Numerical modeling of the solution of the system semilinear heat conduction problem with absorption at a critical parameter

Аннотация

In this paper we study the asymptotic behavior (for t→ ) of solutions of the system semilinear heat conduction problem with absorption at a critical parameter. The asymptotics were established using the method of standart equations. The proofs were carried out using the method of comparison of solutions and the maximum principle. 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. 1 INTRODUCTION u(0,x)=u (x)0, 0 (2) It is very important to establish the value of v(0,x)=v (x)0,xRN 0 numerical parameters which the character of the t and x are, respectively, the temporal and spatial asymptotic behavior of the solution will change. Such coordinates where , 1 =N 2/x2 . values of numeric parameters are called critical or 1 2 i=1 i critical values of the Fujita type. He first established Problem (1)-(2) is the basis for modeling various this for the semi-linear heat equation [1]. At the processes of heat diffusion, magnetic hydrodynamics, critical parameters we can observe new effects such gas and liquid filtration, oil and gas with nonlinear as infinite energy, localization and others. absorption part. As is well known for the numerical In an unlimited heat сconducting environment computation of a nonlinear problem, the choice of the with nonlinear absorption, the heat diffusion system initial approximation is essential, which preserves the in which is described by equation (1)-(2), some initial properties of the final speed of propagation, spatial temperature perturbation u (x)0,v (x)0 is 0 0 localization, bounded and blow-up solutions, which introduced from the outside, of a rather arbitrary guarantees convergence with a given accuracy to the form. As the results below show for critical values of solution of the problem with minimum number of , the asymptotic of the solution of the problem iterations. 1 2 (1)-(2) will be different. In the domain Q={(t,x):t0,xRN} the Under some suitable assumptions, the existence, following system semi linear heat conduction uniqueness and regularity of a weak solution to the equation Cauchy problem (1)-(2) and their variants have been L(u)− u +u−v1 =0, extensively investigated by many authors (see [3–5] 1 t and the references therein). (1) L (v)− +v−u1 =0 2 t Case    i +1 =N/2,i=1,2 u(t,x)=(T+t)ln(T+t)− N 2 g *   x 1   (5) 1 2 (T+t)2  Is called a critical case. Herero Escobedo For the function g the following estimate of solution considered the problem Cauchy to the case * system with source  2  2 x2 Aexp− g ()Hexp− ,2 = ,  4  *  4  T+t u v L(u)− +u+v1 =0, L (v)− +v+u1 =0 1 t 2 t were obtained. Where A, H are constants. and proved that condition blow up solution is 2 +1 For 1+ , the approximate self-similar solution i N/2,i=1,2 N 1 2 is different from (5), which means that for critical A lot of works studied properties of solutions values the asymptotic of the solutions will change for of problem with critical value of parameter and were established asymptotic behavior for t→(see In [22] was considered following Cauchy [1],[2],[19]-[27]). The long time asymptotic of problem for nonlinear heat equation with absorption solutions has been established for the critical value of u =(u+1)−u (6) parameter  for following problem t in areaQ=RN(0,) u = ( u p−2u ) −u* u(x,0)=u (x) for xRN (7) t 0 Authors for critical value of parameter established the u(0,x)=u (x), xRN , 0 the following long time asymptotic of the solution for where critical parameter  = p−1+ p/N in [19]. * the critical exponent = =+1+2/N. * The author in [20] established asymptotic solution for u(x,t)=((T+t)ln(T+t))−kF(;a) (8) double nonlinearity problem =x(T+t)−k/Nln(T+t)k/2 = ( um−1u ) −u* t k =N/(N+2),F(;a)=C (a2−2)1/, 0 + u(0,x)=u 0 (x), xRN C =k/2N(+1)1/,T 1,a0 0 where the value of the numerical parameter a  =m+2/N. * determined from the law of energy conservation In [21] authors were established the long time asymptotic of solutions for the critical value of w (x,t)dx=F(;a)d=M 2 M parameter =1+ for problem (1)-(2) in case one N M =CaN/kwhere part of system. 1 They considered following semi-linear parabolic C =N/2C B(N/2,1+1/)/Г(N/2) 1 0 equation u −u+u=0,t0 (3) В аnd G is beta and gamma of Euler function. t They proved that solution (8) is the long time u(0,x)=u (x)0 (4) 0 asymptotic of the solution to problem (6)-(7) by 2 =N 2 /x2,=1+ constructing lower and upper solutions. The i=1 i N following lower and upper solution with variable a The solution of problem (3)-(4) is “infinity” energy. ((T+t)log(T+t))−kF(;a )u(x,t)((T+t)log(T+t))−kF(;a ) The initial solution is 0a a u (x)=o{exp(−x2)},x→+ − + 0 In [34] authors find asymptotic for the following 0 double nonlinear heat equation They proved that for problem (3)-(4) the long time u = (