On the Qualitative Theory of the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations
Jeli, Roqia Abdullah
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This dissertation presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear second order parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration ut − ( |ux| p−2 ux ) x +buβ = 0, p > 1, β > 0. (1) Nonlinear partial differential equation (1) is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in either slow (p > 2) or fast (1 < p < 2) regime and nonlinear state dependent reaction (b > 0) or absorption (b < 0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β,sign b, and asymptotics of the initial function near its support. In the fast diffusion regime strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are generaliii ization of the methods developed in U.G. Abdulla & J. King, SIAM J. Math. Anal., 32, 2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002), 541-560 and based on rescaling laws for the nonlinear PDE and blow-up techniques for the identification of the asymptotics of the solution near the interfaces, construction of barriers using special comparison theorems in irregular domains with characteristic boundary curves.