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Erschienen in:
2005 | OriginalPaper | Buchkapitel
Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension
verfasst von : Sergei Shmarev
Erschienen in: Trends in Partial Differential Equations of Mathematical Physics
Verlag: Birkhäuser Basel
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We study the Cauchy-Dirichlet problem for the degenerate parabolic equation
with the parameters
a
∊
,
m
> 1,
p
> 0, satisfying the condition
m
+
p
≥ 2. The problem domain
ɛ
is the exterior of the cylinder bounded by a simple-connected surface
S
, supp
u
0
is an annular domain
. We show that the velocity of the outer interface Γ = ∂ {supp
u
(
x, t
)} is given by the formula
where II(
x, t
) is a solution of the degenerate elliptic equation
,depending on
t
as a parameter. It is proved that the solution and its interface Γ preserve their initial regularity with respect to the space variables, and that they are real analytic functions of time
t
. We also show that the regularity of the velocity v is better than it was at the initial instant. For the space dimensions
n
= 1, 2, 3, these results were established in [8]. We propose a modification of the method of [8] that makes it applicable to equations with an arbitrary number of independent variables.