Traveling Wave Solutions to the Allen-Cahn
Equations with Fractional Laplacians
Mingfeng Zhao
University of Connecticut
March 28, 2014
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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Overview
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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Traveling Waves
The reaction-diffusion systems can be used to describe the front
propagation phenomena arises in a variety of contexts such as flames,
chemical reactions, phase transitions, biological invasions, social behaviors etc.. 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(UConn)
∀t > 0, ∀x ∈ Rn .
Traveling Waves with Fractional Laplacians
(1.1)
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Traveling Waves
The reaction-diffusion systems can be used to describe the front
propagation phenomena arises in a variety of contexts such as flames,
chemical reactions, phase transitions, biological invasions, social behaviors etc.. 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(UConn)
Traveling Waves with Fractional Laplacians
4 / 40
Traveling Waves
The reaction-diffusion systems can be used to describe the front
propagation phenomena arises in a variety of contexts such as flames,
chemical reactions, phase transitions, biological invasions, social behaviors etc.. 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(UConn)
Traveling Waves with Fractional Laplacians
4 / 40
Traveling Waves
The reaction-diffusion systems can be used to describe the front
propagation phenomena arises in a variety of contexts such as flames,
chemical reactions, phase transitions, biological invasions, social behaviors etc.. 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(UConn)
Traveling Waves with Fractional Laplacians
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Traveling Front
Figure: Traveling Front
Mingfeng Zhao(UConn)
Traveling Waves with 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
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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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
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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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
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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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) = −
Mingfeng Zhao(UConn)
Rt
−1 f (u)
du is called a double well potential.
Traveling Waves with 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
Mingfeng Zhao(UConn)
∀x ∈ R,
Traveling Waves with Fractional Laplacians
(UL)
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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
∀x ∈ R,
(UL)
Fisher-KPP Model
Mingfeng Zhao(UConn)
Traveling Waves with 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
∀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→−∞
Mingfeng Zhao(UConn)
u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves with 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
∀x ∈ R,
(UL)
Combustion Model
Mingfeng Zhao(UConn)
Traveling Waves with 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
∀x ∈ R,
(UL)
Combustion Model
∃! (µ, u) as the solution to (UL) such that
lim
x→−∞
Mingfeng Zhao(UConn)
u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves with 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
∀x ∈ R,
(UL)
Bistable or Allen-Cahn Model
Mingfeng Zhao(UConn)
Traveling Waves with 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
∀x ∈ R,
(UL)
Bistable or Allen-Cahn Model
∃! (µ, u) as the solution to (1.2) such that
lim
x→±∞
Mingfeng Zhao(UConn)
u(x) = ±1.
Traveling Waves with Fractional Laplacians
11 / 40
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 (1.2) such that
lim
x→±∞
u(x) = ±1.
Moreover, ∃ν± > 0 such that
u 0 (x) ∼ e −ν± |x| ,
Mingfeng Zhao(UConn)
as |x| → ∞.
Traveling Waves with Fractional Laplacians
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Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
12 / 40
Fractional Laplacians
The reaction diffusion systems with an anomalous diffusion such
as super diffusion, which plays important roles in various physical,
chemical, biological and geological processes etc.. Such a super diffusion is related to Lévy process and may be modeled by a fractional
Laplace operator (−∆)s , which is the infinitesimal generator of the
2s-stable symmetric Lévy process.
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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Fractional Laplacians
The reaction diffusion systems with an anomalous diffusion such
as super diffusion, which plays important roles in various physical,
chemical, biological and geological processes etc.. Such a super diffusion is related to Lévy process and may be modeled by a fractional
Laplace operator (−∆)s , which is the infinitesimal generator of the
2s-stable symmetric Lévy process.
For any 0 < s < 1 and u ∈ Cc∞ (Rn ), define
Z
s
(−∆) u(x) =
e 2πix·y (2π|y |)2s û(y ) dy ,
Rn
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
∀x ∈ Rn .
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Integral Form of Fractional Laplacians
Equivalently, we have
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 =
Mingfeng Zhao(UConn)
s22s Γ
n
n+2s
2
π 2 Γ(1 − s)
.
Traveling Waves with Fractional Laplacians
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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
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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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 =
Mingfeng Zhao(UConn)
21−2s Γ(1 − s)
.
Γ(s)
Traveling Waves with Fractional Laplacians
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Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
16 / 40
Traveling Waves with Fractional Laplacians

(−∆)s u(x) − µu 0 (x) = f (u(x)),



 0
u (x) > 0, ∀x ∈ R,



 u(0) = 1 .
2
Mingfeng Zhao(UConn)
∀x ∈ R,
Traveling Waves with Fractional Laplacians
(FL)
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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(UConn)
Traveling Waves with Fractional Laplacians
17 / 40
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→−∞
Mingfeng Zhao(UConn)
u(x) = 0,
and
lim u(x) = 1.
x→∞
Traveling Waves with Fractional Laplacians
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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(UConn)
∀x ∈ R,
Traveling Waves with Fractional Laplacians
(FL)
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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 with Fractional Laplacians
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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→−∞
Mingfeng Zhao(UConn)
x→∞
Traveling Waves with Fractional Laplacians
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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
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∀x ∈ R,
Traveling Waves with Fractional Laplacians
(FL)
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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 with Fractional Laplacians
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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) ∼
Mingfeng Zhao(UConn)
1
|x|1+2s
,
as |x| → ∞.
Traveling Waves with Fractional Laplacians
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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(UConn)
Traveling Waves with Fractional Laplacians
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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 with Fractional Laplacians
20 / 40
Outline
Traveling Waves
Fractional Laplacians
Traveling Waves with Fractional Laplacians
Main Results and Outline of Proof
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
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Main Results
Theorem A [Gui-Zhao(2012)]
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) ∼
Mingfeng Zhao(UConn)
1
|x|1+2s
,
as |x| → ∞.
Traveling Waves with Fractional Laplacians
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Main Results
Theorem B [Gui-Zhao(2012)]
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 with Fractional Laplacians
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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 with Fractional Laplacians
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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 with Fractional Laplacians
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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(UConn)
Traveling Waves with Fractional Laplacians
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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 with Fractional Laplacians
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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) ∼
Mingfeng Zhao(UConn)
1
|x|1+2s
, |vt00 (x)| ∼
1
|x|2+2s
,
as |x| → ∞.
Traveling Waves with Fractional Laplacians
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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(UConn)
Traveling Waves with Fractional Laplacians
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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
Mingfeng Zhao(UConn)
3
wδ,T0 ≥ 0,
T0
∀|x| ≥ R.
Traveling Waves with Fractional Laplacians
25 / 40
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,
Mingfeng Zhao(UConn)
∀|x| ≤ R + 1.
Traveling Waves with Fractional Laplacians
25 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
25 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
26 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
27 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
27 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
27 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
28 / 40
ODE Case
(
−u 00 (x) − µu 0 (x) = f (u(x)),
lim
x→±∞
∀x ∈ R,
u(x) = ±1.
Then
Z y
1 0
2
|u (y )| + µ
|u 0 (x)|2 dx
2
−∞
Z
µ |u 0 (x)|2 dx
= G (u(y )) − G (−1),
= G (1) − G (−1).
(4.1)
R
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
29 / 40
ODE Case
(
−u 00 (x) − µu 0 (x) = f (u(x)),
lim
x→±∞
∀x ∈ R,
u(x) = ±1.
Then
Z y
1 0
2
|u (y )| + µ
|u 0 (x)|2 dx
2
−∞
Z
µ |u 0 (x)|2 dx
= G (u(y )) − G (−1),
= G (1) − G (−1).
(4.1)
R
WOLG, assume u(0) = t0 , then
1 0
|u (0)|2 ≥ G (t0 ) − G (1).
2
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
(4.2)
29 / 40
ODE Case
(
−u 00 (x) − µu 0 (x) = f (u(x)),
lim
x→±∞
∀x ∈ R,
u(x) = ±1.
Then
Z y
1 0
2
|u (y )| + µ
|u 0 (x)|2 dx
2
−∞
Z
µ |u 0 (x)|2 dx
= G (u(y )) − G (−1),
= G (1) − G (−1).
(4.1)
R
WOLG, assume u(0) = t0 , then
1 0
|u (0)|2 ≥ G (t0 ) − G (1).
2
Approach I
Let u 0 (x0 ) = max u 0 (x), then u 00 (x0 ) = 0. By (4.2), we get
x∈R
kf kC ([−1,1])
−f (u(x0 ))
µ=
≤p
.
0
u (x0 )
2[G (t0 ) − G (1)]
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
(4.2)
29 / 40
ODE Case
Approach II
µ|u 0 (0)|2 = µ
0
Z
−∞
0
Z
= −
00
Z
0
f (u(x))u (x) dx −
−∞
Mingfeng Zhao(UConn)
u 00 (x) · u 0 (x) dx
|u 00 (x)|2 dx
−∞
Traveling Waves with Fractional Laplacians
30 / 40
ODE Case
Approach II
µ|u 0 (0)|2 = µ
0
Z
−∞
0
Z
= −
u 00 (x) · u 0 (x) dx
00
0
f (u(x))u (x) dx −
−∞
Z
Z
0
f 0 (u(x))|u 0 (x)|2 dx
≤
|u 00 (x)|2 dx
−∞
Since f (t0 ) = 0
−∞
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
30 / 40
ODE Case
Approach II
µ|u 0 (0)|2 = µ
0
Z
u 00 (x) · u 0 (x) dx
−∞
0
Z
= −
00
0
f (u(x))u (x) dx −
−∞
Z
Z
0
−∞
f 0 (u(x))|u 0 (x)|2 dx
≤
|u 00 (x)|2 dx
Since f (t0 ) = 0
−∞
≤ kf 0 kC ([−1,1]) ·
Mingfeng Zhao(UConn)
G (1) − G (−1)
,
µ
Traveling Waves with Fractional Laplacians
By (4.1).
30 / 40
ODE Case
Approach II
µ|u 0 (0)|2 = µ
0
Z
u 00 (x) · u 0 (x) dx
−∞
0
Z
= −
Z
00
−∞
Z
0
f (u(x))u (x) dx −
0
−∞
f 0 (u(x))|u 0 (x)|2 dx
≤
|u 00 (x)|2 dx
Since f (t0 ) = 0
−∞
≤ kf 0 kC ([−1,1]) ·
G (1) − G (−1)
,
µ
By (4.1).
By (4.2), we have
s
q
µ ≤ kf 0 kC ([−1,1]) ·
Mingfeng Zhao(UConn)
G (1) − G (−1)
.
2[G (t0 ) − G (1)]
Traveling Waves with Fractional Laplacians
30 / 40
Hamiltonian Identity
Z
µ
|u 0 (x)|2 dx = G (1) − G (−1).
R
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
31 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
31 / 40
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
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
31 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
31 / 40
Estimate of Speed

 (−∆)s u(x) − µu 0 (x) = f (u(x)),

lim
x→±∞
Mingfeng Zhao(UConn)
∀x ∈ R,
u(x) = ±1.
Traveling Waves with Fractional Laplacians
32 / 40
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).
(HI )
R
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
32 / 40
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).
(HI )
R
WOLG, assume u(0) = t0 , then
Z
0
−
(−∆)s u(x) · u 0 (x) dx ≥ G (t0 ) − G (1).
(4.3)
−∞
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
32 / 40
Estimate of Speed: 0 < s <
1
2
(Supercritical)
Step I: ∀R > 0, we have
2C1,s −2s
(−∆) u(x) ≤
R
+ C1,s
s
s
Mingfeng Zhao(UConn)
Z
|z|<R
Z
0
1
yu 0 (x + ty )
dydt.
|y |1+2s
Traveling Waves with Fractional Laplacians
33 / 40
Estimate of Speed: 0 < s <
1
2
(Supercritical)
Step I: ∀R > 0, we have
2C1,s −2s
(−∆) u(x) ≤
R
+ C1,s
s
s
Z
|z|<R
Z
0
1
yu 0 (x + ty )
dydt.
|y |1+2s
Step II: By Cauchy-Schwarz’s inequality and the Hamiltonian identity (HI), we get
Z
0
−
(−∆)s u(x)·u 0 (x) dx ≤
−∞
Mingfeng Zhao(UConn)
4C1,s −2s
C1,s
G (1) − G (−1)
·R +
·R 1−2s ·
.
s
1 − 2s
µ
Traveling Waves with Fractional Laplacians
33 / 40
Estimate of Speed: 0 < s <
1
2
(Supercritical)
Step I: ∀R > 0, we have
2C1,s −2s
(−∆) u(x) ≤
R
+ C1,s
s
s
Z
|z|<R
Z
1
0
yu 0 (x + ty )
dydt.
|y |1+2s
Step II: By Cauchy-Schwarz’s inequality and the Hamiltonian identity (HI), we get
Z
0
−
(−∆)s u(x)·u 0 (x) dx ≤
−∞
4C1,s −2s
C1,s
G (1) − G (−1)
·R +
·R 1−2s ·
.
s
1 − 2s
µ
Step III: By (4.3) and take special R, we get
µ ≤ C [G (1) − G (−1)].
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
33 / 40
Estimate of Speed: s =
Z
Step I:
1
1
2
(Critical)
|(−∆) 2 u(x)|2 dx =
R
Mingfeng Zhao(UConn)
Z
|u 0 (x)|2 dx.
R
Traveling Waves with Fractional Laplacians
34 / 40
Estimate of Speed: s =
Z
Step I:
1
2
1
(Critical)
|(−∆) 2 u(x)|2 dx =
R
Z
|u 0 (x)|2 dx.
R
Step II: By (4.3) and Cauchy-Schwarz’s inequality, we have
Z
0
G (t0 ) − G (1) ≤
12 Z
·
|(−∆) u(x)|2 dx
1
2
−∞
Z
0
|u 0 (x)|2 dx
21
−∞
≤
|u 0 (x)|2 dx
=
G (1) − G (−1)
,
µ
R
By the Hamiltonian Identity (HI).
Hence, we get
µ≤
Mingfeng Zhao(UConn)
G (1) − G (−1)
.
G (t0 ) − G (1)
Traveling Waves with Fractional Laplacians
34 / 40
Estimate of Speed:
Z
1
2
< s < 1 (Subcritical)
0
s
2
Z
0
|(−∆) u(x)| dx ≤
Step I:
−∞
Mingfeng Zhao(UConn)
f (u(x)) · (−∆)s u(x) dx.
−∞
Traveling Waves with Fractional Laplacians
35 / 40
Estimate of Speed:
Z
1
2
< s < 1 (Subcritical)
0
s
2
Z
0
|(−∆) u(x)| dx ≤
Step I:
−∞
f (u(x)) · (−∆)s u(x) dx.
−∞
Step II: ∀R > 0, we have
Z
0
f (u(x) · (−∆)s u(x) dx ≤
−∞
Mingfeng Zhao(UConn)
4kf kC ([−1,1])
· R 1−2s
2s − 1
kf 0 kC ([−1,1])
G (1) − G (−1)
+
· R 2−2s ·
.
1−s
µ
Traveling Waves with Fractional Laplacians
35 / 40
Estimate of Speed:
Z
1
2
< s < 1 (Subcritical)
0
s
2
Z
0
|(−∆) u(x)| dx ≤
Step I:
−∞
f (u(x)) · (−∆)s u(x) dx.
−∞
Step II: ∀R > 0, we have
Z
0
f (u(x) · (−∆)s u(x) dx ≤
−∞
4kf kC ([−1,1])
· R 1−2s
2s − 1
kf 0 kC ([−1,1])
G (1) − G (−1)
+
· R 2−2s ·
.
1−s
µ
Step III: By (4.3), Cauchy-Schwarz’s inequality and take special R,
we get
µ ≤ C [G (1) − G (−1)].
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
35 / 40
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(UConn)
Traveling Waves with Fractional Laplacians
36 / 40
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(UConn)
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H.
Traveling Waves with Fractional Laplacians
(4.4)
36 / 40
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.
(4.4)
Step I: ∃R 1 and 0 < H1 , δ0 1 such that
uh (0) = −1 + δ0 , and uh (x) ≥ 1 − δ0 ,
Mingfeng Zhao(UConn)
∀x ≥ R, ∀|h| ≤ H1 .
Traveling Waves with Fractional Laplacians
36 / 40
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.
(4.4)
Step I: ∃R 1 and 0 < H1 , δ0 1 such that
uh (0) = −1 + δ0 , and uh (x) ≥ 1 − δ0 ,
∀x ≥ R, ∀|h| ≤ H1 .
Step II: ∃C > 0 such that
0 < uh0 (x) ≤
Mingfeng Zhao(UConn)
C
,
|x|1+2s
∀|x| ≥ 1, ∀|h| ≤ H1 .
Traveling Waves with Fractional Laplacians
36 / 40
Uniform Decays at Infinity
The estimate (4.4) implies
Z
Z
0
2
lim
|uh (x)| dx =
|u00 (x)|2 dx.
h→0
Mingfeng Zhao(UConn)
R
R
Traveling Waves with Fractional Laplacians
37 / 40
Uniform Decays at Infinity
The estimate (4.4) implies
Z
Z
0
2
lim
|uh (x)| dx =
|u00 (x)|2 dx.
h→0
R
R
If f is balanced, recall that the Hamiltonian identity:
Z
µh
|uh0 (x)|2 dx = Gh (mh+ ) − Gh (mh− ),
R
where mh+ , mh− are zeros of fh and fh = −Gh0 .
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
37 / 40
Uniform Decays at Infinity
The estimate (4.4) implies
Z
Z
0
2
lim
|uh (x)| dx =
|u00 (x)|2 dx.
h→0
R
R
If f is balanced, recall that the Hamiltonian identity:
Z
µh
|uh0 (x)|2 dx = Gh (mh+ ) − Gh (mh− ),
R
where mh+ , mh− are zeros of fh and fh = −Gh0 .So we get
R1
− −1 g (t) dt
µh
=R
.
lim
0
2
h→0 h
R |u0 (x)| dx
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
37 / 40
References
X. Cabré and J. M. Roquejoffre. The infuence of fractional diffusion
in Fisher-KPP equations, Comm. Math. Phys., 320(3):679-722,2013.
X. Cabré and Y. Sire. Nonlinear equations for fractional Laplacians,
II: Existence, uniqueness, and qualitative properties of solutions,
Preprint, arXiv:1111.0796,2011.
X. Cabré and Y. Sire. Nonlinear equations for fractional Laplacians,
I: Regularity, maximum principles, and Hamiltonian estimates, Ann.
Inst. H. Poincaré Anal. Non Linéaire, 31(1):23-53,2014.
X. Cabré and J. Solà-Morales. Layer solutions in a half space
for boundary reactions, Comm. Pure Appl. Math., 58(12):16781732,2005.
L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32(7-9):12451260,2007.
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
38 / 40
C. Gui and T. Huan. Existence and Nonexistence of traveling wave
solution to the non local combustion model and the generalized FisherKPP model, Preprint, 2012.
A. Mellet, J. M. Roquejoffre, and Y. Sire. Existence and asymptotics of
fronts in non local combustion models, Commun. Math. Sci., 12(1):111,2014.
V. A. Volpert, Y. Nec, and A. A. Nepomnyashchy. Exact solutions
in front propagation problems with superdiffusion, Phys. D, 239(34):134-144,2010.
V. A. Volpert, V. A. Volpert, and A. I. Volpert. Traveling Wave Solutions of Parabolic Systems, volume 180 of Translations of Mathematical Monographs, American Mathematical Society, 1994.
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
39 / 40
Acknowledgments
Many thanks to:
I Advisor:
Prof. Changfeng Gui
I Committees:
Prof. Choi and Prof. Yan
I Department staff:
Prof. Gine-Nasdeu, Blei, Solomon, Monique, Tammy
I Course Teachers:
Prof. McKenna, Wu, Huang, Bass, Schiffler, Conrad,
Glaz, Gordina, Teplyaev, Ben-Ari, Rogers, Lozano-Robledo,
Haas, Tollefson, Terwilleger
I Family and Friends
Mingfeng Zhao(UConn)
Traveling Waves with Fractional Laplacians
40 / 40