This innovative monograph contains a detailed asymptotic analysis of hyperbolic systems, systems of parabolic equations, semilinear parabolic equations with singular perturbations. All these equations are mathematical models for various phenomena arising recently in biology, chemistry, and engineering. The main method of investigation is the so-called Vishik-Lyusternik method which was applied before only to linear equations and systems of equations. The monograph offers a successful extension of this method to some nonlinear cases.
The material covered by the monograph is organized as follows. The first part has an introductory character; it contains the authors’ presentation of the Vishik-Lyusternik method and some facts from the theory of evolution equations in Hilbert spaces. Part II is devoted to an investigation of boundary problems with algebraic and dynamic conditions of the telegraph system. The authors determine formally some asymptotic expansions of the solutions of such problems for this system and find out the corresponding boundary layer functions. The main results of this part are existence, uniqueness and high regularity for all terms of these asymptotic expansions. Part III is concerned with the coupling of some boundary value problems, considered in two subdomains of a given domain, with transmission conditions at the interface. In Part IV the authors consider elliptic and hyperbolic regularization of a semilinear heat equation.
The authors designate their monograph for researchers and graduate students.