Singularly Perturbed Stationary Diffusion Model with a Cubic Nonlinearity

Abstract

We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to study the existence of asymptotically Lyapunov stable solutions with internal layers as stationary solutions of the corresponding parabolic problems. Sufficient conditions are established for the existence of such solutions in the form of some conditions on the coefficients of the equation, an asymptotic approximation to the solution of an arbitrary accuracy order with coefficients is constructed in closed form, and the formal constructions are justified. This result can be used for creating efficient numerical algorithms for direct and coefficient inverse problems for stationary equations of the reaction–diffusion–advection type as well as for constructing test examples. Heat and mass transfer problems occurring in chemical industry are pointed out as possible applications of our results.