To Asymptotic of the Solution of the Heat Conduction Problem with Double Nonlinearity with Absorption at a Critical Parameter

We study the asymptotic behavior (for ) of solutions of the Cauchy problem for a nonlinear parabolic equation with a double nonlinearity, describing the diffusion of heat with nonlinear heat absorption at the critical value of the parameter . 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.


Introduction
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.
It is very important to establish the values of numerical parameters at which the nature 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.
In the domain the following Cauchy problem (1) (2) t and x are, respectively, the temporal and spatial coordinates where given numerical parameters, characterizing the nonlinear medium , ( -critical value, N-size of dimension). In case k=1, critical value of parameter was found in [2].
Problem (1.1) -(1.2) is the basis for modeling various processes of nonlinear heat diffusion, magnetic hydrodynamics, gas and liquid filtration, oil and gas, in the theory of non-Newtonian fluids, etc.
Under some suitable assumptions, the existence, uniqueness and regularity of a weak solution to the Cauchy problem (1.1) -(1.2) and their variants have been extensively investigated by many authors (see [3][4][5] and the references therein).
A lot of works studied properties of solutions of problem with critical value of parameter and were established asymptotic behavior for (see [ In [21] authors were established the long time asymptotic of solutions for the critical value of parameter To Asymptotic of the Solution of the Heat Conduction Problem with Double Nonlinearity with Absorption at a Critical Parameter for problem (1.1) -(1.2) in case m=1, p=2.
They considered following semi-linear parabolic equation The solution of problem (3) -(4) is "infinity" energy. The initial data is They proved that for problem (3) -(4) the long time asymptotic of the solutions is the following approximate self-similar solution (5) For function upper and lower bounds were obtained , where A, H -constants.
For , the approximate self-similar differs from (5), which means that for critical values the asymptotic of the solutions will change for . In [22] was considered following nonlinear heat equation with absorption in area (6) for (7) Authors established the long time asymptotic of the solution for the critical exponent . The following asymptotic (8) where the value of the numerical parameter a is determined from the law of energy conservation where В and Г is beta and gamma of Euler function. They proved that solution (8) is the long time asymptotic of the solution to problem (6) - (7) where p > 1, m > 1 or max{0, 1 -(2/N)} <m< 1, is nonnegative bounded and continuous, and proved that for , there exists a secondary critical exponent such that the solution of (1.11)-(1.12) blows up in finite time for the initial data .
The authors of [28] considered the fast diffusion equation for large t, when the degree of absorption is critical for the following equation in area They established a clear lower bound, which eliminates convergence to zero.
In this paper, we study the asymptotic behavior (for ) of solutions of the Cauchy problem (1)-(2) for a nonlinear parabolic equation with a double nonlinearity, describing the diffusion of heat with nonlinear heat absorption at the critical value of the parameter .

Asymptotic of the Solution
Based on the method of standard equation [2], the solution to the problem (1)-(2) will be find in the following form 13) Put (13) in (1) and select For we get following equation (14) We note that for large t by the Hardy theorem on the behavior of the integral [2], the function has the following asymptotic Where Now put , in (14) we get an approximately self-similar equation: To Asymptotic of the Solution of the Heat Conduction Problem with Double Nonlinearity with Absorption at a Critical Parameter (15) We take as a generalized solution of the equation (15) the following function (16) After putting (16) (1) - (2). After putting in (17), we get the following estimates To satisfy these estimates, the following conditions are sufficient These conditions are true due to conditions of the theorem 2.
From the last two theorems it follows that for all large t the self-similar solution is bounded above and below For we get the following estimates

Results of the Numerical Experiments
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.

One Dimensional Case (N=1)
From problem (1) -(2) we have following one dimensional nonlinear heat equation with initial and boundary conditions 2 1 1 1  (18) , For problem (18) we construct the spatial grid x with steps h And temporary grid with replace problem (18) when i=1 goes beyond the points, so following Milne formulas can be used [31] or (21) Scheme (19) is absolutely stable and has the first order of approximation for τ , and the second for h [29].
System of algebraic equation (19) is nonlinear is relative . For solve a system of nonlinear equations (19), we apply an iterative method and obtain following system of algebraic equation [29] (22) where initial and boundary conditions unchanged Now system of algebraic equation (3.1.5) is linear is relative . As the initial iteration for is taken from the previous time step . When counting by an iterative scheme, the accuracy of the iteration is set and the process continues until execution the following conditions Remark. In all numerical calculations we take Let following notation , .
From different scheme (22) we will find tridiagonal matrix coefficients A, B, C, F and solve following system linear equations by method Thomas [29] , .

Two-Dimensional Case (N=2)
From problem (1) -(2) we have following two-dimensional nonlinear heat equation with initial and boundary conditions   (21) is also used. Scheme (24) is stable both according to the initial data and on the right-hand side for any τ and h . Scheme has accuracy 2 2 . In this scheme, the transition from layer l to layer l+1 is carried out in two stages. In first stage find intermediate values for l+1/2 and in second stages find values for l+1 with usages founded value l+1/2 layer.
Rewrite the initial and boundary conditions as follows where [30].
For solve a system of nonlinear equations (3.2.2), we apply an iterative method and obtain following system of algebraic equation (26)   (27) Now system of algebraic equation (26) After we have 3 diagonal matrix equations with following coefficients A, B, C, F and can solve following system linear equations by method Thomas.

Visualization
Notice that very important to found appropriate initial approximation of solution depending on value of numerical parameters. Therefore, an initial and a boundary values are calculated using the founded above following asymptotic ,