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.