Traveling Waves involving the Fractional
Laplacians
Mingfeng Zhao
University of British Columbia
October 28, 2014
Joint work with Changfeng Gui
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
1 / 33
Overview
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
2 / 33
Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
3 / 33
Traveling Waves
The general form of reaction-diffusion system can be represented as
follows:
vt (t, x) − ∆x v (t, x) = f (v (t, x)),
Mingfeng Zhao(UBC)
∀t > 0, ∀x ∈ Rn .
Traveling Waves and Fractional Laplacians
(1.1)
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Traveling Waves
The general form of reaction-diffusion system can be represented as
follows:
vt (t, x) − ∆x v (t, x) = f (v (t, x)),
∀t > 0, ∀x ∈ Rn .
(1.1)
When n = 2, we say v is a traveling wave solution to (1.1) if v
has the special form v (t, x) = u(x − µt) for all (x, t) ∈ R2 .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
4 / 33
Traveling Waves
The general form of reaction-diffusion system can be represented as
follows:
vt (t, x) − ∆x v (t, x) = f (v (t, x)),
∀t > 0, ∀x ∈ Rn .
(1.1)
When n = 2, we say v is a traveling wave solution to (1.1) if v
has the special form v (t, x) = u(x − µt) for all (x, t) ∈ R2 .Then u
satisfies
−u 00 (x) − µu 0 (x) = f (u(x)), ∀x ∈ R.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
4 / 33
Traveling Waves
The general form of reaction-diffusion system can be represented as
follows:
vt (t, x) − ∆x v (t, x) = f (v (t, x)),
∀t > 0, ∀x ∈ Rn .
(1.1)
When n = 2, we say v is a traveling wave solution to (1.1) if v
has the special form v (t, x) = u(x − µt) for all (x, t) ∈ R2 .Then u
satisfies
−u 00 (x) − µu 0 (x) = f (u(x)), ∀x ∈ R.
We are interested in the traveling front and three kinds of nonlinearities: u is bounded and monotone, and
Fisher-KPP, Combustion and Bistable
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
4 / 33
Traveling Front
Figure: Traveling Front
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Traveling Waves and Fractional Laplacians
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Fisher-KPP Model
f (t) > 0 = f (0) = f (1), ∀t ∈ (0, 1),
f 0 (0) > 0
and f 0 (1) < 0
Figure: Fisher-KPP Model
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Traveling Waves and Fractional Laplacians
6 / 33
Combustion Model
∃t0 ∈ (0, 1) such that
(
f (t) = 0, ∀t ∈ [0, t0 ]
f (t) > 0 = f (1),
∀t ∈ (t0 , 1)
and f 0 (−1) < 0
Figure: Combustion Model
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Traveling Waves and Fractional Laplacians
7 / 33
Bistable or Allen-Cahn Model
∃t0 ∈ (−1, 1) such that
(
f (t) < 0 = f (−1), ∀t ∈ (−1, t0 )
f (t) > 0 = f (1), ∀t ∈ (t0 , 1)
and f 0 (±1) < 0
Figure: Bistable Model
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Traveling Waves and Fractional Laplacians
8 / 33
Bistable or Allen-Cahn Model
∃t0 ∈ (−1, 1) such that
(
f (t) < 0 = f (−1), ∀t ∈ (−1, t0 )
f (t) > 0 = f (1), ∀t ∈ (t0 , 1)
and f 0 (±1) < 0
Figure: Bistable Model
The function G (t) = −
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Rt
−1 f (u)
du is called a double well potential.
Traveling Waves and Fractional Laplacians
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Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
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∀x ∈ R,
Traveling Waves and Fractional Laplacians
(UL)
9 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Fisher-KPP Model
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
9 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Fisher-KPP Model
∃ µ∗ such that for any µ ≤ µ∗ , ∃! u such that (µ, u) is a solution
to (UL) such that
lim
x→−∞
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u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves and Fractional Laplacians
9 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Combustion Model
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
10 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Combustion Model
∃! (µ, u) as the solution to (UL) such that
lim
x→−∞
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u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves and Fractional Laplacians
10 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Bistable or Allen-Cahn Model
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
11 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Bistable or Allen-Cahn Model
∃! (µ, u) as the solution to (UL) such that
lim
x→±∞
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u(x) = ±1.
Traveling Waves and Fractional Laplacians
11 / 33
Classical Results

−u 00 (x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(UL)
Bistable or Allen-Cahn Model
∃! (µ, u) as the solution to (UL) such that
lim
x→±∞
u(x) = ±1.
Moreover, ∃ν± > 0 such that
u 0 (x) ∼ e −ν± |x| ,
Mingfeng Zhao(UBC)
as |x| → ∞.
Traveling Waves and Fractional Laplacians
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Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
12 / 33
Integral Form of Fractional Laplacians
For any 0 < s < 1, define
u(x) − u(y )
dy
(−∆) u(x) = Cn,s P.V.
n+2s
Rn |x − y |
Z
Cn,s
u(x + y ) + u(x − y ) − 2u(x)
= −
dy ,
2 Rn
|y |n+2s
s
Z
where
Cn,s =
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s22s Γ
n
n+2s
2
π 2 Γ(1 − s)
.
Traveling Waves and Fractional Laplacians
13 / 33
s-Harmonic Extension
For any 0 < s < 1, the s-Poisson kernel is defined as
Γ n+2s
y 2s
s
2
Py (x) = π
·
∀(x, y ) ∈ Rn+1
n+2s ,
+ .
π 2 Γ(s) [|x|2 + y 2 ] 2
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Traveling Waves and Fractional Laplacians
14 / 33
s-Harmonic Extension
For any 0 < s < 1, the s-Poisson kernel is defined as
Γ n+2s
y 2s
s
2
Py (x) = π
·
∀(x, y ) ∈ Rn+1
n+2s ,
+ .
π 2 Γ(s) [|x|2 + y 2 ] 2
Theorem [Caffarelli-Silvestre(2007)]
For any u ∈ C 2 (Rn ), let u(x, y ) = Pys ∗ u(x) in Rn+1
+ . Then
n+1
2
u ∈ C (R+ ) and

div [y 1−2s ∇u(x, y )] = 0, ∀(x, y ) ∈ Rn+1

+ ,


n
lim u(x, y ) = u(x), ∀x ∈ R ,
y &0

 − lim y 1−2s u y (x, y ) = ds · (−∆)s u(x), ∀x ∈ Rn ,

y &0
where
ds =
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21−2s Γ(1 − s)
s
21−2s Γ(2 − s)
=
·
.
Γ(s)
1−s
Γ(1 + s)
Traveling Waves and Fractional Laplacians
14 / 33
Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
15 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
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∀x ∈ R,
Traveling Waves and Fractional Laplacians
(FL)
16 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(FL)
Fisher-KPP Model
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
16 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(FL)
Fisher-KPP Model
Theorem [Cabré-Roquejoffre(2011)]
For any 0 < s < 1, then there is no (µ, u) as the solution to (FL)
such that
lim
x→−∞
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u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves and Fractional Laplacians
16 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Combustion Model
Mingfeng Zhao(UBC)
∀x ∈ R,
Traveling Waves and Fractional Laplacians
(FL)
17 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Combustion Model
∀x ∈ R,
(FL)
Theorem [Mellet-Roquejoffre-Sire(2011)]
If 1/2 < s < 1, ∃! (µ, u) as the solution to (FL) such that
lim
x→−∞
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u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves and Fractional Laplacians
17 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Combustion Model
∀x ∈ R,
(FL)
Theorem [Mellet-Roquejoffre-Sire(2011)]
If 1/2 < s < 1, ∃! (µ, u) as the solution to (FL) such that
lim
x→−∞
u(x) = 0,
and
lim u(x) = 1.
x→∞
Theorem [Gui-Huan(2012)]
If 0 < s ≤ 1/2, then there is no (µ, u) as the solution to (FL) such
that
lim u(x) = 0, and
lim u(x) = 1.
x→−∞
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x→∞
Traveling Waves and Fractional Laplacians
17 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Bistable or Allen-Cahn Model
Mingfeng Zhao(UBC)
∀x ∈ R,
Traveling Waves and Fractional Laplacians
(FL)
18 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Bistable or Allen-Cahn Model
∀x ∈ R,
(FL)
Theorem [Cabré-Soá Morales(2005), Cabré-Sire(2010)]
For any 0 < s < 1, if
R1
−1 f (t)
dt = 0 (i.e., balanced), then ∃! u
as the solution to (FL) with µ = 0 such that
lim
x→±∞
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u(x) = ±1.
Traveling Waves and Fractional Laplacians
18 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Bistable or Allen-Cahn Model
∀x ∈ R,
(FL)
Theorem [Cabré-Soá Morales(2005), Cabré-Sire(2010)]
For any 0 < s < 1, if
R1
−1 f (t)
dt = 0 (i.e., balanced), then ∃! u
as the solution to (FL) with µ = 0 such that
lim
x→±∞
u(x) = ±1.
Moreover, we have
u 0 (x) ∼
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1
|x|1+2s
,
as |x| → ∞.
Traveling Waves and Fractional Laplacians
18 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(FL)
Bistable or Allen-Cahn Model
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
19 / 33
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
∀x ∈ R,
(FL)
Bistable or Allen-Cahn Model
Theorem [Volper-Nec-Nepomnyashchy(2010)]
For any 0 < s < 1, if f is piece-wise linear, then ∃! (µ, u) as the
solution to (FL) such that
lim
x→±∞
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u(x) = ±1.
Traveling Waves and Fractional Laplacians
19 / 33
Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
20 / 33
Main Results
Theorem A [Gui-Zhao]
For any 0 < s < 1 and any bistable nonlinearity f ∈ C 2 (R), there
exists a unique pair (µ, u) as the solution of the following problem:

(−∆)s u(x) − µu 0 (x) = f (u(x)), ∀x ∈ R,



u 0 (x) > 0, ∀x ∈ R,


 lim u(x) = ±1, and u(0) = 0.
x→±∞
Moreover, we have
u 0 (x) ∼
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1
|x|1+2s
,
as |x| → ∞.
Traveling Waves and Fractional Laplacians
21 / 33
Main Results
Theorem B [Gui-Zhao]
For any 0 < s < 1, any bistable nonlinearity f ∈ C 2 (R) and any
function g ∈ C 2 (R). For any small h, let fh (t) = f (t) + hg (t) ,
and (µh , uh ) be the traveling wave solutions corresponding to fh .
Then there exist some constants H > 0 and C > 0 such that
0 < uh0 (x) ≤
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H.
Moreover, if f is balanced, we have
R1
− −1 g (t) dt
µh
lim
=R
.
0
2
h→0 h
R |u0 (x)| dx
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Traveling Waves and Fractional Laplacians
22 / 33
Uniqueness
Sliding Method:
Take any two solutions (µ1 , u1 ) and (µ2 , u2 ) such that µ1 ≥ µ2 and
u1 (0) = u2 (0). For any t > 0, let
wt (x) = u1 (x + t) − u2 (x),
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∀x ∈ R.
Traveling Waves and Fractional Laplacians
23 / 33
Uniqueness
Sliding Method:
Take any two solutions (µ1 , u1 ) and (µ2 , u2 ) such that µ1 ≥ µ2 and
u1 (0) = u2 (0). For any t > 0, let
wt (x) = u1 (x + t) − u2 (x),
∀x ∈ R.
Step I: ∃T0 1 such that for all t ≥ T0 , we have
wt (x) > 0,
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∀x ∈ R.
Traveling Waves and Fractional Laplacians
23 / 33
Uniqueness
Sliding Method:
Take any two solutions (µ1 , u1 ) and (µ2 , u2 ) such that µ1 ≥ µ2 and
u1 (0) = u2 (0). For any t > 0, let
wt (x) = u1 (x + t) − u2 (x),
∀x ∈ R.
Step I: ∃T0 1 such that for all t ≥ T0 , we have
wt (x) > 0,
∀x ∈ R.
Step II: For all t > 0, we have wt (x) > 0 in R.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
23 / 33
Uniqueness
Sliding Method:
Take any two solutions (µ1 , u1 ) and (µ2 , u2 ) such that µ1 ≥ µ2 and
u1 (0) = u2 (0). For any t > 0, let
wt (x) = u1 (x + t) − u2 (x),
∀x ∈ R.
Step I: ∃T0 1 such that for all t ≥ T0 , we have
wt (x) > 0,
∀x ∈ R.
Step II: For all t > 0, we have wt (x) > 0 in R.
Step III: w0 (x) ≡ 0, i.e., u1 (x) ≡ u2 (x) in R.
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Traveling Waves and Fractional Laplacians
23 / 33
Asymptotic Behaviors at Infinity
For any t > 0, construct a balanced bistable nonlinearity ft such
that ft0 (±1) = − 1t , and the standing wave solution vt such that
vt0 (x) ∼
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1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Traveling Waves and Fractional Laplacians
24 / 33
Asymptotic Behaviors at Infinity
For any t > 0, construct a balanced bistable nonlinearity ft such
that ft0 (±1) = − 1t , and the standing wave solution vt such that
vt0 (x) ∼
1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Upper Bound: For any δ > 0, let wδ,t := δvt0 − u 0 .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
24 / 33
Asymptotic Behaviors at Infinity
For any t > 0, construct a balanced bistable nonlinearity ft such
that ft0 (±1) = − 1t , and the standing wave solution vt such that
vt0 (x) ∼
1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Upper Bound: For any δ > 0, let wδ,t := δvt0 − u 0 .
Step I: ∃T0 , R 1 such that
0
(−∆)s wδ,T0 − µwδ,T
+
0
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3
wδ,T0 ≥ 0,
T0
∀|x| ≥ R.
Traveling Waves and Fractional Laplacians
24 / 33
Asymptotic Behaviors at Infinity
For any t > 0, construct a balanced bistable nonlinearity ft such
that ft0 (±1) = − 1t , and the standing wave solution vt such that
vt0 (x) ∼
1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Upper Bound: For any δ > 0, let wδ,t := δvt0 − u 0 .
Step I: ∃T0 , R 1 such that
0
(−∆)s wδ,T0 − µwδ,T
+
0
3
wδ,T0 ≥ 0,
T0
∀|x| ≥ R.
Step II: ∃δ0 1 such that
wδ0 ,T0 (x) ≥ 1,
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∀|x| ≤ R + 1.
Traveling Waves and Fractional Laplacians
24 / 33
Asymptotic Behaviors at Infinity
For any t > 0, construct a balanced bistable nonlinearity ft such
that ft0 (±1) = − 1t , and the standing wave solution vt such that
vt0 (x) ∼
1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Upper Bound: For any δ > 0, let wδ,t := δvt0 − u 0 .
Step I: ∃T0 , R 1 such that
0
(−∆)s wδ,T0 − µwδ,T
+
0
3
wδ,T0 ≥ 0,
T0
∀|x| ≥ R.
Step II: ∃δ0 1 such that
wδ0 ,T0 (x) ≥ 1,
∀|x| ≤ R + 1.
Lower Bound: Similar arguments (take 0 < T1 , δ1 1).
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
24 / 33
Existence
Continuation Method:
Let f be unbalacned bistable such that G (1) > G (−1) with G 0 (t) =
−f (t). Take a balanced bistable potential G0 such that G0 (t) =
G (t) for all t ∈ [−1, t0 ]. Consider
Gθ = (1 − θ)G0 + θG , and fθ = −Gθ0 ,
∀θ ∈ [0, 1].
Figure: The Graph of Gθ
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
25 / 33
Existence
Continuation Method
Let g be the traveling wave solution corresponding to f0 . Look at
v = uθ − g , so we need to find the zero set of the mapping:
S(θ, µ, v ) = −v +((−∆)s +1)−1 [v +µv 0 +fθ (v +g )−(−∆)s +µg 0 ].
Define the solution set of S:
Σ = {θ ∈ [0, 1] : ∃ µ and v s.t. S(θ, µ, v ) = 0} .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
26 / 33
Existence
Continuation Method
Let g be the traveling wave solution corresponding to f0 . Look at
v = uθ − g , so we need to find the zero set of the mapping:
S(θ, µ, v ) = −v +((−∆)s +1)−1 [v +µv 0 +fθ (v +g )−(−∆)s +µg 0 ].
Define the solution set of S:
Σ = {θ ∈ [0, 1] : ∃ µ and v s.t. S(θ, µ, v ) = 0} .
Step I: Nondegeneracy of Linearized Equation
The first eigenfunction is simple for the fractional Laplacians.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
26 / 33
Existence
Continuation Method
Let g be the traveling wave solution corresponding to f0 . Look at
v = uθ − g , so we need to find the zero set of the mapping:
S(θ, µ, v ) = −v +((−∆)s +1)−1 [v +µv 0 +fθ (v +g )−(−∆)s +µg 0 ].
Define the solution set of S:
Σ = {θ ∈ [0, 1] : ∃ µ and v s.t. S(θ, µ, v ) = 0} .
Step I: Nondegeneracy of Linearized Equation
The first eigenfunction is simple for the fractional Laplacians.
Step II: Boundedness of (µ, v )
It’s equivalent to show the speeds µθ ’s are uniformly bounded.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
26 / 33
Estimate of Speed
Theorem
For any 0 < s < 1, let f be a bistable nonlinearity with f = −G 0 ,
then there exists some constant C > 0 which only depends on s,
the upper bound of kG kC 2 ([−1,1]) and the positive lower bound of
G (t0 ) − G (1) such that
C −1 |G (1) − G (−1)| ≤ |µ| ≤ C |G (1) − G (−1)|.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
27 / 33
Hamiltonian Identity
Z
µ
|u 0 (x)|2 dx = G (1) − G (−1).
R
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
28 / 33
Hamiltonian Identity
Z
µ
|u 0 (x)|2 dx = G (1) − G (−1).
R
Let u be the s-harmonic extension of u, consider
Z
1 ∞ 2
v (x) =
u x (x, y ) − u 2y (x, y ) y 1−2s dy
2 0
Z x
0
2
−ds −µ
|u (t)| dt + G (u(x)) − G (−1) .
−∞
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
28 / 33
Hamiltonian Identity
Z
µ
|u 0 (x)|2 dx = G (1) − G (−1).
R
Let u be the s-harmonic extension of u, consider
Z
1 ∞ 2
v (x) =
u x (x, y ) − u 2y (x, y ) y 1−2s dy
2 0
Z x
0
2
−ds −µ
|u (t)| dt + G (u(x)) − G (−1) .
−∞
Z
Step I: lim
|x|→∞
Mingfeng Zhao(UBC)
∞
|∇u(x, y )|2 y 1−2s dy = 0.
0
Traveling Waves and Fractional Laplacians
28 / 33
Hamiltonian Identity
Z
µ
|u 0 (x)|2 dx = G (1) − G (−1).
R
Let u be the s-harmonic extension of u, consider
Z
1 ∞ 2
v (x) =
u x (x, y ) − u 2y (x, y ) y 1−2s dy
2 0
Z x
0
2
−ds −µ
|u (t)| dt + G (u(x)) − G (−1) .
−∞
Z
Step I: lim
|x|→∞
∞
|∇u(x, y )|2 y 1−2s dy = 0.
0
Step II: v 0 (x) ≡ 0 in R.
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
28 / 33
Estimate of Speed

 (−∆)s u(x) − µu 0 (x) = f (u(x)),

lim
x→±∞
Mingfeng Zhao(UBC)
∀x ∈ R,
u(x) = ±1.
Traveling Waves and Fractional Laplacians
29 / 33
Estimate of Speed

 (−∆)s u(x) − µu 0 (x) = f (u(x)),

lim
x→±∞
∀x ∈ R,
u(x) = ±1.
Then
Z y
Z y
s
0
−
(−∆) u(x) · u (x) dx+ µ
|u 0 (x)|2 dx = G (u(y )) − G (−1),
−∞
−∞
Z
0
2
µ |u (x)| dx = G (1) − G (−1)
R
WOLG, assume u(0) = t0 , then
Z
0
−
(−∆)s u(x) · u 0 (x) dx ≥ G (t0 ) − G (1) .
−∞
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
29 / 33
Uniform Decays at Infinity
For any bistable nonlinearity f and any function g ∈ C 2 (R). For
any small h, let fh (t) = f (t) + hg (t), and (µh , uh ) be the traveling
wave solutions corresponding to fh .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
30 / 33
Uniform Decays at Infinity
For any bistable nonlinearity f and any function g ∈ C 2 (R). For
any small h, let fh (t) = f (t) + hg (t), and (µh , uh ) be the traveling
wave solutions corresponding to fh . Then there exist some constants
C > 0 and 0 < H 1 such that
0 < uh0 (x) ≤
Mingfeng Zhao(UBC)
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H.
Traveling Waves and Fractional Laplacians
(4.1)
30 / 33
Uniform Decays at Infinity
For any bistable nonlinearity f and any function g ∈ C 2 (R). For
any small h, let fh (t) = f (t) + hg (t), and (µh , uh ) be the traveling
wave solutions corresponding to fh . Then there exist some constants
C > 0 and 0 < H 1 such that
0 < uh0 (x) ≤
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H.
(4.1)
Step I: There exist C 1 and 0 < H1 , δ0 1 such that
0 < uh0 (x) ≤
Mingfeng Zhao(UBC)
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H1 .
Traveling Waves and Fractional Laplacians
30 / 33
Uniform Decays at Infinity
For any bistable nonlinearity f and any function g ∈ C 2 (R). For
any small h, let fh (t) = f (t) + hg (t), and (µh , uh ) be the traveling
wave solutions corresponding to fh . Then there exist some constants
C > 0 and 0 < H 1 such that
0 < uh0 (x) ≤
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H.
(4.1)
Step I: There exist C 1 and 0 < H1 , δ0 1 such that
0 < uh0 (x) ≤
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H1 .
Step II: By Hamiltonian identity, we get
R1
− −1 g (t) dt
µh
lim
=R
.
0
2
h→0 h
R |u0 (x)| dx
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
30 / 33
Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Some Interesting Open Problems
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
31 / 33
Some Interesting Open Problems
I
For a fixed unbalanced bistable nonlinearity, how does the speed
µs depend on s? For example, what will happen when s & 0
or s % 1?
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
32 / 33
Some Interesting Open Problems
I
For a fixed unbalanced bistable nonlinearity, how does the speed
µs depend on s? For example, what will happen when s & 0
or s % 1?
I
For the combustion model with 0 < s ≤ 12 , how does x depend
on t in the reaction-diffusion equation? How about the higher
dimensional case?
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
32 / 33
Some Interesting Open Problems
I
For a fixed unbalanced bistable nonlinearity, how does the speed
µs depend on s? For example, what will happen when s & 0
or s % 1?
I
For the combustion model with 0 < s ≤ 12 , how does x depend
on t in the reaction-diffusion equation? How about the higher
dimensional case?
I
De Giorgi type conjecture for the balanced bistable nonlinearity
with (−∆)s .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
32 / 33
Some Interesting Open Problems
I
For a fixed unbalanced bistable nonlinearity, how does the speed
µs depend on s? For example, what will happen when s & 0
or s % 1?
I
For the combustion model with 0 < s ≤ 12 , how does x depend
on t in the reaction-diffusion equation? How about the higher
dimensional case?
I
De Giorgi type conjecture for the balanced bistable nonlinearity
with (−∆)s .
I
Modica type estimate for (−∆)s .
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
32 / 33
Thank You!
Mingfeng Zhao(UBC)
Traveling Waves and Fractional Laplacians
33 / 33