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.

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具有三次非线性的奇异扰动稳态扩散模型

我们考虑具有三次非线性的多维奇异摄动平稳扩散模型。对于此类模型，将经典的渐近分析方法扩展到多维问题的情况下，采用改进的边界函数渐近方法和基于比较原理的微分不等式渐近方法来研究存在性渐近 Lyapunov 稳定解的内层作为相应抛物线问题的平稳解。以方程系数的某些条件的形式为此类解的存在建立了充分条件，以封闭形式构造了具有系数的任意精度阶解的渐近逼近，并证明了形式构造是合理的。该结果可用于为反应-扩散-平流类型的平稳方程的直接和系数逆问题创建有效的数值算法以及构建测试示例。化学工业中发生的传热和传质问题被指出作为我们结果的可能应用。