Asymptotic of Local Solutions to Burger’s Equation with Nonlinear Reaction –Diffusion process
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2021-09-28
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Abstract
In this thesis, we study the second order degenerate singular parabolic
equations that describe the nonlinear Burger equation with the nonlinear
reaction and weak diffusion forces. This model has applications in many
areas of science and engineering , such as fluid flow, chemical reactions,
and population evolution in mathematical biology. Our goal in this thesis is
to study the qualitative behavior of local solutions and development of the
interfaces with a finite speed of propagation in the irregular domain.
In the first part, we introduce the local solutions of the equation in the
regions where diffusion dominates over the equation limits, finding an
approximate solution along the boundary curves, establishing determinants
and constraints that achieve the existence and uniqueness of the solution, and
applying the comparison theorem in the irregular fields with boundary
curves. We also discussed the analysis of the growth in the interfaces, when
both the reaction and the diffusion are in balance and both are more powerful
than the nonlinear Burger term so that the interface function may expand or
shrink or remain stationary.
In the second part of the thesis, we concentrate on proving the existence of
a solution to the traveling wave equation when the Burger term and the
reaction are higher than the diffusion.
Among the methods used for solving the technique of rescaling and blow-up
techniques for the identification of the asymptotics of the barriers and
application of the comparison theorem in non-cylindrical domains with
characteristic boundary curves.
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