Some Free Boundary Problems for The Nonlinear Degenerate Multidimensional Parabolic Equations Modeling Reaction-Diffusion Processes

Abstract

This dissertation presents a full classification of the short-time behavior of the interfaces or free boundaries for the nonlinear second order degenerate multidimensional parabolic partial differential equation (PDE) ut −∆u m +buβ = 0, x ∈ R N ,0 < t < T (1) with m > 0, β > 0,b ∈ R, arising in various applications in fluid mechanics, filtration of oil or gas in a porous media, plasma physics, reaction-diffusion equations in chemical kinetics, population dynamics in mathematical biology etc. as a mathematical model of nonlinear diffusion phenomena in the presence of the absorption or release of energy. Cauchy problem with compactly supported and nonnegative initial function u0 such that supp u0 = {|x| < R}, u0 ∼ C(R− |x|) α , as |x| → R−0, with C,α > 0 is analyzed. There is a finite speed of propagation property, and interface or free boundary emerge from the boundary of the support of the initial function either in slow diffusion regime (m > 1), or in fast diffusion regime (0 < m < 1) accompanied with strong absorption (b > 0,0 < β < m). Interface surface t = η(x) may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction or absorption terms near the boundary of support, expressed in terms of the parameters m, β,α,sign b and C. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface. In the fast diffusion regime (0 < m < 1) with weak absorption (b > 0, β ≥ m) or reaction (b < 0, β ≥ 1), there is an infinite speed of propagation, and interfaces are absent. In all these cases we prove explicit asymptotic formula for the solution at infinity. The methods of proof are based on rescaling and blow-up techniques to establish the asymptotics of solution along some interface type manifolds, followed by application of the comparison theorems in non-cylindrical domains with non-smooth and characteristic boundary manifolds. The latter is developed in U.G. Abdulla, Trans. Amer. Math. Soc. 357, 1, 2005, 247-265, while the former is based on the generalization of the methods developed in U.G. Abdulla & J. King, SIAM J. Math. Anal., 32, 2, 2000, 541-560 & U.G. Abdulla, Nonlinear Analysis, 50, 2, 2002, 541-560.