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2015-2015

N/A

The project calls for the development of a new direction in the study of nonlinear singularly perturbed problems in which the degenerate equation (with the small parameter equal to zero) has multiple roots (identically or in some subregion of the domain of consideration). Such cases arise in many applied problems, in particular, in problems of chemical kinetics in the presence of fast reactions, and are not sufficiently studied. In recent years, the scientific group representing this project has developed new approaches in the asymptotic study of nonlinear singularly perturbed problems - the construction of asymptotic solutions and their justification. In particular, methods have been developed for constructing asymptotic approximations of contrast structures - solutions with internal transition layers. We develop an asymptotic method of differential inequalities that uses the formal asymptotics for constructing upper and lower solutions in the justification of the asymptotics. These methods proved to be very effective in many nonlinear singularly perturbed problems, including for some classes of singularly perturbed problems with multiple roots of the degenerate equation. The purpose of this project is the development of these methods in relation to a new more complex class of tasks. It is supposed to work in the following directions: 1. Development of methods for constructing asymptotic approximations of solutions of nonlinear singularly perturbed differential reaction-diffusion equations in which the degenerate equation has multiple roots. 2. The development of the asymptotic method of differential inequalities for the proof of existence theorems and estimates of the remainder terms of the asymptotics of solutions of problems of this type. 3. Investigation of stability issues and the formation of such solutions. More... »

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