Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations

Abstract

A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

Singularly perturbed reaction diffusion equations with time delay

“…Recently, many scholars such as Ni and Wei [2] , Alfaro et al [3] , Khasminskii and Yin [4] , Marques [5] , and Bobkova [6] have done a great deal of work. Mo et al considered a class of nonlinear boundary value problems for differential equations [7] , reaction diffusion equations [8][9][10] , elliptic equations [11] , shock layer solutions [12] , solitary waves [13] , laser pulses [14] , ocean sciences [15][16][17] , and atmospheric physical problems [18][19] .…”

A class of boundary value problems for third-order differential equation with a turning point

“…Recently, many scholars such as Ni and Wei [2] , Bartier [3] , Libre et al [4] , and Guarguaglini and Natalini [5] have done much work. Using the method of differential inequalities and others, Mo et al considered also a class of reaction diffusion problems [6] , the shock wave [7] , the soliton wave [8][9][10][11] , the laser pulse [12][13] , the ocean science [14][15][16] , and the atmospheric physics [17][18][19] , etc. In this paper, using a special method, we construct a class of singularly perturbed differential equation boundary value problems with a turning point.…”

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

“…Recently, many scholars, such as Ni and Wei [2] , Zhang [3] , Khasminskii and Yin [4] , Marques [5] , and Bobkova [6] , have done a great deal of work. Using the differential inequalities and other methods, Mo et al considered the singularly perturbed nonlinear boundary value problems for the ordinary differential equations [7] , the reaction diffusion equations [8][9][10] , the boundary value problems of the elliptic equations [11] , the ecological problems [12] , the shock layer solution of the nonlinear equations for the singularly perturbed problems [13][14] , and the problems of atmospheric physics [15][16][17][18] . In this paper, using a special singularly perturbed method, we study a class of initial boundary value problems for the reaction equations with two parameters.…”