Date of Award

12-2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Ugur Abdulla

Second Advisor

Hamid Rassoul

Third Advisor

Jian Du

Fourth Advisor

Kanishka Perera

Abstract

Dissertation pursues analysis of the short-time evolution of interfaces or free boundaries for the non-negative solutions of the nonlinear degenerate second order parabolic partial differential equation (PDE) ut = ( u m ) xx +b ( u γ ) x , x ∈ R,t > 0; m > 1, γ > 0,b ∈ R (1) modeling diffusion-convection processes arising in fluid or gas flow in a porous media, plasma physics, population dynamics in mathematical biology and other applications. Due to the implicit degeneration (m > 1), PDE (1) it possesses a property of the finite speed of propagation and develops interfaces or free boundaries separating the region where a solution is positive from the region where it vanishes. The direction of the movement of interfaces and their asymptotic properties depends on the relative strength of the diffusion and convection terms (m vs. γ), direction of the movement of the convection (sign b), asymptotics of the initial function near its support, and whether left or right interface curve is under consideration. Classification of the direction of the movement of the interfaces is presented by Alvarez, Diaz & Kersner, 1986. Dissertation presents a classification of the short-time asymptotics of the interfaces and local solutions near the interfaces depending on the parameters of the model. Proof methods are based on the general theory of the nonlinear reaction-diffusion equations in non-cylindrical and non-smooth domains (Abdulla, J. Diff. Eq., 164, 2, 2000), scaling laws for the identification of the asymptotics of solutions along with construction of local barriers using special comparison theorems as it is developed in Abdulla & King, SIAM J. Math. Anal., 32, 2, 2000; Abdulla, Nonlinear Analysis, 50, 4, 2002.

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