Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations

Abstract

Eidel’man, Butuzov, and Ivasishen. In particular, Ladyzhenskaya and Ural’tseva established sufficient conditions for the existence and uniqueness of classical and generalized solutions of main boundary-value problems in the linear case and substantiated the applicability of the Fourier method to the solution of these problems [1, pp. 295‐299]. Analogous results on the existence and uniqueness of solutions of the corresponding problems for singularly perturbed equations were obtained by Oleinik in [2]. In [3], Butuzov proposed an original method for the solution of boundary-value problems for singularly perturbed partial differential equations. According to the method of angular boundary functions developed by Butuzov, asymptotic solutions of boundary-value problems are constructed in the form of regular and boundary-layer parts. In this case, functions of the boundary-layer part are solutions of certain differential equations with constant coefficients that satisfy certain boundary conditions. In the present paper, we consider the first boundary-value problem for a degenerate, linear, singularly perturbed parabolic system. The statement of the problem is close to problems investigated by Mitropol’skii and Khoma for regularly perturbed quasilinear and nonlinear equations of the hyperbolic type [4, pp. 137‐ 225]. In the solution of the first boundary-value problem for hyperbolic equations with slowly varying coefficients, Feshchenko and Shkil’ used the Fourier method [5, pp. 226‐245]. However, the question of the convergence of the corresponding series and the possibility of their term-by-term differentiation remains open. Note that equations with slowly varying coefficients can be reduced to singularly perturbed equations by a change of the independent variable. Consider the problem