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

Aripov M,Mukimov A,Mirzayev B

doi:10.13189/ms.2019.070507

Abstract

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 ᵝ. 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.

Highlights

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
We study the asymptotic behavior 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 β

Summary

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. The initial data is solution with variable a ((T + t)log(T+ t))−k F (ξ ;a− ) ≤ u(x, t) ≤ They proved that for problem (3) - (4) the long time solution the following nonlinear parabolic equation asymptotic of the solutions is the following approximate self-similar solution ut − div(um−1 Du λ−1 Du) = f (x)u p in u(t, x). Mu et al [26] studied the secondary critical exponent for the following p-Laplacian equation with slow decay initial values: They established a clear lower bound, which eliminates convergence to zero. We study the asymptotic behavior (for t → ∞ ) 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 β

Asymptotic of the Solution

Results of the Numerical Experiments

Visualization