On a singularly perturbed problem of the nonlinear thermal conductivity in the case of balanced nonlinearity

On the basis of the modified asymptotic method of boundary functions and the asymptotic method of differential inequalities, the question of the existence of Lyapunov-stable stationary solutions with internal layers of the nonlinear heat equation in the case of nonlinear dependence of the power of thermal sources from temperature is investigated. The main conditions of the existence of such solutions are discussed. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and suggest an efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use an asymptotic method of differential inequalities. The main complexity is related to the description of the transition surface in whose neighborhood the internal layer is localized. We use a more efficient method for localizing the transition surface, which permits one to develop an approach to a more complicated case of balanced nonlinearity. The results can be used to create a numerical algorithm which uses the asymptotic analyses to construct space-non-uniform meshes while describing internal layer behaviour of the solution. As an illustration, we consider a problem on the plane that allows us to visualize the numerical calculations. Numerical and asymptotic solutions of zero order are compared for different values of the small parameter.