A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.

We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.

基于渐近分析方法的进一步发展和使用，提出了一种研究非线性热传导多维奇异摄动问题的新方法。我们研究了非线性传热方程的具有边界和内部过渡层（稳态热结构）的经典李雅普诺夫稳态解的存在性问题。我们建议构建过渡层局部化表面的渐近近似的有效算法。为了证明构造的形式渐近，我们使用比较原理。

我们考虑应用渐近分析的结果来解决从热结构内层的已知位置重构导热系数的温度依赖性的逆问题。