Transition Fronts of Nonlocal Dispersal Evolution ... · Reasons for Nonlocal Dispersal and...

Transition Fronts of Nonlocal Dispersal EvolutionEquations in Heterogeneous Media

Wenxian ShenAuburn University

IMA Workshop on Dynamics and Differential EquationsDedicated to the memory of George Sell

June 22 - 25, 2016

Collaborators

Zhongwei Shen

Postdoc at University of Alberta

Outline of Talk

I Models and Problems

I Traveling Waves in Homogeneous Media

I Traveling Waves in Periodic Media

I Transitions Fronts in Heterogeneous Media

Models and ProblemsDispersal Evolution Equations

ut(t, x) = Au + f (t, x , u), x ∈ IRN (1)

• u(t, x) – state variable of a dispersal evolution system

• ut(t, x) = change due to dispersal + change due to “reaction”

• f (t, x , u) = change due to “reaction”

• Au = change due to dispersal

• ∃ several models for change due to dispersal, e.g., randomdispersal, nonlocal dispersal

Models and ProblemsRandom Dispersal

Random dispersal – internal interaction of organisms occursbetween adjacent locations randomly

— Au = ∆uut = ∆u + f (t, x , u)

— random dispersal evolution equation

Models and ProblemsNonlocal Dispersal

Nonlocal dispersal – internal interaction of organisms isnonlocal

— Au =∫

IRN k(y − x)[u(t, y)− u(t, x)]dy

∫IRN k(z)dz = 1

Models and ProblemsNonlocal Dispersal

k(·) ∈ C 1, k(·) ≥ 0, k(−z) = k(z), supp(k) is compact

– nonlocal dispersal evolution equation

ut =

∫IRN

k(y − x)[u(t, y)− u(t, x)]dy + f (t, x , u)

Models and ProblemsAssumptions on f (t, x , u)

ut = Au + f (t, x , u), x ∈ IRN (1)

• f (t, x , u) is smooth in t, x , and u

• f (t, x , 0) = f (t, x , 1) = 0 for all t and x

=⇒• u(t, x) = 0 and u(t, x) = 1 are two trivial states

• For any s ∈ IR and u0 ∈ Cbunif(IR

N , IR), (1) has a unique(local) solution u(t, x ; s, u0) with u(s, x ; s, u0) = u0(x).

Models and ProblemsSome Important Examples of f (t, x , u)

• monostable: population dynamics, chemical kinetics, . . . ;

• bistable: population dynamics (Allee effect), phase transition,signal propagation, . . . ;

• ignition: flame propagation, . . . .

=⇒ u(t, x ; s, u0) exists globally for any u0 ≥ 0

Models and ProblemsProblems

Evolution/spreading of species, flame, etc.related to the transition between two states or phases.

Mathematically, u(t, x ; s, u0)→ ? as t − s →∞, where theinitial data u0 is compactly supported or front-like.

This is closely related to entire (global-in-time) front-like solutionsconnecting 0 and 1: existence, stability and uniqueness.

Models and ProblemsReasons for Nonlocal Dispersal and Heterogeneous Media

• In applications, the movement or internal interaction of theorganisms of many evolution systems occur between adjacentlocations as well as non-adjacent locations. It is more properto model the dynamics of such systems by certain nonlocaldispersal evolution equations.

• In applications, spatial and temporal variations exist, e.g.,growth rates change when time elapses and are different indifferent regions, therefore, we must take temporal and spatialvariations into consideration in order to have a betterunderstanding of the phenomena.


• In applications, the movement or internal interaction of theorganisms of many evolution systems occur between adjacentlocations as well as non-adjacent locations. It is more properto model the dynamics of such systems by certain nonlocaldispersal evolution equations.• In applications, spatial and temporal variations exist, e.g.,

growth rates change when time elapses and are different indifferent regions, therefore, we must take temporal and spatialvariations into consideration in order to have a betterunderstanding of the phenomena.


• Mathematically, many tools for random dispersal evolutionequations are difficult to be applied for nonlocal dispersalevolutions;

some fundamental theory (for example, spectraltheory) for nonlocal dispersal evolution equations is less welldeveloped than random dispersal ones; many tools andconcepts in homogeneous media cannot be applied toheterogeneous media; the development of concepts,techniques, etc. are of interest.


• Mathematically, many tools for random dispersal evolutionequations are difficult to be applied for nonlocal dispersalevolutions; some fundamental theory (for example, spectraltheory) for nonlocal dispersal evolution equations is less welldeveloped than random dispersal ones;

many tools andconcepts in homogeneous media cannot be applied toheterogeneous media; the development of concepts,techniques, etc. are of interest.


• Mathematically, many tools for random dispersal evolutionequations are difficult to be applied for nonlocal dispersalevolutions; some fundamental theory (for example, spectraltheory) for nonlocal dispersal evolution equations is less welldeveloped than random dispersal ones; many tools andconcepts in homogeneous media cannot be applied toheterogeneous media;

the development of concepts,techniques, etc. are of interest.


• Mathematically, many tools for random dispersal evolutionequations are difficult to be applied for nonlocal dispersalevolutions; some fundamental theory (for example, spectraltheory) for nonlocal dispersal evolution equations is less welldeveloped than random dispersal ones; many tools andconcepts in homogeneous media cannot be applied toheterogeneous media; the development of concepts,techniques, etc. are of interest.

Models and ProblemsA Formal Relation Between Random and Nonlocal Dispersals

k(z) = kδ(z) = kδ(z) = 1δNk0( zδ )

supp(k0(·)) = B(0, 1),∫

IRN k0(z)dz = 1, k0(z) is symmetric

Models and ProblemsA Formal Relation Between Random and Nonlocal Dispersals

k(z) = kδ(z) = kδ(z) = 1δNk0( zδ )

u ∈ C∞(IRN),∫IRN kδ(y − x)[u(y)− u(x)]dy

=∫

IRN k0(z)[u(x + δz)− u(x)]dz

=∫

IRN k0(z)[δ∂iu(x)zi + δ2∂iju(x)zizj + · · · ]dz

= Cδ2∆u + · · ·

C = 12

∫IRN k0(z)z21dz = 1

2

∫B(0,1) k0(z)z21dz

Models and ProblemsDifferences between Random and Nonlocal Dispersal

ut = Au + f (t, x , u), x ∈ IRN (1)

u(t, x ; s, u0) – solution of (1) with u(s, x ; s, u0) = u0(x), s ∈ IRand u0 ∈ Cb

unif(IRN , IR).

• Au(t, x) = ∆u(t, x) =⇒ u(t, x ; s, u0) is smoother in x thanu0(x) for t > s

• Au(t, x) = [∫

IRN k(y − x)u(t, y)dy − u(t, x)] 6=⇒ u(t, x ; s, u0) issmoother in x than u0(x) for t > s

• Additional difficulties arise in the study of nonlocal dispersalevolution equations

Traveling Waves in Homogeneous MediaDefinition

Considerut = Au + f (u), x ∈ IR

f (0) = f (1) = 0

Study the asymptotic behavior of u(t, x ; u0) with front-like initialdata u0 =⇒ Look for front-like solutions connecting 0 and 1.

Traveling wave: entire solution of the form u(t, x) = φ(x − ct)with φ(−∞) = 1 and φ(∞) = 0,

Traveling Waves in Homogeneous MediaRandom Dispersal Case: methods for existence

ut = uxx + f (u), x ∈ IR

u(t, x) = φ(x − ct): speed-profile pair (c , φ) satisfies{φxx + cφx + f (φ) = 0,

φ(−∞) = 1, φ(∞) = 0.

Phase-plane method (Aronson-Weinberger 1975; Fife-McLeod1980; · · · ):

It does not work in periodic media and does not work for thenonlocal dispersal.


PDE method (Berestycki-Nicolaenko-Scheurer 1985; · · · ):φxx + cφx + f (φ) = 0, x ∈ (−a, a),

φ(−a) = 1, φ(a) = 0,

φ(0) = 12

⇒ (ca, φa)a→∞=⇒ (c , φ).

The difficulty is to show that {ca} is uniformly bounded.This method can be applied to periodic media with nontrivialadaption, but is difficult to be applied to the nonlocal dispersalequations directly because of the lack of regularity of the solutions.


Dynamical system method (Xinfu Chen 1997; W. Shen 1999; · · · )• pick appropriate wave-like initial data u0(·);• let the solutions evolve,u(t, ·; s) = u(t, ·; s, u0(· − xs)) = u(t − s, ·; 0, u0(· − xs));

• consider the limit of u(t, x ; s) := u(t − s, ·; 0, u0(· − xs)) ass → −∞ (u(0, 0; s, u0(· − xs)) = θ)• show the limit is the profile of some traveling wave.

This method can be applied to heterogeneous media and nonlocaldispersal evolution equations with nontrivial adaption.

Traveling Waves in Homogeneous MediaRandom Dispersal Case: existing results

• Monostable: ∃c∗ > 0 s.t. ∀ c ≥ c∗, there is a unique(up totranslation) φc (Fisher 1937; Kolmogorov-Petrowsky-Piscunov 1937; Aronson-Weinberger 1975, 1978;· · · ); stability (Sattinger 1976; Kirchgassner 1999, · · · );

• Bistable : unique c∗ > 0, and unique (up to translation) φ(Aronson-Weinberger 1975, 1978; Fife-McLeod 1977. . . );stability (Sattinger 1976; Fife-McLeod 1977; · · · );

• Ignition: unique c∗ > 0, and unique (up to translation) φ(Kanel 1960, 1961, 1962, 1964; Aronson-Weinberger 1975,1978; Fife-McLeod 1977; · · · ); stability (Sattinger 1976;Roquejofree 1994; · · · ).

Traveling Waves in Homogeneous MediaNonlocal Dispersal Case: methods for existence

ut =

∫IRk(y − x)u(t, y)dy − u(t, x) + f (u), x ∈ IR

Regularization method: (Bates-Fife-Ren-Wang 1991; Coville 2003;Coville-Dupaigne 2005; · · · )Consider

ut = εuxx +∫

IR k(y − x)u(t, y)dy − u(t, x) + f (u)

Show the existence of traveling wave solution u(t, x) = φε(x − cε)of the perturbed equation

Show the limits φε(·)→ φ(·) and cε → c exist andu(t, x) = φ(x − ct) is the traveling wave solution of the nonlocalevolution equation

Dynamical system method: (X. Chen 1997; Shen-Zhang 2012; · · · )

Traveling Waves in Homogeneous MediaNonlocal Dispersal Case: existing results

• Monostable: ∃c∗ > 0 s.t. c ≥ c∗ and unique(up totranslation) φc (Coville-Dupaigne 2007; · · · );

• Bistable : unique c∗ > 0, and unique (up to translation) φ(Bates-Fife-Ren-Wang 1991; X. Chen 1997; · · · );

• Ignition: unique c∗ > 0, and unique (up to translation) φ(Coville 2003; · · · ).

Traveling Waves in Homogeneous MediaRemarks

I Asymptotic behavior of solutions with compact support is alsostrongly related to the so called threshold behavior

I Threshold behavior in the random dispersal case with bistablenonlineariity. Let uL0 initial data indexed by L and uL(t, x) becorresponding solutions to ut = uxx + fB(u). Then, ∃L∗ s.t.(Kanel 64; Zlatos 05; Du-Matano 08; Polacik 15)

where θ = θ(·) is a nontriivial stationary solution withlim|x |→∞ θ(x) = 0.

I No result in the nonlocal equations.

Traveling Waves in Periodic MediaDefinition

Considerut = Au + f (t, x , u),

where f (t, x , 0) = 0 = f (t, x , 1) and

f (t + T , x ; u) = f (t, x + L, u) = f (t, x , u).

Study the asymptotic behavior of u(t, x ; u0) with front-likeinitial data u0 with front-like initial data u0.

No traveling wave of the form φ(x − ct).

Look for something more general: solutions which

propagate as traveling waves;

change the shape “periodically”.


Periodic traveling wave: entire solution of the formu(t, x) = φ(x − ct, t, x) (Nolen-Xin 2005, Nadin 2009), oru(t, x) = ψ(x − ct, t, ct) (Shen 2004), with φ and ψ connecting 0and 1, and T -periodic in the second argument, and L-periodic inthe third argument.

p is the smallest number such that both pT and p Lc are integers

(such p may not exist).


f (t, x , u) = f (t, u), u(t, x) = φ(t, x − ct) (Weinberger1982;Alikakos-Bates-Chen 1999; · · · ).

f (t, x , u) = f (x , u), u(t, x) = φ(x − ct, x) (Xin 1992;Hamel-Berestycki 2002; · · · ), or u(t, x) = ψ(x − ct, ct) (Matano2003; Shen 2004; · · · ).

Traveling Waves in Periodic MediaExisting Results in the Random Dispersal Case

Time periodic media f (t, x , u) = f (t, u)

• monostable: existence (Weinberger 1982, Liang-Yi-Zhao2006), stability and uniqueness (W. Shen 2011);

• bistable: Alikakos-Bates-Chen 1999;

• ignition: W.Shen-Z.Shen 2014.


Space periodic media f (t, x , u) = f (x , u)

• monostable: existence (Weinberger 2002;Berestycki-Hamel-Roques 2005; Nadin 2009; Liang-Zhao2010); stability and uniqueness (Hamel-Roques 2011);

• bistable: existence-small or large period, and stability(Ding-Hamel-Zhao 2014);

• ignition: existence (Berestycki-Hamel 2002); stability anduniqueness (Mellet-Nolen-Roquejoffre-Ryzhik 2009).


General periodic media f (t, x , u)

• monostable: existence (Nadin 2009, W. Shen 2011)

• bistable: open!

• ignition: open!

Traveling Waves in Periodic MediaExisting Results in the Nonlocal Dispersal Case

Time periodic media f (t, x , u) = f (t, u)

• monostable: Shen-Zhang 2012;

• bistable: W. Shen-Z. Shen 2015;

• ignition: W.Shen-Z.Shen 2014.

Space periodic media f (t, x , u) = f (x , u)

• monostable: existence (Coville-Juan-Salome 2013;Shen-Zhang 2012); stability and uniqueness (Shen-Zhang2012);

• bistable: open!

• ignition: open!

General periodic media f (t, x , u)

• monostable: existence (Rawal-Shen-Zhang 2015)

• bistable: open!

• ignition: open!

Traveling Waves in Periodic MediaRemarks

• In the random dispersal case, most results in one spacedimension media can be extended to hiigher space dimensionmedia.

• In the nonlocal dispersal case, it is not clear whether theresults in one space dimension media can be extended tohigher space dimension media.

• One reason: the eigenvalue problem{−ut + ∆u + a(t, x)u = λu, x ∈ RN

u(t + T , x) = u(t, x + piei) = u(t, x)

has always a principal eigenvalue, but the eigenvalue problem{−ut +

∫RN k(y − x)u(t, y)dy − u(t, x) + a(t, x)u = λu, x ∈ RN

u(t + T , x) = u(t, x + piei) = u(t, x)

may not have a principal eigenvalue when N ≥ 3

Transition Fronts in Heterogeneous MediaDefinition

Considerut = Au + f (t, x , u) (1)

f (t, x , 0) = 0 = f (t, x , 1).

Study the asymptotic behavior of u(t, x ; u0).

Characterization of traveling wave u(t, x) = φ(x − ct) in the casef (t, x , u) = f (u): profile: φ; front location: ct; uniform limits:

u(t, x + ct) = φ(x)→

{1, x → −∞,0, x →∞

uniformly in t ∈ R.

In general, look for something much more general.

Transition Fronts in Heterogeneous MediaDefinition

Transition front (W.Shen 99, 04, Matano 03, Berestycki-Hamel07): An entire solution u(t, x) is called a transition front if thereexists a function X : IR→ IR (interface location function) suchthat

u(t, x + X (t))→

{1, x → −∞,0, x →∞

uniformly in t.

Transition Fronts in Heterogeneous MediaRemarks and Problems

Basic problems: existence, stability, and uniqueness of transitionfronts.

Remarks: The regularity of interface location function X (t) andthe front profiles plays an important in the study of stability anduniqueness, but no regularity, even continuity, of the interfacelocation function X (t) and the front profiles are assumed in thedefinition.

More problems:Existence of “smooth” interface location function (interfacelocation function is not unique: if X (t) is an interface locationfunction, then X (t) + ξ(t) is also an interface location function forany bounded function ξ(t))

Regularity of front profiles in the nonlocal dispersal case (theregularity of front profiles follows from the regularity of solutionsfor parabolic equations)

Transition Fronts in Heterogeneous MediaMore assumptions

(HG) ∃ C 2 functions fB : [0, 1]→ IR and fM : [0, 1]→ IR s.t.

fB(u) ≤ f (t, x , u) ≤ fM(u), (t, x , u) ∈ R× R× [0, 1];

moreover, the following conditions hold:

I there exist θ1 ∈ (0, 1) such that fu(t, x , u) ≤ 0 for all(t, x , u) ∈ R× R× [θ1, 1];

I fB is of standard bistable type, that is,fB(0) = fB(θ) = fB(1) = 0 for some θ ∈ (0, 1), fB(u) < 0 for

u ∈ (0, θ), fB(u) > 0 for u ∈ (θ, 1) and∫ 10 fB(u)du > 0;

moreover, ut = J ∗ u − u + fB(u) admits a traveling waveφB(x − cBt) with φB(−∞) = 1, φB(∞) = 0 and cB 6= 0;

I fM is of standard monostable type, that is, fM(0) = fM(1) = 0and fM(u) > 0 for u ∈ (0, 1).

Transition Fronts in Heterogeneous MediaMore assumptions

fM(u)

fB(u)

Transition Fronts in Heterogeneous MediaRegularity of Transition Fronts in the Nonlocal Dispersal Case

Theorem 1. (Regularity) (W. Shen-Z. Shen, to appear in JDDE)(1) (Regularity of the interface) Suppose (HG). Let u(t, x) be anarbitrary transition front with interface location function X (t)satisfying

X (t)− X (t0) ≤ c(t − t0 + T ), t ≥ t0 (2)

for some c and T > 0. Then, there are constants0 < cmin ≤ cmax <∞ and a continuously differentiable functionX (t) satisfying

cmin ≤ ˙X (t) ≤ cmax, t ∈ IR

such thatsupt∈IR|X (t)− X (t)| <∞.

In particular, X (t) is also an interface location function of u(t, x).


(2) (Regularity of the front profile) If ∃ θ0 ∈ (0, θ1) and κ0 > 0 s.t.

fu(t, x , u) ≤ 1− κ0, (t, x , u) ∈ R× IR× [0, θ0]. (3)

Let u(t, x) be an arbitrary transition front. Then, for any t ∈ IR,u(t, x) is continuously differentiable in x and

sup(t,x)∈IR×IR

|ux(t, x)| <∞.

(bistable and ignition type f (t, x , u) satisfies (3))


(3) (Regularity of the front profile) Let u(t, x) be an arbitrarytransition front satisfying

u(t, x) ≤ Cer |x−y |u(t, y), (t, x , y) ∈ IR× R× IR (4)

for some C > 0 and r > 0. Then, u(t, x) is regular in space in thefollowing sense: for any t ∈ IR, u(t, x) is continuouslydifferentiable in x and satisfies

sup(t,x)∈IR×IR

|ux(t, x)|u(t, x)

<∞.

(transition front with monostable type f (t, x , u) satisfies (4))

Transition Fronts in Heterogeneous MediaTransition Fronts of Monostable Equations with Nonlocal Dispersal

(HM) Assume that f (t, x , u) = f (t, u) and there exists a C 1

function g : [0, 1]→ R satisfying

I g(0) = g(1) = 0 < g(u) for u ∈ (0, 1),

I gu(0) = 1, gu(1) ≥ −1, gu is decreasing and∫ 10

u−g(u)u2

du <∞ such that

a(t)g(u) ≤ f (t, u) ≤ a(t)u, (t, u) ∈ R× [0, 1],

where a(t) := fu(t, 0) satisfies ainf := inft∈IR a(t) > 0.

For κ > 0, let

cκ(t) =

∫IR k(y)eκydy − 1

κt +

1

κ

∫ t

0a(s)ds, t ∈ IR.


a(t)u

a(t)g(u)


Theorem 2. (W. Shen-Z. Shen, CPAA, 2016)Assume (HG) and (HM).(1) (Existence) There exists κ0 > 0 such that for any κ ∈ (0, κ0],there is a transition front uκ(t, x) with interface location functionXκ(t) = cκ(t) and satisfying the following properties

(i) uκ(t, x) is decreasing in x for any t ∈ IR;

(ii) there holds limx→∞uκ(t,x+Xκ(t))

e−κx = 1 uniformly in t ∈ IR;

(2) (Stability) Let κ0 be as in (1). For κ ∈ (0, κ0], let uκ(t, x) bethe transition front in (1). Let u0 : R→ [0, 1] be uniformlycontinuous and satisfy

lim infx→−∞

u0(x) > 0 and limx→∞

u0(x)

uκ(t0, x)= 1

for some t0 ∈ R. Then, there holds the limit

limt→∞

supx∈R

∣∣∣∣u(t, x ; t0, u0)

uκ(t, x)− 1

∣∣∣∣ = 0.

Transition Fronts in Heterogeneous MediaTransition Fronts of Bistable Equations with Nonlocal Dispersal

(HB) Assume that f (t, x , u) = f (t, u) and there exists θ∗ ∈ [θ, θ]such that for all t ∈ IR

f (t, u) < 0, u ∈ (0, θ∗) and f (t, u) > 0, u ∈ (θ∗, 1).

Transition Fronts in Heterogeneous MediaTransition Fronts of Bistable Equations with Nonlocal Dispersal

Theorem 3. (W. Shen-Z. Shen, submitted) Assume (HG), (HB).

(1) (Existence and uniqueness) There exists a unique spacedecreasing transition front u(t, x).

(2) (Stability) Let u(t, x) be a space decreasing transition front.Let u0 : R→ [0, 1] be uniformly continuous and satisfieslim infx→−∞ u0(x) > θ1 and lim supx→∞ u0(x) < θ0. Then, thereexist t0 = t0(u0) ∈ R, ξ = ξ(u0) ∈ R, C = C (u0) > 0 and ω∗ > 0(independent of u0) such that

supx∈R|u(t, x ; t0, u0)− u(t, x − ξ)| ≤ Ce−ω∗(t−t0) ∀ t ≥ t0.

Transition Fronts in Heterogeneous MediaTransition Fronts of Ignition Type Equations with Nonlocal Dispersal

(HI) Assume that f (t, x , u) = f (t, u) and there are θ ∈ (0, 1) (theignition temperature), fmin ∈ C 1,α([0, 1]) and a Lipschitzcontinuous function fmax : [0, 1]→ IR satisfying

fmin(u) = 0 = fmax(u), u ∈ [0, θ] ∪ {1},0 < fmin(u) ≤ fmax(u), u ∈ (θ, 1),

f ′min(1) < 0

such that

fmin(u) ≤ f (t, u) ≤ fmax(u), (t, u) ∈ R× [0, 1].

Transition Fronts in Heterogeneous MediaTransition Fronts of Ignition Type Equations with Nonlocal Dispersal

Theorem 4. (W.Shen-Z.Shen, submitted) Assume (HG) and (HI).(1) (Existence) There exist a transition front u(t, x) and acontinuously differentiable interface location function X (t) s.t.

I 0 < c1 ≤ X (t) ≤ c2 <∞ (asymptotic speed limt→∞X (t)t may

not exist);I space monotonicity: ux < 0;I the profile φ(t, x) = u(t, x + X (t)) is as in the graph.

(2) (Stability) Let β0 > 0. Suppose t0 ∈ R and u0 ∈ Cbunif(R,R)

satisfy{u0 : R→ [0, 1], u0(−∞) = 1;

∃C > 0 s.t. |u0 − u(t0, x)| ≤ Ce−β0(x−X (t0)) for x ∈ R.

Then, there exist C = C (u0) > 0, ζ∗ = ζ∗(u0) ∈ R andr = r(β0) > 0 such that

supx∈R|u(t, x ; t0, u0)− u(t, x − ζ∗)| ≤ Ce−r(t−t0) ∀ t ≥ 0.

Transition Fronts in Heterogeneous MediaProof for Existence

Dynamical system method:

• pick appropriate wave-like initial data u0(·);• let the solutions evolve, u(t, ·; s) := u(t, ·; s, u0(· − xs))

(s < 0, u(0, 0; s, u0(· − xs)) = θ);

• consider the limit of u(t, x ; s) as s → −∞• show the limit is a transition front

Difficulties:

• Control of the width of the transition region as time elapses• The regularity of limiting front

Transition Fronts in Heterogeneous MediaProof for Regularity

Let u(t, x) be an arbitrary transition front with interface locationfunction X (t) of

ut =

∫Rk(y − x)u(t, y)dy − u(t, x) + f (t, u(t, x)).

Question: is it continuously differentiable in space?

• propagation estimate (a rough characterization of thepropagation of u(t, x));

• modified interface location function (a better characterizationof the propagation of u(t, x));

• finishing the proof.


Propagation estimate: there exist 0 < c1 ≤ c2 <∞ and T1,2 > 0such that

c1(t − t0 − T1) ≤ X (t)− X (t0) ≤ c2(t − t0 + T2), t ≥ t0.

• main idea of proof (fB(u) ≤ f (t, u) ≤ fM(u)): bistable TWsof ut = J ∗ u − u + fB(u) push X (t) move to the right;monostable TWs of ut = J ∗ u − u + fM(u) control the rightmovement of X (t).


Modified interface location function: there exists a C 1 functionX : R→ R satisfying 0 < cmin ≤ ˙X (t) ≤ cmax <∞ for all t ∈ Rsuch that

supt∈R|X (t)− X (t)| <∞.

S1. To find a continuous modification. Fix some T > 0.I at t = 0 = T+

0 ,

Z+(t;0)=X (0)+c2(T+T2)+c12t, T+

1 =inf{t≥0|X (t)≥Z+(t;0)}.

I at t = T+1 ,

Z+(t;T+1 )=X (T+

1 )+c2(T+T2)+c12(t−T+

1 ), T+2 =inf{t≥T+

1 |X (t)≥Z+(t;T+1 )}.

I at t = T+n−1,

Z+(t;T+n−1)=X (T+

n−1)+c2(T+T2)+c12(t−T+

n−1), T+n =inf{t≥T+

n−1|X (t)≥Z+(t;T+n−1)}.

Define Z+(t) = Z+(t;T+n−1), t ∈ [T+

n−1,T+n ), n ≥ 1.

Similarly, we find Z−(t), and then modify.


S2. To find a continuously differentiable modification as required.Suppose, w.l.o.g, that X (t) is continuous.

I Assume, w.l.o.g, that X (t) is continuously differentiable with0 < cmin ≤ X (t) ≤ cmax <∞ for all t ∈ R.


Space regularity: ux(t, x) exists and sup(t,x)∈R×R |ux(t, x)| <∞.

Let vη(t, x) = u(t,x+η)−u(t,x)η . Then,

vηt = k ∗ vη − vη + aη(t, x)vη, so

vη(t, x) = vη(t0, x)e−

∫ tt0(1−aη(s,x))ds

+

∫ t

t0

bη(τ, x)e−∫ tτ (1−a

η(s,x))dsdτ ,

where aη(t, x) = f (t,u(t,x+η))−f (t,u(t,x))u(t,x+η)−u(t,x) → fu(t, u(t, x)) and

bη(t, x) =∫R

k(x−y+η)−k(x−y)η u(t, y)dy →

∫R k ′(x − y)u(t, y)dy .

To control vη(t, x) as η → 0, there are two problems:

P1. the behavior of aη(t, x), or fu(t, u(t, x));

P2. the possible blow-up behavior of vη(t0, x) = u(t0,x+η)−u(t0,x)η

as η → 0.


P1. the behavior of aη(t, x), or fu(t, u(t, x)). Fix x and look atu(t, x) when t changes. Fix h� 1 and break the transition frontinto 3 parts for each t according to

L(t) = (−∞,X (t)−h],M(t) = [X (t)−h,X (t)+h], R(t) = [X (t)+h,∞).

tfirst(x) = min{t ∈ R|x ∈M(t)}tlast(x) = max{t ∈ R|x ∈M(t)}.T := supx∈R[tlast(x)− tfirst(x)] ≤ 2h

cmin.


P2. the possible blow-up behavior of vη(t0, x) as η → 0.

I We have∣∣∣∣vη(t0, x)e−

∫ tt0(1−aη(s,x))ds

∣∣∣∣ ≤ C (T )1

|η|e−(tfirst(x)−t0).

I Small trick: tfirst(x)− t0 = 1|η| .

Transition Fronts in Heterogeneous MediaRemarks in the nonlocal dispersal case

I f (t, x , u) = f (x , u) and f is monostale type, existence oftransition fronts (Lim-Zlatos, 2014)

I f (t, x , u) = f (x , u) and f is bistable or ignition type, open

I f (t, x , u) depends on both t and x , open

Transition Fronts in Heterogeneous MediaExisting Results in the Random Dispersal Case

Monostable case:

• f (t, x , u) = f (t, u): existence (W.Shen 11, Nadin-Rossi 12);stability and uniqueness up to space shift (W. Shen 11);

• f (t, x , u) = f (x , u): existence(Nolen-Roquejoffre-Ryzhik-Zlatos 12, Zlatos 12);


Bistable case:

• f (t, x , u) = f (t, u): existence, stability and uniqueness up tospace shift (W.Shen 99, 04, 06);

• f (t, x , u) = f (x , u): existence, stability and uniqueness under

the restriction∫ 10 minx∈R f (x , u)du > 0 (Nolen-Ryzhik 09,

Mellet-Nolen-Roquejoffre-Ryzhik 09). Iff (x , u) = u(u − θ(x))(1− u), then this restriction is the caseif θ(x), x ∈ R are uniformly close to 0.


Ignition case:

• f (t, x , u) = f (t, u): existence, stability and uniqueness up tospace shift (W.Shen-Z.Shen 14).

• f (t, x , u) = f (x , u): existence (Nolen-Ryzhik 09,Mellet-Roquejoffre-Sire 10, Zlatos 13); stability anduniqueness up to time shift (Mellet-Nolen-Roquejoffre-Ryzhik09, Zlatos 13);

Thank You!

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