Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

0.32 MB·English

by  Wenxian Shen

REGULARITY AND STABILITY OF TRANSITION FRONTS IN
NONLOCAL EQUATIONS WITH TIME HETEROGENEOUS IGNITION
NONLINEARITY
5
1
0 WENXIANSHEN ANDZHONGWEI SHEN
2

n
Abstract. Thepresent paperis devotedtotheinvestigation of various propertiesof tran-
a
sition fronts in nonlocal equationsin heterogeneous media of ignition type,whose existence
J
hasbeenestablishedbytheauthorsofthepresentpaperinapreviouswork. Itisfirstshown
9
thatthetransition front iscontinuouslydifferentiableinspacewith uniformlyboundedand

uniformlyLipschitzcontinuousspacepartialderivative. Thisisthefirsttimethatregularity
]
P of transition fronts in nonlocal equations is ever studied. It is then shown that the tran-
A sition front is uniformly steep. Finally, asymptotic stability, in the sense of exponentially
attracting front like initial data, of thetransition front is studied.
.
h
t
a
m
[
Contents

1
v 1. Introduction 1
9 2. Regularity of transition fronts 6
2
3. Uniform steepness 13
0
4. Stability of transition fronts 17
2
0 5. Asymptotic stability of transition fronts 23
1. Appendix A. Comparison principles 32
0 References 33
5
1
:
v
i
X 1. Introduction
r
a Consider
u = J ∗u−u+f(t,x,u), (1.1)
t
where J is the dispersal kernel and [J ∗u](x) = J(x−y)u(y)dy = J(y)u(x−y)dy, and
R R
the reaction term f is of monostable type, bistable type or ignition type. Such an equation,
R R
introduced as a substitute for the classical reaction-diffusion equation
u = ∆u+f(t,x,u), (1.2)
t
has been used to model various diffusive processes with jumps (see e.g. [12] for some back-
ground). While a large amount of literature has been carried out to the understanding of
(1.2), its nonlocal version (1.1) has attracted a lot of attention recently andsome results have
2010 Mathematics Subject Classification. 35C07, 35K55, 35K57, 92D25.
Key words and phrases. transition front, regularity, stability.
1
2 WENXIANSHENANDZHONGWEISHEN
been established. For (1.1) in the homogeneous media, traveling waves, i.e., solutions of the
form u(t,x) = φ(x−ct) with (c,φ) satisfying
J ∗φ−φ+cφ +f(φ)= 0, φ(−∞) = 1, φ(∞) = 0,
x
have been obtained (see [1, 6, 7, 8, 9, 10, 18] and references therein). The study of (1.1) in
the heterogeneous media is rather recent and results concerning front propagation are very
limited. In [11, 22, 23, 24], the authors investigated (1.1) in the space periodic monostable
media and proved the existence of spreading speeds and periodic traveling waves. In [17],
Rawal, ShenandZhangstudiedtheexistence ofspreadingspeedsandtraveling waves of (1.1)
inthespace-timeperiodicmonostablemedia. For(1.1)inthespaceheterogeneousmonostable
media,Berestycki, CovilleandVostudiedin[2]theprincipaleigenvalue, positivesolutionand
long-timebehaviorofsolutions,whileLimandZlatoˇs provedin[14]theexistenceoftransition
fronts in the sense of Berestycki-Hamel (see [3, 4]). In [5], Berestycki and Rodr´ıguez studied
(1.1) with a barrier nonlinearity of monostable type or bistable type, and proved that while
propagation always occur in the monostable case, it may be obstructed in the bistable case.
For (1.1) in the time heterogeneous media of ignition type, the authors of the present paper
proved in [21] the existence of transition fronts.
In the present paper, we continue to study (1.1) in the time heterogeneous media based
on the work done in [21]. Recall that, an entire solution u(t,x) of (1.1) is called a transition
front in the sense of Berestycki-Hamel (see [3, 4]) if u(t,−∞) = 1 and u(t,∞) = 0 for any
t ∈ R, and for any ǫ ∈ (0,1) there holds
supdiam{x ∈ R|ǫ ≤ u(t,x) ≤ 1−ǫ} < ∞.
t∈R
Equivalently, an entire solution u(t,x) of (1.1) is called a transition front if there exists a
function X :R → R such that
lim u(t,x+X(t)) = 1 and lim u(t,x+X(t)) = 0 uniformly in t ∈ R.
x→−∞ x→∞
We remark that neither the definition of transition front nor the equation (1.1) itself guar-
antees any space regularity of transition fronts beyond continuity. Also, the transition fronts
constructed in [14] and [21] are only uniformly Lipschitz continuous in space; it is not known
if they are continuously differentiable in space. One of the main goals of the present paper
is to investigate the space regularity of transition fronts constructed in [21]. It should be
pointed out that space regularity is of fundamental importance in further studying various
important properties, such as uniform steepness and stability, of transition fronts.
Now, let us focus on (1.1) in the time heterogeneous media of ignition type, i.e.,
u = J ∗u−u+f(t,u), (t,x) ∈ R×R, (1.3)
t
where the convolution kernel J satisfies
(H1) J 6≡ 0, J ∈ C1(R), J(x) = J(−x) ≥ 0 for all x ∈ R, J(x)dx = 1, |J′(x)|dx < ∞
R R
and
R R
J(x)eλxdx < ∞, ∀λ > 0; (1.4)
R
Z
and the time heterogeneous nonlinearity f(t,u) satisfies
REGULARITY AND STABILITY OF TRANSITION FRONTS 3
(H2) f :R×[0,∞) → R is continuously differentiable and satisfies the following conditions:
• there are θ ∈ (0,1) (the ignition temperature), f ∈ C1,α([0,1]) and a Lipschitz
min
continuous function f : [0,1] → R satisfying
max
f (u) = 0= f (u), u∈ [0,θ]∪{1},
min max
0 < f (u) ≤ f (u), u∈ (θ,1),
min max
f′ (1) < 0
min
such that
f (u) ≤ f(t,u)≤ f (u), (t,u) ∈ [0,1].
min max
• f(t,u)< 0 for (t,u) ∈ R×(1,∞)
• first-order partial derivatives are uniformly bounded, i.e.,
sup |f (t,u)| <∞ and sup |f (t,u)| < ∞
t u
(t,u)∈R×[0,1] (t,u)∈R×[0,∞)
• there exists θ˜∈ (θ,1) such that f (t,u) ≤ 0 for all t ∈R and u∈ [θ˜,1].
u
For convenience and later use, let us first summarize the main results obtained in [21]. To
this end, consider the following homogeneous equation
u = J ∗u−u+f (u), (t,x) ∈ R×R, (1.5)
t min
where f , given in (H2), is of ignition type. Assume (H1) and (H2). It is proven in [8] that
min
there are a unique c∗ > 0 and a unique C1 function φ= φ : R → (0,1) satisfying
min min
J ∗φ−φ+c∗ φ′+f (φ) = 0,
min min (1.6)
(φ′ < 0, φ(0) = θ, φ(−∞)= 1 and φ(∞) = 0.
That is, φ is the normalized wave profileand φ (x−c∗ t) is the traveling wave of (1.5).
min min min
Moreover, using the equation in (1.6), it is not hard to see that φ′ is uniformly Lipschitz
min
continuous, that is,
φ′ (x)−φ′ (y)
sup min min < ∞. (1.7)
x−y
x6=y(cid:12) (cid:12)
(cid:12) (cid:12)
The following proposition is proved in [21].
(cid:12) (cid:12)
(cid:12) (cid:12)
Proposition 1.1 ([21]). Suppose (H1)-(H2).
(1) For s < 0, there exists a unique y ∈ R with y → −∞ as s → −∞ such that the
s s
classical solution u(t,x;s) of (1.3) with initial data u(s,x;s) = φ (x−y ) satisfies
min s
the normalization u(0,0;s) = θ and the following properties:
(i) u(t,−∞;s) = 1, u(t,∞;s) = 0 and u(t,x;s) is strictly decreasing in x;
(ii) let X (t;s) be such that u(t,X (t;s);s) = λ for any λ ∈(0,1); there exist c >
λ λ min
0, c > 0, and a twice continuously differentiable function X(·;s) : [s,∞) → R
max
satisfying
0 < c ≤ X˙(t;s) ≤ c < ∞, s < 0, t ≥ s and sup |X¨(t;s)| < ∞
min max
s<0,t≥s
such that
∀λ ∈ (0,1), sup |X(t;s)−X (t;s)| < ∞
λ
s<0,t≥s
4 WENXIANSHENANDZHONGWEISHEN
and there exist exponents c > 0 and shifts h > 0 such that
± ±
u(t,x;s) ≥ 1−ec−(x−X(t;s)+h−) if x ≤ X(t;s)−h ,
−
u(t,x;s) ≤ e−c+(x−X(t;s)−h+) if x ≥ X(t;s)+h
+
for all s < 0, t ≥ s;
(iii) u(t,x;s) is uniformly Lipschitz continuous in space, that is,
u(t,y;s)−u(t,x;s)
sup < ∞. (1.8)
y−x
x6=y (cid:12) (cid:12)
s<0,t≥s(cid:12) (cid:12)
(cid:12) (cid:12)
(2) There is a transition front u(cid:12)(t,x) that is strictly(cid:12)decreasing in space and uniformly
Lipschitz continuous in space, that is,
u(t,y)−u(t,x)
sup < ∞,
y−x
x6=y (cid:12) (cid:12)
t∈R (cid:12) (cid:12)
and a continuously differentia(cid:12)ble function X :(cid:12)R→ R satisfying the following proper-
(cid:12) (cid:12)
ties:
(i) there holds
X(t;s) → X(t), u(t,x;s) → u(t,x) and u (t,x;s) → u (t,x)
t t
locally uniformly in (t,x) ∈R×R as s→ −∞ along some subsequence;
(ii) X˙(t) ∈ [c ,c ] for all t ∈R, where c and c are as in (1)(ii);
min max min max
(iii) there hold
u(t,x) ≥ 1−ec−(x−X(t)+h−) if x ≤ X(t)−h ,
−
u(t,x) ≤ e−c+(x−X(t)−h+) if x ≥ X(t)+h
+
for all s < 0, t ≥ s, where c and h are as in (1)(ii).
± ±
In the present paper, we intend to improve the uniform Lipschitz continuity in space of
u(t,x) in Proposition 1.1(2), and then, study other important properties of u(t,x) such as
uniform steepness and stability. To do so, we further assume
(H3) f(t,u) is twice continuously differentiable in u and satisfies
sup |f (t,u)| < ∞.
uu
(t,u)∈R×[0,1]
Our first main result concerning space regularity of u(t,x) is stated in the following theo-
rem.
Theorem 1.2. Suppose (H1)-(H3). Let u(t,x) be the transition front in Proposition 1.1(2).
Then, foranyt ∈ R, u(t,x)iscontinuouslydifferentiable inx. Moreover, u (t,x)isuniformly
x
bounded and uniformly Lipschitz continuous in x, that is,
u (t,x)−u (t,y)
x x
sup |u (t,x)| < ∞ and sup <∞, (1.9)
x
(t,x)∈R×R x6=y (cid:12) x−y (cid:12)
t∈R (cid:12) (cid:12)
(cid:12) (cid:12)
respectively.
(cid:12) (cid:12)
REGULARITY AND STABILITY OF TRANSITION FRONTS 5
We remark that since u(t,x) is strictly decreasing in x, the uniform bound of u (t,x) in
x
(1.9) is equivalent to inf(t,x)∈R×Rux(t,x) > −∞. With the regularity, the profile function
φ(t,x) = u(t,x+X(t)) satisfies the following evolution equation
φ = J ∗φ−φ+X˙(t)φ +f(t,φ),
t x
which could be used to construct transition fronts if X(t) can be first constructed (see [16]
for the work on (1.2) in time heterogeneous monostable media).
Next, we study the uniform steepness of the transition front. We prove
Theorem 1.3. Suppose (H1)-(H3). Let u(t,x) and X(t) be as in Proposition1.1(2). Then,
for any M > 0, there holds
sup sup u (t,x) < 0.
x
t∈Rx∈[X(t)−M,X(t)+M]
Asimpleconsequenceof Theorem1.2 andTheorem1.3 isthat theinterface location atany
constant value between 0 and 1 is continuously differentiable with finite speed (see Corollary
3.3).
Finally, we studythestability of transition fronts. Let Cb (R,R) bethe spaceof bounded
unif
and uniformly continuous functions on R. For u ∈ Cb (R,R), denote by u(t,x;t ,u ) the
0 unif 0 0
unique solution of (1.3) with initial data u(t ,·;t ,u ) = u . To state the result, we enhance
0 0 0 0
the last assumption in (H2) and assume
(H4) there exist θ˜∈(θ,1) and β˜> 0 such that f (t,u) ≤ −β˜ for all (t,u) ∈ R×[θ˜,2]. Also,
u
f(t,u) =0 for (t,u) ∈ R×(−∞,0).
Let M > 0 be such that for any t ∈ R
1
1+θ˜ θ
u(t,x) ≥ if x−X(t) ≤ −M and u(t,x) ≤ if x−X(t) ≥ M , (1.10)
1 1
2 2
where θ˜ is as in (H4). Such an M exists by Proposition1.1(2)(iii). For given α > 0, let
1
Γ :R → [0,1] be a smooth nonincreasing function satisfying
α
1, x ≤ −M −1,
1
Γ (x) = (1.11)
α (e−α(x−M1), x ≥ M1+1.
This function is introduced for making up the lack of asymptotic stability of the equilibrium
0 (see e.g. [15, 20]). We prove
Theorem 1.4. Suppose (H1)-(H4).
(1) There is α > 0 such that for any 0 < α ≤ α , there are ǫ = ǫ (α), ω = ω(α),
0 0 0 0
and A = A(α) satisfying that for any u : R → [0,1], u ∈ Cb (R,R), if there exist
0 0 unif
t ∈ R, ǫ ∈ (0,ǫ ], ζ± such that
0 0 0
u(t ,x−ζ−)−ǫΓ (x−ζ−−X(t ))
0 0 α 0 0
(1.12)
≤u (x) ≤ u(t ,x−ζ+)+ǫΓ (x−ζ+−X(t ))
0 0 0 α 0 0
for all x ∈ R, then, there holds
u(t,x−ζ−(t))−q(t)Γ (x−ζ−(t)−X(t))
α
≤ u(t,x;t ,u ) ≤ u(t,x−ζ+(t))+q(t)Γ (x−ζ+(t)−X(t))
0 0 α
6 WENXIANSHENANDZHONGWEISHEN
for all x ∈ R and t ≥ t , where
0
Aǫ
ζ±(t) = ζ±± (1−e−ω(t−t0)) and q(t) =ǫe−ω(t−t0).
0 ω
(2) Let u(t,x) and X(t) be as in Proposition1.1(2). Let β > 0. Suppose t ∈ R and
0 0
u ∈ Cb (R,R) satisfy
0 unif
u :R → [0,1], u (−∞)= 1;
0 0
(∃C > 0 s.t. |u0−u(t0,x)| ≤ Ce−β0(x−X(t0)) for x ∈ R.
Then, there exist ω > 0 and ǫ˜ > 0 such that for any ǫ ∈ (0,ǫ˜ ] there are ζ± =
0 0
ζ±(ǫ,u ) ∈ R such that
0
u(t,x−ζ−)−ǫe−ω(t−t0) ≤u(t,x;t ,u )≤ u(t,x−ζ+)+ǫe−ω(t−t0)
0 0
for all x ∈ R and t ≥ t .
0
Based on Theorem 1.4 and the “squeezing technique” (see e.g. [7, 15, 19, 20]), we obtain
the asymptotic stability.
Theorem 1.5. Suppose (H1)-(H4). Let u(t,x) and X(t) be as in Proposition1.1(2). Let
β > 0. Suppose t ∈ R and u ∈ Cb (R,R) satisfy
0 0 0 unif
u :R → [0,1], u (−∞)= 1;
0 0
(∃C > 0 s.t. |u0−u(t0,x)| ≤ Ce−β0(x−X(t0)) for x ∈ R.
Then, there exist C = C(u ) > 0, ζ = ζ (u ) ∈R and r = r(β )> 0 such that
0 ∗ ∗ 0 0
sup|u(t,x;t ,u )−u(t,x−ζ )| ≤ Ce−r(t−t0)
0 0 ∗
x∈R
for all t ≥ t .
0
We point out, allowing the solution to develop into the shape satisfying the condition in
Theorem 1.4 at a later time, Theorem 1.4(2) and Theorem 1.5 are true for more general
initial data (see Corollary 4.2 and Corollary 5.3).
The rest of the paper is organized as follows. In Section 2, we study the space regularity
of u(t,x) and prove Theorem 1.2. In Section 3, we study the uniform steepness of u(t,x) and
prove Theorem 1.3. In Section 4, we study the stability of u(t,x) and prove Theorem 1.4.
In Section 5, we study the asymptotic stability of u(t,x) and prove Theorem 1.5. We also
include an appendix, Appendix A, on comparison principles for convenience.
2. Regularity of transition fronts
In this section, we study the regularity of u(t,x) and prove Theorem 1.2. Throughout this
section, weassume(H1)-(H3). ToproveTheorem1.2, wefirstinvestigate thespaceregularity
of u(t,x;s). We have
Theorem 2.1. For any s < 0 and t ≥ s, u(t,x;s) is continuously differentiable in x. More-
over,
REGULARITY AND STABILITY OF TRANSITION FRONTS 7
(i) u (t,x;s) is uniformly bounded, that is,
x
sup |u (t,x;s)| < ∞;
x
x6=y
s<0,t≥s
(ii) u (t,x;s) is uniformly Lipschitz continuous in space, that is,
x
u (t,x;s)−u (t,y;s)
x x
sup < ∞.
x−y
x6=y (cid:12) (cid:12)
s<0,t≥s (cid:12) (cid:12)
(cid:12) (cid:12)
Assuming Theorem 2.1, let us pr(cid:12)ove Theorem 1.2. (cid:12)
Proof of Theorem 1.2. It follows from Proposition 1.1(2)(i), Theorem 2.1, Arzela`-Ascoli the-
orem andthe diagonal argument. More precisely, besidesu(t,x;s) → u(t,x) and u (t,x;s) →
t
u (t,x) locally uniformly as in Proposition 1.1(2)(i) we also have
t
u (t,x;s) → u (t,x) locally uniformly in (t,x) ∈ R×R (2.1)
x x
as s → −∞ along some subsequence. The properties of u(t,x) then inherit from that of
u(t,x;s). (cid:3)
In the rest of this section, we prove Theorem 2.1.
Proof of Theorem 2.1. (i) Setting
u(t,x+η;s)−u(t,x;s)
vη(t,x;s) := .
η
By (1.8), supx∈R,η6=0 |vη(t,x;s)| < ∞. Clearly, vη(t,x;s) satisfies
s<0,t≥s
vη(t,x;s) = J(x−y)vη(t,y;s)dy−vη(t,x;s)+aη(t,x;s)vη(t,x;s), (2.2)
t
R
Z
where
f(t,u(t,x+η;s))−f(t,u(t,x;s))
aη(t,x;s) =
u(t,x+η;s)−u(t,x;s)
is uniformly bounded by (H2). Setting
J(x−y+η)−J(x−y)
bη(t,x;s) := J(x−y)vη(t,y;s)dy = u(t,y;s)dy,
R R η
Z Z
we see that supx∈R,η6=0 |bη(t,x;s)| < ∞, since J′ ∈ L1(R) and u(t,x;s) ∈ (0,1).
s<0,t≥s
The solution of (2.2) is given by
t
vη(t,x;s) = vη(s,x;s)e−Rst(1−aη(τ,x;s))dτ + bη(r,x;s)e−Rrt(1−aη(τ,x;s))dτdr. (2.3)
Zs
Notice as η → 0, the following pointwise limits hold:
φ (x+η−y )−φ (x−y )
vη(s,x;s) = min s min s → φ′ (x−y ),
η min s
aη(t,x;s) → f (t,u(t,x;s)) and
u
bη(t,x;s) → J′(x−y)u(t,y;s)dy,
R
Z
8 WENXIANSHENANDZHONGWEISHEN
where φ is as in (1.6). Then, setting η → 0 in (2.3), we conclude from the dominated con-
min
vergence theorem that for any s < 0, t ≥ s and x ∈ R, thelimit u (t,x;s) = lim vη(t,x;s)
x η→0
exists and
t
ux(t,x;s) = φ′min(x−ys)e−Rst(1−fu(τ,u(τ,x;s)))dτ + b(r,x;s)e−Rrt(1−fu(τ,u(τ,x;s)))dτdr, (2.4)
Zs
where b(t,x;s) = J′(x − y)u(t,y;s)dy = J′(y)u(t,x − y;s)dy. In particular, for any
R R
s < 0 and t ≥ s, u(t,x;s) is continuously differentiable in x. The uniform boundedness of
R R
u (t,x;s), i.e., sup |u (t,x;s)| < ∞, then follows from (1.8).
x x6=y x
s<0,t≥s
(ii) Since u (t,x;s) is uniformly bounded by (i), we trivially have
x
u (t,x;s)−u (t,y;s)
x x
∀δ > 0, sup < ∞.
x−y
|x−y|≥δ (cid:12) (cid:12)
s<0,t≥s (cid:12) (cid:12)
(cid:12) (cid:12)
Thus, to show the uniform Lipschitz c(cid:12)ontinuity of u (t,x;s),(cid:12)it suffices to show the local
x
uniform Lipschitz continuity, i.e.,
u (t,x;s)−u (t,y;s)
x x
∀δ > 0, sup < ∞. (2.5)
x−y
|x−y|≤δ (cid:12) (cid:12)
s<0,t≥s (cid:12) (cid:12)
(cid:12) (cid:12)
To this end, we fix δ > 0. Let X(t;(cid:12)s) and X (t;s) for λ(cid:12)∈ (0,1) be as in Proposition
λ
1.1(1)(ii) and define
L = δ+ sup X (t;s)−X(t;s) and L = δ+ sup X (t;s)−X(t;s) ,
1 θ 2 θ˜
s<0,t≥s s<0,t≥s
(cid:12) (cid:12) (cid:12) (cid:12)
where θ˜∈ (θ,1) is give(cid:12)n in (H2). Notice(cid:12)L < ∞ and L < ∞ by(cid:12) the uniform exp(cid:12)onential
1 2
decaying estimates in Proposition 1.1(1)(ii). Then, for any x ∈ R and |η| ≤ δ we have
• if x ≥ X(t;s)+L , then x+η ≥ x−δ ≥ X (t;s), which implies that u(t,x+η;s) ≤ θ
1 θ
by monotonicity, and hence
f (t,u(t,x+η;s)) = 0; (2.6)
u
• if x ≤ X(t;s)−L , then x+η ≤ x+δ ≤ X (t;s), which implies that u(t,x+η;s) ≥ θ˜
2 θ˜
by monotonicity, and hence by (H2),
f (t,u(t,x+η;s)) ≤ 0. (2.7)
u
According to (2.6) and (2.7), we consider time-dependent disjoint decompositions of R into
R = R (t;s)∪R (t;s)∪R (t;s),
l m r
where
R (t;s) = (−∞,X(t;s)−L ),
l 2
R (t;s) = [X(t;s)−L ,X(t;s)+L ] and (2.8)
m 2 1
R (t;s) = (X(t;s)+L ,∞).
r 1
For s < 0 and x ∈ R, let t (x ;s) be the first time that x is in R (t;s), that is,
0 first 0 0 m
t (x ;s) = min t ≥ s x ∈ R (t;s) ,
first 0 0 m
(cid:8) (cid:12) (cid:9)
(cid:12)
REGULARITY AND STABILITY OF TRANSITION FRONTS 9
and t (x ;s) be the last time that x is in R (t;s), that is,
last 0 0 m
t (x ;s)= max t ∈ R x ∈R (t ;s) and x ∈/ R (t,s) for t > t .
last 0 0 0 m 0 0 m 0
Since X˙(t;s) ≥ c > 0 by (cid:8)Proposi(cid:12)tion 1.1(1)(ii), if x ∈ R (s;s), then x ∈ R(cid:9)(t;s) for all
min (cid:12) 0 l 0 l
t > s. In this case, t (x ;s) and t (x ;s) are not well-defined, but it will not cause any
first 0 last 0
trouble. We see that t (x ;s) and t (x ;s) are well-defined only if x ∈/ R (s;s). As a
first 0 last 0 0 l
simple consequence of X˙(t;s) ∈ [c ,c ] in Proposition 1.1(1)(ii) and the fact that the
min max
length of R (t;s) is L +L , we have
m 1 2
T = T(δ) := sup t (x ;s)−t (x ;s) < ∞. (2.9)
last 0 first 0
s<0,x0∈/Rl(s;s)
(cid:2) (cid:3)
Moreover, we see that for any |η| ≤ δ,
f (t,u(t,x +η;s)) = 0 if t ∈ [s,t (x ;s)],
u 0 first 0
(2.10)
f (t,u(t,x +η;s)) ≤ 0 if t ≥ t (x ;s).
u 0 last 0
We now show that
u (t,x +η;s)−u (t,x ;s)
x 0 x 0
sup < ∞. (2.11)
η
x0∈R,0<|η|≤δ (cid:12) (cid:12)
s<0,t≥s (cid:12) (cid:12)
(cid:12) (cid:12)
Using (2.4), we have (cid:12) (cid:12)
u (t,x +η;s)−u (t,x ;s)
x 0 x 0
η
= φ′min(x0+η−ys)−φ′min(x0−ys)e−Rst(1−fu(τ,u(τ,x0+η;s)))dτ
η
(I)
+| φ′ (x −y )e−Rst(1−fu(τ,u{(τz,x0+η;s)))dτ −e−Rst(1−fu(τ,u(τ,}x0;s)))dτ
min 0 s η
(II)
+| t b(r,x0 +η;s)−b(r,x0;s)e−{zRrt(1−fu(τ,u(τ,x0;s)))dτdr }
η
Zs
(III)
| t e−Rrt(1−fu(τ,u({τz,x0+η;s)))dτ −e−Rrt(1−fu(τ,u}(τ,x0;s)))dτ
+ b(r,x ;s) dr.
0
η
Zs
(IV)
Hence, it suffice to| bound terms (I)-(IV). To do{szo, we need to consider three c}ases: x ∈
0
R (s;s), x ∈ R (s;s) and x ∈ R (s;s). We here focus on the last case, i.e., x ∈ R (s;s),
l 0 m 0 r 0 r
which is the most involved one. The other two cases are simpler and can be treated similarly.
Also, for fixed s < 0 and x ∈ R (s;s), we will focus on t ≥ t (x ;s); the case with
0 r last 0
t ∈ [t (x ;s),t (x ;s)] or t ≤ t (x ;s) will be clear. Thus, we assume x ∈ R (s;s) and
first 0 last 0 first 0 0 r
t ≥ t (x ;s).
last 0
10 WENXIANSHENANDZHONGWEISHEN
We will frequently use the following estimates: for any |η˜|≤ δ there hold
e−Rrtfirst(x0;s)(1−fu(τ,u(τ,x0+η˜;s)))dτ = e−(tfirst(x0;s)−r), r ∈ [s,tfirst(x0;s)]
e−Rrtlast(x0;s)(1−fu(τ,u(τ,x0+η˜;s)))dτ ≤ eTsup(t,u)∈R×[0,1]|1−fu(t,u)|, r ∈ [tfirst(x0;s),tlast(x0;s)]
e−Rrt(1−fu(τ,u(τ,x0+η˜;s)))dτ ≤ e−(t−r), r ∈ [tlast(x0;s),t].
(2.12)
They are simple consequences of (2.9) and (2.10). Set
φ′ (x)−φ′ (y)
C := sup |1−f (t,u)|, C := sup min min , C := sup|φ′ (x)|
0 (t,u)∈R×[0,1] u 1 x6=y(cid:12) x−y (cid:12) 2 x∈R min
(cid:12) (cid:12)
(cid:12) (cid:12)u(t,x;s)−u(t,y;s)
C := sup |f (t,u)|× sup |u (t,x(cid:12);s)|, C = sup (cid:12) .
3 uu x 4
(t,u)∈R×[0,1] x∈R x6=y (cid:12) x−y (cid:12)
s<0,t≥s s<0,t≥s(cid:12) (cid:12)
(cid:12) (cid:12)
Note that all these constants are finite. In fact, C < ∞ by (H2),(cid:12)C < ∞ by (1.7), C (cid:12)< ∞
0 1 3
by (H3) and Theorem 2.1(i), and C < ∞ by Proposition 1.1(1)(iii).
4
We are ready to bound (I)-(IV). For the term (I), using (1.7) and (2.12), we see that
|(I)| ≤ C1e−Rst(1−fu(τ,u(τ,x0+η;s)))dτ
= C1e− Rstfirst(x0;s)+Rttfilarsstt((xx00;;ss))+Rttlast(x0;s) (1−fu(τ,u(τ,x0+η;s)))dτ (2.13)
≤ C e−((cid:2)tfirst(x0;s)−s)eC0Te−(t−tlast(x0;s)) ≤(cid:3) C eC0T.
1 1
Fortheterm(II),wehavefromTaylorexpansionofthefunctionη 7→ e−Rst(1−fu(τ,u(τ,x0+η;s)))dτ
at η = 0 that
e−Rst(1−fu(τ,u(τ,x0+η;s)))dτ −e−Rst(1−fu(τ,u(τ,x0;s)))dτ
|(II)| ≤ C
2
η
(cid:12) (cid:12)
(cid:12) t (cid:12)
≤ C2(cid:12)(cid:12)e−Rst(1−fu(τ,u(τ,x0+η∗;s)))dτ fuu(τ,u(τ,x0 +η∗;s)(cid:12)(cid:12))ux(τ,x0+η∗;s) dτ,
Zs (cid:12) (cid:12)
where η is between 0 and η, and hence, |η |(cid:12)≤ δ. We see (cid:12)
∗ ∗ (cid:12) (cid:12)
t
f (τ,u(τ,x +η ;s))u (τ,x +η ;s) dτ ≤ C (t−s).
uu 0 ∗ x 0 ∗ 3
Zs (cid:12) (cid:12)
It then follows from ((cid:12)2.12) that (cid:12)
(cid:12) (cid:12)
|(II)| ≤ C C e−(tfirst(x0;s)−s)e−(t−tlast(x0;s))(t−s)
2 3
= C C e−(tfirst(x0;s)−s)e−(t−tlast(x0;s))
2 3
× (t−t (x ;s))+(t (x ;s)−t (x ;s))+(t (x ;s)−s)
last 0 last 0 first 0 first 0 (2.14)
≤ C2(cid:2)C3 e−(t−tlast(x0;s))(t−tlast(x0;s))+T +e−(tfirst(x0;s)−s)(tfirst(x0(cid:3);s)−s)
2
(cid:2) (cid:3)
≤ C C +T .
2 3
e
(cid:18) (cid:19)