A free boundary value problem related to the combustion of a solid: flux boundary conditions
Authors: Cannon John R. , Matheson Alec L. 
Published: December 2016

We demonstrate the existence, uniqueness, and continuous dependence upon the data for the solution

(
u
,
v
,
s
)

\left ( u, v, s \right )

of the free boundary value problem:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals alpha u Subscript x x Baseline comma v Subscript t Baseline equals beta v Subscript x x comma Baseline Baseline 0 greater-than x greater-than s left-parenthesis t right-parenthesis comma 0 greater-than t less-than-or-equal-to upper T">

u
t

=
α

u

x
x

,

v
t

=
β

v

x
x
,

0
>
x
>
s

(
t
)

,

0
>
t
≤
T

{u_t} = \alpha {u_{xx}}, \qquad {v_t} = \beta {v_{xx,}} \qquad 0 > x > s\left ( t \right ), \qquad 0 > t \le T

\]
,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x comma 0 right-parenthesis equals phi left-parenthesis x right-parenthesis comma v left-parenthesis x comma 0 right-parenthesis equals psi left-parenthesis x right-parenthesis comma 0 less-than-or-equal-to x less-than-or-equal-to s left-parenthesis 0 right-parenthesis equals b">

u

(
x
,
0
)

=
ϕ

(
x
)

,

v

(
x
,
0
)

=
ψ

(
x
)

,

0
≤
x
≤
s

(
0
)

=
b

u\left ( x, 0 \right ) = \phi \left ( x \right ), \qquad v\left ( x, 0 \right ) = \psi \left ( x \right ), \qquad 0 \le x \le s\left ( 0 \right ) = b

\]
.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus alpha u Subscript x Baseline left-parenthesis 0 comma t right-parenthesis equals f left-parenthesis t right-parenthesis comma minus beta v Subscript x Baseline left-parenthesis 0 comma t right-parenthesis equals g left-parenthesis t right-parenthesis comma 0 greater-than t less-than-or-equal-to upper T">

−
α

u
x

(
0
,
t
)

=
f

(
t
)

,

−
β

v
x

(
0
,
t
)

=
g

(
t
)

,

0
>
t
≤
T

- \alpha {u_x}\left ( 0, t \right ) = f\left ( t \right ), \qquad - \beta {v_x}\left ( 0, t \right ) = g\left ( t \right ), \qquad 0 > t \le T

\]
,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha u Subscript x Baseline left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis equals minus left-parenthesis gamma plus u left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis ModifyingAbove s With dot left-parenthesis t right-parenthesis comma beta v Subscript x Baseline left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis equals left-parenthesis mu minus v left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis ModifyingAbove s With dot left-parenthesis t right-parenthesis comma 0 greater-than t less-than-or-equal-to upper T">

α

u
x

(
s

(
t
)

,
t
)

=
−

(
γ
+
u

(
s

(
t
)

,
t
)

)

s
˙

(
t
)

,

β

v
x

(
s

(
t
)

,
t
)

=

(
μ
−
v

(
s

(
t
)

,
t
)

)

s
˙

(
t
)

,

0
>
t
≤
T

\alpha {u_x}\left ( s\left ( t \right ), t \right ) = - \left ( \gamma + u\left ( s\left ( t \right ), t \right ) \right )\dot s\left ( t \right ), \qquad \beta {v_x}\left ( s\left ( t \right ), t \right ) = \left ( \mu - v\left ( s\left ( t \right ), t \right ) \right )\dot s\left ( t \right ), \qquad 0 > t \le T

\]
,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove s With dot left-parenthesis t right-parenthesis equals nu left-parenthesis v left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis e x p left-parenthesis negative delta slash v left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis upper F left-parenthesis u left-parenthesis s left-parenthesis t right-parenthesis comma t right-parenthesis right-parenthesis comma 0 greater-than t less-than-or-equal-to upper T">

s
˙

(
t
)

=
ν

(
v

(
s

(
t
)

,
t
)

)

e
x
p

(
−
δ

/

v

(
s

(
t
)

,
t
)

)

F

(
u

(
s

(
t
)

,
t
)

)

,

0
>
t
≤
T

\dot s\left ( t \right ) = \nu \left ( v\left ( s\left ( t \right ), t \right ) \right ) exp\left ( - \delta /v\left ( s\left ( t \right ), t \right ) \right )F\left ( u\left ( s\left ( t \right ), t \right ) \right ), \qquad 0 > t \le T

\]
, where

α
,
β
,
γ
,
δ

\alpha , \beta , \gamma , \delta

, and

μ
\mu

are positive constants related to the physical constants.