## On the Qualitative Theory of the Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes

##### Abstract

We consider nonlinear second order degenerate or singular parabolic equation
ut − a(um)xx + buβ + c(up)x = 0, a, m, β, p > 0, b, c ∈ R
describing reaction-diffusion-convection processes arising in many areas of science
and engineering, such as filtration of oil or gas in porous media, transport of
thermal energy in plasma physics, flow of chemically reacting fluid, evolution of
populations in mathematical biology etc. We apply the methods developed in
U.G. Abdulla, Journal of Differential Equations, 164, 2(2000), 321-354 for the
reaction-diffusion equation (c = 0) and prove the existence, uniqueness, boundary
regularity and comparison theorems for the initial-boundary value problems in non-cylindrical
domains with non-smooth boundary curves under the minimal restriction on the boundary. Constructed weak solutions are continuous up to the non-smooth
boundary if at each interior point the left modulus of the lower (respectively upper)
semicontinuity of the left (respectively right) boundary curve satisfies an upper
(respectively lower) H¨older condition near zero with H¨older exponent ν > 1
2
. The
value 1
2
is critical as in the classical theory of heat equation, and is independent
of nonlinearity parameters m, β, p, and from the degeneration or singularity of the
PDE. General theory is applied to the problem on the initial development and
asymptotics of the interfaces and local solutions near the interfaces for the reaction-diffusion-convection
equation with compactly supported initial function. Depending
on the relative strength of three competing forces such as diffusion, convection, and
reaction, the interface may expand, shrink or remain stationary. The methods used
are rescaling and blow-up techniques for the identification of the asymptotics of
the solution along the class of interface type curves, construction of the barriers and
application of the comparison theorem in non-cylindrical domains with characteristic
boundary curves, as they are developed in papers U.G. Abdulla & J.King, SIAM J.
Math. Anal., 32, 2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002),
541-560.