Semi-waves with Λ-shaped free boundary for nonlinear Stefan problems: Existence
Authors: Du Yihong , Gui Changfeng , Wang Kelei , Zhou Maolin 
Published: March 2021

We show that for a monostable, bistable or combustion type of nonlinear function

f
(
u
)

f(u)

, the Stefan problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column Blank 2nd Column a m p semicolon u Subscript t Baseline minus normal upper Delta u equals f left-parenthesis u right-parenthesis comma u greater-than 0 3rd Column a m p semicolon 4th Column a m p semicolon for x element-of normal upper Omega left-parenthesis t right-parenthesis subset-of double-struck upper R Superscript n plus 1 Baseline comma 2nd Row 1st Column Blank 2nd Column a m p semicolon u equals 0 and u Subscript t Baseline equals mu StartAbsoluteValue nabla Subscript x Baseline u EndAbsoluteValue squared 3rd Column a m p semicolon 4th Column a m p semicolon for x element-of partial-differential normal upper Omega left-parenthesis t right-parenthesis comma EndLayout">

{

a
m
p
;

u
t

−
Δ
u
=
f
(
u
)
,

u
>
0

a
m
p
;

a
m
p
;
for
 
 
x
∈
Ω
(
t
)
⊂

R

n
+
1

,

a
m
p
;
u
=
0
 
and
 

u
t

=
μ

|

∇
x

u

|

2

a
m
p
;

a
m
p
;
for
 
 
x
∈
∂
Ω
(
t
)
,

\left \{ \begin {aligned} &u_t-\Delta u=f(u),\; u>0 & &\text {for}~~x\in \Omega (t)\subset \mathbb {R}^{n+1},\\ & u=0~\text {and}~u_t=\mu |\nabla _x u|^2 && \text {for}~~x\in \partial \Omega (t), \end {aligned} \right .

\]
has a traveling wave solution whose free boundary is

Λ
\Lambda

-shaped, and whose speed is

κ
\kappa

, where

κ
\kappa

can be any given positive number satisfying

κ
>

κ
∗

\kappa >\kappa _*

, and

κ
∗

\kappa _*

is the unique speed for which the above Stefan problem has a planar traveling wave solution. To distinguish it from the usual traveling wave solutions, we call it a semi-wave solution. In particular, if

α
∈
(
0
,
π

/

2
)

\alpha \in (0, \pi /2)

is determined by

sin
⁡
α
=

κ
∗

/

κ

\sin \alpha =\kappa _*/\kappa

, then for any finite set of unit vectors

{

ν
i

:
1
≤
i
≤
m
}
⊂

R

n

\{\nu _i: 1\leq i\leq m\}\subset \mathbb R^n

, there is a

Λ
\Lambda

-shaped semi-wave with traveling speed

κ
\kappa

, with traveling direction

−

e

n
+
1

=
(
0
,
.
.
.
,
0
,
−
1
)
∈

R

n
+
1

-e_{n+1}=(0,...,0, -1)\in \mathbb {R}^{n+1}

, and with free boundary given by a hypersurface in

R

n
+
1

\mathbb {R}^{n+1}

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript n plus 1 Baseline equals phi left-parenthesis x 1 comma period period period comma x Subscript n Baseline right-parenthesis equals normal upper Phi Superscript asterisk Baseline left-parenthesis x 1 comma period period period comma x Subscript n Baseline right-parenthesis right-parenthesis plus upper O left-parenthesis 1 right-parenthesis as StartAbsoluteValue left-parenthesis x 1 comma period period period comma x Subscript n Baseline right-parenthesis EndAbsoluteValue right-arrow normal infinity comma">

x

n
+
1

=
ϕ
(

x
1

,
.
.
.
,

x
n

)
=

Φ
∗

(

x
1

,
.
.
.
,

x
n

)
)
+
O
(
1
)
 as 

|

(

x
1

,
.
.
.
,

x
n

)

|

→
∞
,

x_{n+1}=\phi (x_1,..., x_n)=\Phi ^*(x_1,...,x_n))+O(1)\text { as }|(x_1,..., x_n)|\to \infty ,

\]
where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Superscript asterisk Baseline left-parenthesis x 1 comma period period period comma x Subscript n Baseline right-parenthesis colon-equal minus left-bracket max Underscript 1 less-than-or-equal-to i less-than-or-equal-to m Endscripts nu Subscript i Baseline dot left-parenthesis x 1 comma period period period comma x Subscript n Baseline right-parenthesis right-bracket cotangent alpha">

Φ
∗

(

x
1

,
.
.
.
,

x
n

)

≔

−

[

max

1
≤
i
≤
m

ν
i

⋅
(

x
1

,
.
.
.
,

x
n

)
]

cot
⁡
α

\Phi ^*(x_1,..., x_n)\colonequals - \left [\max _{1\leq i\leq m} \nu _i\cdot (x_1,..., x_n)\right ]\cot \alpha

\]
is a solution of the eikonal equation

|

∇
Φ

|

=
cot
⁡
α

|\nabla \Phi |=\cot \alpha

on

R

n

\mathbb R^n

.