We show that for a monostable, bistable or combustion type of nonlinear function
f
(
u
)
f(u)
, the Stefan problem
\[
{
a
m
p
;
u
t
−
Δ
u
=
f
(
u
)
,
u
>
0
a
m
p
;
a
m
p
;
for
x
∈
Ω
(
t
)
⊂
R
n
+
1
,
a
m
p
;
u
=
0
and
u
t
=
μ
|
∇
x
u
|
2
a
m
p
;
a
m
p
;
for
x
∈
∂
Ω
(
t
)
,
\left \{ \begin {aligned} &u_t-\Delta u=f(u),\; u>0 & &\text {for}~~x\in \Omega (t)\subset \mathbb {R}^{n+1},\\ & u=0~\text {and}~u_t=\mu |\nabla _x u|^2 && \text {for}~~x\in \partial \Omega (t), \end {aligned} \right .
\]
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
\[
x
n
+
1
=
ϕ
(
x
1
,
.
.
.
,
x
n
)
=
Φ
∗
(
x
1
,
.
.
.
,
x
n
)
)
+
O
(
1
)
as
|
(
x
1
,
.
.
.
,
x
n
)
|
→
∞
,
x_{n+1}=\phi (x_1,..., x_n)=\Phi ^*(x_1,...,x_n))+O(1)\text { as }|(x_1,..., x_n)|\to \infty ,
\]
where
\[
Φ
∗
(
x
1
,
.
.
.
,
x
n
)
≔
−
[
max
1
≤
i
≤
m
ν
i
⋅
(
x
1
,
.
.
.
,
x
n
)
]
cot
α
\Phi ^*(x_1,..., x_n)\colonequals - \left [\max _{1\leq i\leq m} \nu _i\cdot (x_1,..., x_n)\right ]\cot \alpha
\]
is a solution of the eikonal equation
|
∇
Φ
|
=
cot
α
|\nabla \Phi |=\cot \alpha
on
R
n
\mathbb R^n
.