8), (10) we obtain the estimates for the solution of the problem (1)- (2).

Slow Diffusion (Case
It should be noted, that the functions (10).
Then the classical solution ( ) u, v of the problem (1)-(2) exists globally and it satisfies estimation Proof. Theorem 1 is proved by comparing solution methods. As comparable functions we take the functions (12). Then, according (8) and From these expressions, we find, that for execution, the inequalities Lu 0, Lv 0 ≤ ≤ + + are enough to fulfill the The proof of the theorem 1 is completed. Thus, we obtained FSPD property of the solutions of the Cauchy problem (1), (2).

The Asymptotic Behavior of
Self-similar Solutions of the Problem (8), (9), (11) We introduce the notations: . Then the following theorems are valid.
we obtain the identity  2) just one of the inequalities (20) are valid and therefore, from (19) it follows that graph of the Accordingly, the function v ( ) i η (i=1,2) has a limit at η → +∞ .

Fast Diffusion (Case
Then the classical solution ( ) u, v of the problem (1)-(2) exists globally and it satisfies estimation The proof of the Theorem 4 is similar to the proof of the Theorem 1.
To solve the system of nonlinear equations apply iterative methods and obtained: To apply the tridiagonal matrix algorithm sufficient to require that the coefficients (26) satisfy the conditions For the iterative scheme (26) tridiagonal matrix algorithm is stable and gives the unique solution.

Numerical Analysis of Solutions
It is important to choose a proper initial approximation that preserves its nonlinearity properties. On the basis of the above qualitative studies, we produced numerical calculations.
Since we have an asymptotic solution (21) and (23), we can use the numerical data of the asymptotic solution for the initial approximation.
The numerical results show quick convergence of the iterative process to the solution of the Cauchy problem (1)-(2), due to the successful choice of the initial approximation. Below, some numerical experiment results for different numerical parameter values are presented.
Programs for the numerical solution of nonlinear systems not in divergence form developed in MATLAB. Programs are compact. By the user is entered the necessary numerical parameters. At the end of the file automatically displays the calculated results in the form of matrices and graphic. In the same place by running animation can trace the evolution of the process in time.
Let us give some results of numerical experiments. Grid step is small enough h=0.05, the number of nodes N=1000 and the iteration accuracy is defined 3 10 − = ε . The score was conducted until t=2 with a step 0.02 = κ . The initial approximation was taken in the form: Results of the numerical experiments and graphs show that the self-similar solutions are very appropriate approximations. Fig.1 shows a compactly supported solution of the problem (1)-(2). In Fig.2 show the properties of the solution of the problem (1)-(2), vanishing at infinity.
Number of iterations is not exceeding of 6 for different values of numerical parameters.

Conclusions
The property a FSPD of the Cauchy problem for a cross diffusion parabolic system not in divergence form based on comparison method is considered. An asymptotic behavior of self-similar solutions for slow and fast diffusion cases are established. It is shown that coefficients of main terms of asymptotic of solution satisfy to some system of a nonlinear algebraic equation.
Results of computational experiments show that the self-similar solutions are very appropriate approximation. And the iterative method based on the Picard's method is effective for the solution of nonlinear problems and retains the nonlinear effects, if we will use as an initial approximation the solutions of self-similar equations, constructed by the method of nonlinear splitting and by the method of standard equation [14][15][16][17][18][19][20][21].