This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.
Citation: Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity[J]. Electronic Research Archive, 2021, 29(3): 2269-2291. doi: 10.3934/era.2020116
[1] | Shao-Xia Qiao, Li-Jun Du . Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, 2021, 29(3): 2269-2291. doi: 10.3934/era.2020116 |
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This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.
In the present paper, we focus on the nonlocal dispersal equation with spatially dependent bistable nonlinearity
ut(x,t)=J∗u(x,t)−u(x,t)+f(x,u),x,t∈R, | (1) |
where the nonlinearity
(F) The nonlinearity
{f(x,u)=f1(u)forx≥0,u∈[0,1],f(x,u)=f2(u)forx≤−L,u∈[0,1],f2(u)≤f(x,u)≤f1(u),fu(x,u)<1forx∈[−L,0],u∈[0,1], | (2) |
where
fi∈C1,1([0,1]),fi(0)=fi(1)=0,f′i(0),f′i(1)<0,fi<0in(0,θi),fi>0in(θi,1),0<θ1<θ2<1,and∫10fi(s)ds>0,i=1,2. |
(J) The kernel function
{J(x)=J(−x),J(x)≥0,∫RJ(y)dy=1,∫R|J′(y)|dy<∞,∫RJ(y)e−λydy<∞for allλ>0. | (3) |
A typical example is
The nonlocal dispersal equation has attracted so much attention because of its extensive applications to account for diffusion phenomena involving jumps in biology, physics and chemistry [1,12]. Traveling waves as one kind of special solutions with invariant profile and fixed speed as well as entire solutions have been adequately investigated. It is well-known that the importance of the study of entire solutions of reaction-diffusion equations (nonlocal dispersal equations) is frequently recalled in the literature. Since the pioneering works of Hamel and Nadirashvili [13,14], there have been tremendous advances in studying the existence of entire solutions for various models. In particular, when the nonlinearity is homogeneous (i.e.
However, for the inhomogeneous nonlinearity, several works are devoted to transition fronts (see [3,19]) of nonlocal dispersal equations [17,20] and forced waves in shifting habitats [25]. Traveling waves and spreading speed of monostable nonlocal dispersal equations with space periodic nonlinearity were studied in [21]. Li et al. [16] further obtained the existence of entire solutions for space periodic nonlinearity. Particularly, Eberle [10,11] constructed a heteroclinic orbit connecting two traveling waves for bistable local dispersal equation 1 in cylinders. Meanwhile, Berestycki and Rodríguez [5] considered a bistable nonlocal dispersal equation with a gap in one dimension. As far as we know, there is no result on entire solutions for the nonlocal dispersal equation 1 with inhomogeneous nonlinearity
In this paper, we aim to construct an entire solution connecting traveling waves with the two different nonlinearities, motivated by [10,11]. By constructing suitable sub- and super-solutions, we establish the existence and uniqueness of the entire solution behaving as the traveling wave coming from one side and eventually going to the other side, which is different with the one constructed in [23] for the case
The crucial part of this paper is to figure out the long time asymptotic behavior of the entire solution. Since the lack of compactness of the nonlocal operator, we can not use the Lyapunov function argument as [10,11] to show that the entire solution converges to a translation of the other traveling wave as time goes to positive infinity. However, inspired by the idea in [7], we use a "squeezing" technique to address this issue. Furthermore, we also apply sub- and super-solutions method with comparison principle to establish Lyapunov stability of the entire solution. This can be done because that we can obtain a positive estimate on the derivative of the entire solution with respect to
Now we state the main results of this paper as follows.
Theorem 1.1. Let assumptions
u(x,t)−ϕ1(x+c1t)→0ast→−∞uniformly inx∈R |
and
u(x,t)−ϕ2(x+c2t+β)→0ast→+∞uniformly inx∈R |
for some
{J∗ϕi−ϕi+fi(ϕi)−ciϕ′i=0,ϕi(−∞)=0,ϕi(+∞)=1,0<ϕi<1. | (4) |
Theorem 1.2. The entire solution
|v(x,t,v0)−u(x+a,t+t0)|<ϵ |
for
The rest of this paper is organized as follows. In Section 2, we recall some results of the bistable traveling waves for homogeneous nonlinearities and prove the comparison principle for 1. Section 3 is devoted to constructing the unique entire solution. In Section 4, we study the asymptotic profile as time goes to positive infinity. Finally, we establish Lyapunov stability of the entire solution in Section 5.
In this section, some known results on the traveling waves of 4 are outlined and the comparison principle is established.
It follows from Theorem 2.7 in [2] and Theorem 2.7 in [23] that 4 admits a solution
{β0eλ0z≤ϕ1(z)≤α0eλ0z,z≤0,β1e−λ1z≤1−ϕ1(z)≤α1e−λ1z,z>0, | (5) |
where
cλ0=∫RJ1(y)e−λ0ydy−1+f′1(0)andcλ1=∫RJ1(y)e−λ1ydy−1+f′1(1), |
respectively. Moreover, we have
{˜β0eλ0z≤ϕ′1(z)≤˜α0eλ0z,z≤0,˜β1e−λ1z≤ϕ′1(z)≤˜α1e−λ1z,z>0 | (6) |
for some constants
|f1(u+v)−f1(u)−f1(v)|≤Lfuvfor0≤u,v≤1. |
We show that the following comparison principle holds by a contradiction argument.
Proposition 1. Suppose that assumptions
{∂u(x,t)∂t−(∫RJ(x−y)[u(y,t)−u(x,t)]dy)−f(x,u(x,t))≥0,∀(x,t)∈R×(0,T],u(x,0)≥0,x∈R, |
{∂v(x,t)∂t−(∫RJ(x−y)[v(y,t)−v(x,t)]dy)−f(x,v(x,t))≤0,∀(x,t)∈R×(0,T],v(x,0)≤0,x∈R, |
respectively. Then,
u(x,t)≥v(x,t)inR×[0,T]. |
Furthermore, if
Proof of Proposition 1. Let
limn→+∞ˇw(xn,tn)=infR×[0,ϵ0T]ˇw(x,t)<0. |
Observe that
ˇwt(x,t)=Zˇw(x,t)+eZtˉwt(x,t)≥Zˇw(x,t)+eZt∫RJ(x−y)[ˉw(y,t)−ˉw(x,t)]dy+eZt[f(x,u(x,t))−f(x,v(x,t))]=Zˇw(x,t)+∫RJ(x−y)[ˇw(y,t)−ˇw(x,t)]dy+fu(x,uθ(x,t))ˇw(x,t). |
where
ˇw(xn,tn)−ˇw(xn,0)≥∫tn0[J∗ˇw(xn,s)−ˇw(xn,s)+Zˇw(xn,s)+fu(x,uθ(x,s))ˇw(xn,s)]ds≥∫tn0[J∗ˇw(xn,s)+(‖fu(x,u)‖∞−fu(x,uθ(x,s)))infR×[0,ϵ0T]ˇw(x,t)]ds. |
Letting
infR×[0,ϵ0T]ˇw(x,t)≥(1+‖fu(x,u)‖∞−minR×[0,1]|fu(x,uθ(x,s))|)ϵ0TinfR×[0,ϵ0T]ˇw(x,t). |
Since
infR×[0,ϵ0T]ˇw(x,t)≥(1+‖fu(x,u)‖∞−minR×[0,1]|fu(x,uθ(x,s))|)ϵ0TinfR×[0,ϵ0T]ˇw(x,t)>infR×[0,ϵ0T]ˇw(x,t). |
Thus, we get a contradiction. Therefore, we obtain
infR×[0,ϵ0T]ˇw(x,t)≥(1+‖fu(x,u)‖∞−minR×[0,1]|fu(x,uθ(x,s))|−ϵinfR×[0,ϵ0T]ˇw(x,t))ϵ0TinfR×[0,ϵ0T]ˇw(x,t). |
Here, we can also choose
(1+‖fu(x,u)‖∞−minR×[0,1]|fu(x,uθ(x,s))|−ϵinfR×[0,ϵ0T]ˇw(x,t))ϵ0T<1. |
This finishes the proof.
In this section, we focus on the construction of the entire solution which behaves like a traveling wave approaching from infinity. The main idea is to establish suitable sub- and super-solutions, which are defined as follows.
W−(x,t)={ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t)),x≥0,0,x<0, |
and
W+(x,t)={ϕ1(x+c1t+ξ(t))+ϕ1(−x+c1t+ξ(t)),x≥0,2ϕ1(c1t+ξ(t)),x<0, |
here
˙ξ(t)=Meλ(c1t+ξ),t<−T,ξ(−∞)=0, |
where
ξ(t)=1λln11−c−11Meλc1t. |
For the function
c1t+ξ(t)≤0for−∞<t≤−T, |
where
Now we verify that
Lu=ut−(J∗u−u)−f(x,u). |
We first deal with the sub-solution
LW−(x,t)=(c1−˙ξ(t))[ϕ′1(x+c1t−ξ(t))−ϕ′1(−x+c1t−ξ(t))]−∫+∞0J(x−y)[ϕ1(y+c1t−ξ(t))−ϕ1(−y+c1t−ξ(t))]dy+[ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))]−f(x,W−)=−˙ξ(t)[ϕ′1(x+c1t−ξ(t))−ϕ′1(−x+c1t−ξ(t))]+∫0−∞J(x−y)[ϕ1(y+c1t−ξ(t))−ϕ1(−y+c1t−ξ(t))]dy+f1(ϕ1(x+c1t−ξ(t)))−f1(ϕ1(−x+c1t−ξ(t)))−f(x,W−). |
Recall that
LW−(x,t)≤−˙ξ(t)[ϕ′1(x+c1t−ξ(t))−ϕ′1(−x+c1t−ξ(t))]+f1(ϕ1(x+c1t−ξ(t)))−f1(ϕ1(−x+c1t−ξ(t)))−f1(ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))). |
Then we continue to show
Case 1. For
ϕ′1(x+c1t−ξ(t))−ϕ′1(−x+c1t−ξ(t))>m[ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))] |
for some
LW−(x,t)≤−˙ξ(t)[ϕ′1(x+c1t−ξ(t))−ϕ′1(−x+c1t−ξ(t))]+Lfϕ1(−x+c1t−ξ(t)))[ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))]≤[ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))][−mMeλ(c1t+ξ(t))+Lfα0eλ0(−x+c1t−ξ(t))]=[ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))]eλ(c1t+ξ(t))[−mM+Lfα0eλ0(−x−2ξ(t))]. |
Therefore, if we choose
Case 2. For
LW−(x,t)≤−Meλ(c1t+ξ(t))[~β1e−λ1(x+c1t−ξ(t))−˜αeλ0(−x+c1t−ξ(t))]+Lfα0eλ0(−x+c1t−ξ(t))[α1e−λ1(x+c1t−ξ(t))−β0eλ0(−x+c1t−ξ(t))]≤−eλ0(−x+c1t−ξ(t))[M~β1e(λ0−λ1)xe−(λ0+λ1−λ)c1t+(λ+λ0+λ1)ξ(t))−M˜αeλ(c1t+ξ(t))−Lfα0]. |
Since
For
LW−(x,t)≤Meλ(c1t+ξ(t))ϕ′1(−x+c1t−ξ(t))+f1(ϕ1(x+c1t−ξ(t)))−f1(ϕ1(−x+c1t−ξ(t)))−f1(ϕ1(x+c1t−ξ(t))−ϕ1(−x+c1t−ξ(t))). |
Moreover, if
LW−(x,t)≤Meλ(c1t+ξ(t))ϕ′1(−x+c1t−ξ(t))+[f′1(1)−f′(0)]ϕ1(−x+c1t−ξ(t))+o(ϕ1(−x+c1t−ξ(t)))≤eλ0(−x+c1t−ξ(t))[M˜α0eλ(c1t+ξ(t))+(f′1(1)−f′(0))−o(1)]. |
Since
LW−(x,t)≤−eλ0(−x+c1t−ξ(t))[M~β1e(λ0−λ1)xe−(λ0+λ1−λ)c1t+(λ+λ0+λ1)ξ(t))−M˜αeλ(c1t+ξ(t))−Lfα0]≤−eλ0(−x+c1t−ξ(t))[M~β1e(λ0−λ1)Le(λ−2λ0)c1t+(λ+2λ0)ξ(t))−M˜αeλ(c1t+ξ(t))−Lfα0]. |
Thanks to
LW−(x,t)≤0. |
We intend to testify the super-solution
Step 1.
LW+(x,t)=2(c1+˙ξ(t))ϕ′1(c1t+ξ(t))−∫RJ(x−y)W+(y,t)dy+2ϕ1(c1t+ξ(t))−f(x,2ϕ1(c1t+ξ(t)))≥2(c1+˙ξ(t))ϕ′1(c1t+ξ(t))−f1(2ϕ1(c1t+ξ(t)))−∫+∞0J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))]dy. |
Denote
I=∫+∞0J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))]dy. |
Then, it follows that
I=∫−c1t−ξ(t)0J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))]dy+∫+∞−c1t−ξ(t)J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))]dy≤α0eλ0(c1t+ξ(t))∫+∞0J(x−y)[eλ0y+e−λ0y]dy+2∫+∞−c1t−ξ(t)J(x−y)dy. |
Let
Jλ0=∫+∞0J(y)[eλ0y+e−λ0y]dy. |
It follows from assumption
J(x)≤KJe2λ0xforx≥0. |
Furthermore, there hold
I≤Jλ0α0eλ0(c1t+ξ(t))+2KJ∫+∞−c1t−ξ(t)e2λ0(x−y)dy≤Jλ0α0eλ0(c1t+ξ(t))+2KJλ0e2λ0(c1t+ξ(t)). |
Therefore, according to
LW+(x,t)≥2(c1+˙ξ(t))ϕ′1(c1t+ξ(t))−Jλ0α0eλ0(c1t+ξ(t))−2KJλ0e2λ0(c1t+ξ(t))−f1(2ϕ1(c1t+ξ(t)))≥(2c1~β0−α0Jλ0−2f′1(0)β0−o(1))eλ0(c1t+ξ(t))+(2M~β0−2KJλ0)e2λ0(c1t+ξ(t)). |
Since
M>KJ~β0λ0andJλ0<2c1~β0+2|f′1(0)|β0α0, |
consequently, we have
Step 2. For
LW+(x,t)=(c1+˙ξ(t))[ϕ′1(x+c1t+ξ(t))+ϕ′1(−x+c1t+ξ(t))]−∫RJ(x−y)W+(y,t)dy+[ϕ1(x+c1t+ξ(t))+ϕ1(−x+c1t+ξ(t))]−f1(W+)=˙ξ(t)[ϕ′1(x+c1t+ξ(t))+ϕ′1(−x+c1t+ξ(t))]+∫0−∞J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))−2ϕ1(c1t+ξ(t))]dy+f1(ϕ1(x+c1t+ξ(t)))+f1(ϕ1(−x+c1t+ξ(t)))−f1(ϕ1(x+c1t+ξ(t))+ϕ1(−x+c1t+ξ(t))). |
Denote
II=∫0−∞J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))−2ϕ1(c1t+ξ(t))]dy. |
We consider two cases.
Case 1.
II=∫0c1t+ξ(t)J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))−2ϕ1(c1t+ξ(t))]dy+∫c1t+ξ(t)−∞J(x−y)[ϕ1(y+c1t+ξ(t))+ϕ1(−y+c1t+ξ(t))−2ϕ1(c1t+ξ(t))]dy≥∫0c1t+ξ(t)J(x−y)Cλ0eλ0(c1t+ξ(t))[eλ0y+e−λ0y−2−(2+e(λ0+η)y+e−(λ0+η)y)Cηeη(c1t+ξ(t))]dy≥−(2+Jη)Cλ0Cηe(λ0+η)(c1t+ξ(t)). |
The second inequality follows from
|ϕ1(x)−Cλ0eλ0x|≤Cηe(λ0+η)xforx≤0and some0<η<λ0, |
which can be easily obtained by 5 and 6. As a consequence,
LW+(x,t)≥Me(λ+λ0)(c1t+ξ(t))~β0(eλ0x+e−λ0x)−(2+Jη)Cλ0Cηe(λ0+η)(c1t+ξ(t))−Lfα20e2λ0(c1t+ξ(t)))≥e(λ+λ0)(c1t+ξ(t))[2M~β0−(2+Jη)Cλ0Cηe(η−λ)(c1t+ξ(t))−Lfα20e(λ0−λ)(c1t+ξ(t))]≥e(λ+λ0)(c1t+ξ(t))[2M~β0−(2+Jη)Cλ0Cη−Lfα20]. |
The last inequality holds since
M≥(2+Jη)Cλ0Cη+Lfα202~β0. |
Case 2. Here
II≥−∫0c1t+ξ(t)J(x−y)Cλ0eλ0(c1t+ξ(t))(2+e(λ0+η)y+e−(λ0+η)y)Cηeη(c1t+ξ(t))dy≥−Cλ0CηKJe(λ0+η)(c1t+ξ(t))∫0c1t+ξ(t)e−2λ0(x−y)(3+e−(λ0+η)y)dy≥−Cλ0CηKJeλ0(−x+c1t+ξ(t))−λ0x(3λ0eη(c1t+ξ(t))+1η). |
Moreover, if
LW+(x,t)≥Meλ(c1t+ξ(t))(~β1e−λ1(x+c1t+ξ(t))+~β0eλ0(−x+c1t+ξ(t)))−Lfα0eλ0(−x+c1t+ξ(t))−Cλ0CηKJeλ0(−x+c1t+ξ(t))(3λ0eη(c1t+ξ(t))+1η)≥eλ(c1t+ξ(t))+λ0(−x+c1t+ξ(t))[M~β1e(λ0−λ1)x−(λ0+λ1)(c1t+ξ(t))−3Cλ0CηKJλ0e(η−λ)(c1t+ξ(t))−(Lfα0+Cλ0CηKJη)e−λ(c1t+ξ(t))]. |
Remember that
M~β1−3Cλ0CηKJλ0−(Lfα0+Cλ0CηKJη)≥0. | (7) |
This yields that
If
f1(ϕ1(x+c1t+ξ(t)))+f1(ϕ1(−x+c1t+ξ(t)))−f1(W+)=[f′1(0)−f′1(1)]ϕ1(−x+c1t+ξ(t))+o(ϕ1(−x+c1t+ξ(t))). |
Meanwhile,
LW+(x,t)≥eλ(c1t+ξ(t))+λ0(−x+c1t+ξ(t))(M~β1e(λ0−λ1)x−(λ0+λ1)(c1t+ξ(t))−3Cλ0CηKJλ0e(η−λ)(c1t+ξ(t)))+([f′1(0)−f′1(1)]β0−Cλ0CηKJηe−λ0x)eλ0(−x+c1t+ξ(t)). |
Then
[f′1(0)−f′1(1)]β0−Cλ0CηKJηe−λ0x≥0. |
When
LW+(x,t)≥eλ(c1t+ξ(t))+λ0(−x+c1t+ξ(t))[M~β1e(λ0−λ1)x−(λ0+λ1)(c1t+ξ(t))−3Cλ0CηKJλ0e(η−λ)(c1t+ξ(t))−(Lfα0+Cλ0CηKJη)e−λ(c1t+ξ(t))]≥eλ(c1t+ξ(t))+λ0(−x+c1t+ξ(t))[M~β1e(λ0−λ1)M−2λ0(c1t+ξ(t))−3Cλ0CηKJλ0e(η−λ)(c1t+ξ(t))−(Lfα0+Cλ0CηKJη)e−λ(c1t+ξ(t))]. |
Similar to the case
Proof of Theorem 1.1. Let
{(un)t(x,t)=J∗un(x,t)−un(x,t)+f(x,un),x∈R,t>−n,un(x,−n)=W−(x,−n),x∈R. |
Since
W−(x,t)≤un(x,t)≤W+(x,t)forx∈R,t∈[−n,−T]. |
Moreover, since
{v′(t)=Lη−mv(t),t>−n,v(−n)=Mη, |
where
Proposition 2. Let
|u(x+η,t)−u(x,t)|≤M′η, |
and
|∂u(x+η,t)∂t−∂u(x,t)∂t|≤M"η |
for
The proof is similar to that of Proposition 3.1 in [23] by virtue of the fact
As for the uniqueness, since it is easy to see
In this section, we are going to show the entire solution established in previous section converges to a shift of
ω<min{|f′1(0)|4,|f′2(0)|4,|f′1(1)|4,|f′2(1)|4}, |
there hold
ϕ1(x),ϕ2(x)≤σ2forx≤−Xandϕ1(x),ϕ2(x)≥1−σ2forx≥X, |
besides,
f′1(s),f′2(s)≤−ωfors∈[0,σ]∪[1−σ,1]. |
Theorem 4.1. Let assumptions
u(x,t)−ϕ1(x+c1t)→0ast→−∞uniformly inx∈R. | (8) |
Then
u(x,t)−ϕ2(x+c2t+β)→0ast→+∞uniformly inx∈R |
for some
Lemma 4.2. Suppose assumptions
u(x,t)≥max{ϕ2(x+c2t−β−)−δ−e−ωt,0} |
for
Proof of Lemma 4.2. In fact, it follows from 8 that there exists
|u(x,t−)−ϕ1(x+c1t−)|≤ϵ−2 |
for any
˜u(x,t)=u(x,t+t−)andu_(x,t)=max{ϕ2(ξ−(x,t))−Cve−ωt,0}, |
where
ξ−(x,t)=x+c2(t+t−)−β0+Cve−ωt−Cvfort≥0,x∈R. |
We first show
ϕ1(x+c1t−)≥1−ϵ−2andϕ2(x+c2t−)≥1−ϵ−2forx≥x1. |
As
1−ϵ−2≤u(x,t)≤1and1−ϵ−2≤ϕi≤1,i=1,2 |
for
|ϕi−u(x,t)|≤ϵ−,i=1,2fort≥t−,x≥x1. |
Meanwhile, there exists
0<ϕi(x+cit−)≤ϵ−2,i=1,2forx≤x2. |
Particularly, there holds
|u(x,t−)−ϕ2(x+c2t−)|≤ϵ−forx∈R∖(x1,x2). |
In addition,
ϕ2(x+c2t–β0)≤u(x,t−)forx∈[x1,x2]. |
From all the discussion above we obtain
Lu_(x,t)=u_t(x,t)−(J∗u_(x,t)−u_(x,t))−f(x,u_)≤0fort≥0,x∈R. |
In fact, we only need to consider
Lu_(x,t)=(c2−Cvωe−ωt)ϕ′2(ξ−(x,t))+Cvωe−ωt−∫RJ(x−y)ϕ2(ξ−(y,t))dy+ϕ2(ξ−(x,t))−f(x,u_(x,t))≤−Cvωe−ωtϕ′2(ξ−(x,t))+f2(ϕ2(ξ−(x,t)))−f2(ϕ2(ξ−(x,t))−Cve−ωt)+Cvωe−ωt. |
In the following, we continue the proof in three cases.
Case
ϕ2(ξ−)−Cve−ωt≤ϕ2(ξ−)≤σ. |
Then, by
Lu_(x,t)≤(f′2(0)+2ω)Cve−ωt≤0. |
Case
1−σ≤1−σ2−Cv≤ϕ2(ξ−)−Cve−ωt≤ϕ2(ξ−)<1. |
Then, since
Lu_(x,t)≤(f′2(1)+2ω)Cve−ωt≤0. |
Case
Lu_(x,t)≤−τ0Cvωe−ωt+‖f′2‖∞Cve−ωt+Cvωe−ωt≤(‖f′2‖∞Cv++Cvω−τ0Cvω)e−ωt. |
Choose
Similarly, we can prove the following lemma.
Lemma 4.3. Under the assumptions of Lemma 4.2, there exist
min{ϕ1(x+c1t+β+)+C+e−ωt,1}≥u(x,t) |
for
Now we are in a position to establish an important lemma.
Lemma 4.4. Under the assumptions of Lemma 4.2, there exist
min{ϕ2(x+c2t+β)+Ce−ωt,1}≥u(x,t) |
for
Proof of Lemma 4.4. Define
ξ+(x,t)=x+c2(t+T)+β++V(t),V(t)=KV∫t0v(s)ds, |
and
v(t)=(ϵ1+C3)e−ωt−C3e−λ1c2t |
for
c2λ1=∫RJ(x−y)e−λ1ydy−1+f′2(1). |
Then
v(0)=ϵ1,0<v(t)<σ2fort≥0 |
and
˙v(t)=−ωv(t)−(w−λ1c2)C3eλ1c2tfort≥0. |
Now we intend to show
|f1(s)−f2(s)|≤C0|1−s|fors∈[0,1]. |
Since
ϕ1(x+c1T+β+)<ϵ12andϕ2(x+c2T)<ϵ1forx<x3 |
as well as
ϕ2(x+c2T−β−)≥1−ϵ1andϕ2(x+c2T)≥1−ϵ1 |
for
max{C+,δ−}e−ωT<ϵ12andX−c2T≤−x0. |
Therefore, it is obvious that
u(x,T)≤min{ϕ2(x+c2T)+ϵ1,1}≤¯u(x,0)forx∈R∖(x3,x4). |
Furthermore, since
ϕ2(x+c2T+β+)≥u(x,T)forx∈[x3,x4]. |
This yields that
In the following, we are going to prove
L¯u(x,t)=¯ut(x,t)−(J∗¯u(x,t)−¯u(x,t))−f(x,¯u)≥0fort≥0,x∈R. |
Here we only need to consider
L¯u(x,t)=˙ξ+(x,t)ϕ′2(ξ+(x,t))+˙v(t)−∫RJ(x−y)ϕ2(ξ+(y,t))dy+ϕ2(ξ+(x,t))−f(x,ϕ2(ξ+(x,t))+v(t)) |
=˙V(t)ϕ′2(ξ+(x,t))+˙v(t)+f2(ϕ2(ξ+(x,t)))−f(x,ϕ2(ξ+(x,t))+v(t))≥{˙V(t)ϕ′2(ξ+(x,t))+˙v(t)+f2(ϕ2(ξ+(x,t)))−f2(ϕ2(ξ+(x,t))+v(t))forx<−L,˙V(t)ϕ′2(ξ+(x,t))+˙v(t)+f2(ϕ2(ξ+(x,t)))−f1(ϕ2(ξ+(x,t))+v(t))forx≥−L. |
We continue to prove
Case 1. If
L¯u(x,t)≥˙V(t)ϕ′2(ξ+(x,t))+˙v(t)+f1(ϕ2(ξ+(x,t)))−f1(ϕ2(ξ+(x,t))+v(t))−C0|1−ϕ2(ξ+(x,t))|≥˙V(t)ϕ′2(ξ+(x,t))+˙v(t)−(f′1(1)+ω)˙v(t)−C2C0e−λ1(x+c2(t+T)+β++V(t))≥˙V(t)ϕ′2(ξ+(x,t))+˙v(t)−(f′1(1)+ω)˙v(t)−C2C0e−λ1c2teλ1(L−c2T). |
Particularly, choose
˙v(t)=−ωv(t)+C2C0e−λ1c2teλ1(L−c2T), |
which means
C3=C2C0eλ1(L−c2T)λ1c2−ω. |
As a consequence,
L¯u(x,t)≥−(f′1(1)+2ω)v(t)≥0. |
In addition, if
L¯u(x,t)≥−ωv(t)−(w−λ1c2)C3eλ1c2t−(f′2(1)+ω)v(t)≥−(f′2(1)+2ω)v(t)≥0. |
Case 2. If
Case 3. For
L¯u(x,t)≥˙V(t)ϕ′2(ξ+(x,t))+˙v(t)−‖f′2‖∞v(t)≥˙V(t)τ0˙V(t)−ωv(t)−‖f′2‖∞v(t)≥(KVτ0−ω−‖f′2‖∞)v(t), |
where
KV≥ω+‖f′2‖∞τ0 |
that
Now we are ready to show that the entire solution converges to a shift of
minx∈[−n,n]{u1(x,1)−u2(x,1)}≥ζ(n)∫10[u1(y,0)−u2(y,0)]dyform≥1. |
In fact, it follows from the comparison principle that
˘wt(x,t)≥K˘w(x,t)+J∗˘w(x,t)−˘w(x,t)+f(x,u(x,t))−f(x,v(x,t))≥J∗˘w(x,t)+[K−(1+maxs∈[−1,2]|fs(x,s)|)]˘w(x,t)≥J∗˘w(x,t), |
which implies that
˘w(x,T)≥(TN0)NJ∗˘w(x,0)≥(TN0)Nc(M)∫10˘w(y,0)dy, |
where
c(M)=minx∈[−M−1,M+1]J(x)>0. |
Thus, there exists a positive function
u(x,t)−v(x,t)≥η(|x|,t)∫10[u(y,0)−v(y,0)]dyforx∈R,t>0, |
which implies the assumption (A) holds.
Now, we start to prove the following lemma, which plays an important role in the proof of Theorem 4.1.
Lemma 4.5. Suppose that assumptions
ϕ2(x+c2τ+ξ)−δ≤u(x,t)≤ϕ2(x+c2τ+ξ+h)+δforx∈R, | (9) |
then for every
˜ξ(t)∈[ξ−δγCm,ξ+h+δγCm],˜δ(t)≤e−ω(t−τ−1)[δ+ϵ0min{h,1}],˜h(t)≤τh−γϵ0Cmmin{h,1}+2δγCm |
such that 9 holds with
Proof of Lemma 4.5. In view of Lemmas 4.2 and 4.4, it is easy to see that
ϕ2(x+c2t−β−)−δ−e−ωt≤u(x,t)≤ϕ2(x+c2t+β)+Ce−ωt. |
Furthermore, as in the proofs of Lemmas 4.2 and 4.4,
ϕ2(x+c2t−β0+Cve−ωt−Cv)−δ−e−ωt≤u(x,t) |
and
ϕ2(x+c2t+β++KVω−KVωe−ωt)+Ce−ωt≥u(x,t). |
Denote
κ=max{−β0−γ},h=β++KVω−κ. |
In addition, by the definition of
ϕ2(x+c2t+γe−ωt)−Cme−ωt≤ˇu(x,t)≤ϕ2(x+c2t+h+γe−ωt)+Cme−ωt. |
Let
∫10[ϕ2(y+¯h)−ϕ2(y)]dy≥2ϑ0¯h. |
Therefore, at least one of the following two inequalities is true
(i)∫10[ˇu(y,0)−ϕ2(y)]dy≥ϑ0¯h,(ii)∫10[ϕ2(y+¯h)−ˇu(y,0)]dy≥ϑ0¯h. |
Next, we consider the case (i) since the case (ii) is similar. According to the assumption (A), for
ˇu(x,1)−[ϕ2(x+γe−ω)−Cme−ω]≥ζ∫10[ˇu(y,0)−(ϕ2(y)−Cm)]dy≥ζϑ0¯h. |
Now define
ϵ0=min{σ2,γ2Cm,minx∈[−M−c2−γ,M+c2+γ]ζϑ0γ2Cmϕ′2(x)}. |
Accordingly, there exists
ϕ2(x+γe−ω+2ϵ0¯hγCm)−ϕ2(x+γe−ω)=ϕ′2(x+γe−ω+2ϵ0¯h˜θγCm)2ϵ0¯hγCm≤ζϑ0¯h |
for all
ˇu(x,1)≥ϕ2(x+γe−ω+2ϵ0¯hγCm)−Cme−ω |
for
For
ϕ2(x+γe−ω)≥ϕ2(x+γe−ω+2ϵ0¯hγCm)−ϵ0¯h. |
Then, it follows that
u(x,1)≥ϕ2(x+κ+γe−ω+2ϵ0¯hγCm)−ϵ0¯h−Cme−ωforx∈R. |
Note that
u(x,1+t′)≥ϕ2(x+c2t+κ+γe−ω+2ϵ0¯hγCm+γe−ωt′)−pe−ωt′≥ϕ2(x+c2(t′+1)+κ+ϵ0¯hγCm−δγCm)−(δ+ϵ0¯h)e−ωt′. |
Thus we finish the proof by setting
t=1+t′,˜ξ(t)=γϵ0¯hCm,˜δ=(δ+ϵ0¯h)e−ω(t−1)˜h=[h+δγCme−ωt]−˜ξ(t)=h−ϵ0γCm¯h+δγCm[2−e−ωt]. |
Now we shall prove Theorem 4.1.
Proof of Theorem 4.1. We shall divide the proof in three steps.
Step 1. Following from Lemmas 4.2 and 4.4, there exist
ϕ2(x+c2T∗−M∗)−Cm≤u(x,T∗)≤ϕ2(x+c2T∗+M∗)+Cmforx∈R. | (10) |
Here,
ϵ∗=min{σ2,ϵ04}andk0=ϵ0γCm−2ϵ∗γCm≥γϵ02Cm>0. |
Meanwhile, fix
e−ω(t∗−1)[1+ϵ0ϵ∗]≤1−k∗. |
Then, replace
By 10, applying Lemma 4.5 with
ˆδ(T0+t∗)≤[ϵ∗+ϵ0]e−ωt∗≤ϵ∗andˆh(T0+t∗)≤h0−γCmϵ0+2ϵ∗γCm≤h0−k∗. |
Repeat the same process, it yields that 9 holds for
Step 2. In this step, we use a mathematical induction to show that for every nonnegative integer
τ=Tk:=T1+kt∗,δ=δk:=(1−k∗)kϵ∗,h=hk:=(1−k∗)k. |
It is obvious that the assertion holds for
ˆξ∈[ξl−δlγCm,ξl+δlγCm],ˆδ≤(δl+ϵ0hl)=[1−k∗]lϵ∗(1+ϵ0ϵ∗)e−ω(t∗−1)≤(1−k∗)l+1ϵ∗,ˆh≤hl−hlϵ0γCm+2δlγCm=[1−k∗]l[1−ϵ0γCm+2ϵ∗γCm]=[1−k∗]l+1 |
by the definition of
Step 3. So far, we have known that 9 holds for
Now we define
δ(t)=δk,ξ(t)=ξk−δkγCm,h(t)=hk+2δkγCm |
for
ϕ2(x+c2t+ξ(t))−δ(t)≤u(x,t)≤ϕ2(x+c2t+ξ(t+h(t))+δ(t)fort≥T1,x∈R. |
It follows from the definition of
δ(t)=δk=[1−k∗]kϵ∗≤ϵ∗exp[(t−T1t∗−1)ln(1−k∗)]fort≥T1,h(t)=hk+2δkγCm≤[1+2ϵ∗γCm]exp[(t−T1t∗−1)ln(1−k∗)]fort≥T1. |
Moreover, since for any
ξ(t)∈[ξ(τ)−δ(τ)γCm,ξ(τ)+h(τ)+δ(τ)γCm], |
there holds
|ξ(t)−ξ(τ)|≤h(τ)+2δ(τ)γCm, |
which implies that
|ξ(∞)−ξ(τ)|≤h(τ)+2δ(τ)γCm≤[1+4ϵ∗2δ(τ)γCm]e[(t−T1t∗−1)ln(1−k∗)]fort≥T1. |
Therefore, we have that
|u(x,t)−ϕ2(x+c2t+ξ(∞))|→0ast→+∞. |
Furthermore, the convergence is exponential. Then we complete the proof.
We investigate the Lyapunov stability of the entire solution obtained in Theorem 1.1 in this section. That is, the aim here is to prove Theorem 1.2. The following lemma plays an important role in proving Theorem 1.2.
Lemma 5.1. Let
ut(x,t)≥Kφfor anyt≥Tφandx∈Ωφ(t), |
where
Ωφ(t)={x∈R:φ≤u(x,t)≤1−φ}. |
Proof of Lemma 5.1. It is easy to choose
Ωφ(t)⊂{x∈R:|x+c2t|≤Mφ}⊂{x∈R:x≤−1}. |
Now suppose there exist sequences
ut(tk,xk)→0ask→+∞. |
Here only two cases happen,
For the former case, denote
uk(x,t)=u(x+xk,t+tk). |
By Proposition 2,
uk→u∗ask→+∞ |
for some function
∂u∗(x,t)∂t≡0fort≥0. |
However, this is impossible because by Theorem 4.1
u∗(x,t)=ϕ2(x+c2t+β+a)for somea∈[−Mη,Mη]. |
For the second case,
uk(x,t):=u(x+xk,t). |
Then, each
uk→u∗ask→+∞ |
for some function
u∗(x,t)−ϕ2(x+β+x∗+c2t)→0ast→+∞. |
This ends the proof.
Proof of Theorem 1.2. We first define a pair of sub- and super-solutions as follows.
U±(x,t)=u(x+a,t+t0±˜δϖ(1−e−ωt))±ϖe−ωt, |
where
U−(x,t)≤v(x,t)≤U+(x,t)forx∈R,t≥0. | (11) |
In view of that for all
supx∈R,t∈R|u(x,t)−u(x,t+τ)|≤supx∈R,t∈R|ut(x,t)|ϖ≤ϵ2. | (12) |
It then follows from 11 and 12 that
|v(x,t,v0)−u(x+a,t+t0)|<ϵ |
for
Now we prove the claim. We show
LU+(x,t)=U+t(x,t)−(J∗U+(x,t)−U+(x,t))−f(x,U+)=˜δϖωe−ωt−ϖωe−ωt+f(x,u)−f(x,U+)≥0. |
We go further to show
Case 1. For any
LU+(x,t)≥−min{|fu(x,0)|2,|(fu(x,1)|2}ϖe−ωt−ωϖe−ωt≥ϖe−ωt(2ω−ω)≥0. |
Case 2. For
LU+(x,t)≥˜δϖωKσe−ωt−ϖωe−ωt−‖fu(x,u)‖∞ϖe−ωt≥ϖe−ωt(˜δωKσ−ω−‖fu(x,u))‖∞). |
Let
˜δ≥ωKσω+‖fu(x,u)‖∞. |
Thus, we have
Remark 1. In this paper, we have considered the existence, uniqueness, asymptotic behavior and Lyapunov stability of entire solutions of the nonlocal dispersal equation 1 under the assumption
Remark 2. The method used here can be also applied to consider the bistable lattice differential equations with
˙ui(t)=ui+1(t)+ui−1(t)−2ui(t)+fi(ui(t)),i∈Z,t∈R. | (13) |
The existence, uniqueness, asymptotic behavior and the Lyapunov stability of entire solutions to 13 can be similarly obtained.
While for bistable random diffusion equations with
ut=uxx+f(x,u),x∈R,t∈R, |
the construction of the entire solution behaving as the traveling wave pertaining to
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1. | Xing-Xing Yang, Guo-Bao Zhang, Entire Solutions for an Inhomogeneous Bistable Discrete Diffusive Equation, 2023, 46, 0126-6705, 10.1007/s40840-022-01451-5 |