Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity

In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space u_t - J\ast u +u+d(u(t,x))= \int_{\mathbb{R}^n} f_\beta (y) b(u(t-\tau,x-y)) dy, u(s,x)=u_0(s,x), s\in[-\tau,0], \ x\in \mathbb{R}^n} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function. These rates are optimal and the stability results significantly develop the existing studies for nonlocal dispersion equations.


Introduction
For the gradient flow to an order parameter describing the state of a solid material, for example, a perfect crystal with two different orientations, it is usually described by a convolution model of phase transition in the form [2,4,8,23,24] where x = (x 1 , x 2 , · · · , x n ) ∈ R n , J(x) is a non-negative and radial kernel with unit integral, and (J * u)(t, x) = R n J(x − y)u(t, y)dy. (1.2) One example is the Cauchy law by taking J(x) = 1 1+|x| 2 which implies its Fourier transform mentioned above with α = 1. In this case, the behavior of the solutions to the nonlocal dispersion equation is almost identical to the fractional diffusion equation [4,8,23,24] u t = J * u − u ⇔ u t = K∆ α/2 u.
Equation (1.1) represents also the dynamical population model of single species in ecology [13], where u(t, x) is the density of population at location x and time t, and J(x − y) is thought of as the probability distribution of jumping from location y to location x, and J * u = R n J(x − y)u(t, y)dy is the rate at which individuals are arriving to position x from all other places, while −u(x, t) = − R n J(x − y)u(t, x)dy stands the rate at which they are leaving the location x to travel to all other places. In this case, under the consideration of the effects from birth rate and death rate, the equation (1.1) is usually written as follows where b(u(t − τ, x) is the birth rate function, d(u(t, x)) is the death rate function, and τ > 0 is the mature age of the single species, which is usually called the time-delay. Furthermore, if we consider the distribution of all matured population, the effect of birth rate is then involved in whole space R n [20,39,47], and the equation is expressed as where f β (y), with β > 0, is the heat kernel in the form of (1.7) When τ = 0 (no time-delay), then the above equation is reduced to x − y))dy, u(0, x) = u 0 (x), x ∈ R n . (1.8) We will also discuss how the time-delay τ effects the property of the solutions. For the equation (1.1) in 1D case, when F (u) is bistable, namely, two constant equilibria u − and u + both are the stable nodes (the typical example is the Huxley equation with F (u) = u(u − a)(1 − u) for 0 < a < 1), Bates et al [2] and Chen [6] proved that the traveling waves are globally stable as t → +∞. In this paper, we consider another important type of equations with monostable nonlinearity. The typical example in this case is Fisher-KPP equation with F (u) = u(1 − u). Hence, throughout this paper, we assume that the death rate d(u) and birth rate b(u) capture the following characters of monostable nonlinearity: (H 1 ) There exist u − = 0 and u + > 0 such that d(0) = b(0) = 0, d(u + ) = b(u + ), and d(u), b(u) ∈ C 2 [0, u + ]; ( These characters are summarized from the classical Fisher-KPP equation, see also the monostable reaction-diffusion equations in ecology, for example, the Nicholson's blowflies equation [37,38,39,47] with d(u) = δu and b(u) = pue −au , p > 0, δ > 0, a > 0 and u − = 0 and u + = 1 a ln p δ > 0 under the consideration of 1 < p δ ≤ e; and the age-structured population model [19,20,39,42,44] with d(u) = δu 2 and b(u) = pe −γτ u, δ > 0, p > 0, γ > 0, and u − = 0 and u + = p δ e −γτ . Clearly, under the hypothesis (H 1 )-(H 3 ), both u − = 0 and u + > 0 are constant equilibria of the equation (1.7), and u − = 0 is unstable and u + is stable for the spatially homogeneous equation associated with (1.7), this is why we call the equation (1.7), including (1.1) and (1.3) and (1.8), as monostable.
Without loss of generality, we can always assume e = e 1 = (1, 0, · · · , 0) by rotating the coordinates. Thus, the planar traveling wavefront φ(x · e 1 + ct) = φ(x 1 + ct) satisfies, for τ ≥ 0, and (1.9) is reduced to, for τ ≥ 0, (1.12) The main purpose of this paper is to study the global asymptotic stability of planar traveling wavefronts of the equations (1.7) and (1.8) with or without time-delay, respectively, in particular, in the case of the critical wave φ(x 1 + c * t). Here the number c * is called the critical speed (or the minimum speed) in the sense that a traveling wave φ( The nonlocal dispersion equation (1.1) has been extensively studied recently. Chasseigne et al [4] and Cortazar et al [8] showed that the linear nonlocal dispersion equation (1.1) (with F = 0) is almost equivalent to the linear diffusion equation, and the asymptotic behavior of the solutions to the linear equation of nonlocal dispersion is exactly the same to the corresponding linear diffusion equation. Ignat and Rossi [23,24] further obtained the asymptotic behavior of the solutions to the nonlinear equation (1.1). García-Melián and Quirós [17] investigated the blow up phenomenon of the solution to the equation (1.1) with F (u) = u p , and gave the Fujita critical exponent. Regarding the structure of special solutions to (1.1) like traveling wave solutions, early in 1997 Bates et al [2] and Chen [6] established the existence of the traveling waves for (1.1) with bistable nonlinearity, and proved their global stability. For (1.1) with monostable nonlinearity, recently Coville and his collaborators [9,10,11,12] studied the existence and uniqueness (up to a shift) of traveling waves. See also the existence/nonexistence of traveling waves by Yagisita [53] and the existence of almost periodic traveling waves by Chen [5]. However, the stability of traveling waves for the nonlocal equation (1.7) (including (1.1) and (1.3)) with monostable nonlinearity is almost not related, except a special case for the fast waves with large wave speed to the 1D age-structured population model by Pan et al [44]. As we know, such a problem is also very significant but challenging, because the equations of Fisher-KPP type possess an unstable node, different from the bistable case, this unstable node usually causes a serious difficulty in the stability proof, particularly, for the critical traveling waves. The main interest in this paper is to investigate the stability of traveling waves to (1.7) with τ > 0 and (1.8) with τ = 0. An easy to follow method will be introduced for the stability proof to the nonlocal dispersion equations.
In this paper, we will first investigate the linearized equation of (1.7), and derive the optimal decay rates of the solution to the linearized equation by means of Fourier transform. This is a crucial step for get the optimal convergence for the nonlocal stability of traveling waves. Then, we will technically establish the global existence and comparison principles of the solution to the n-D nonlinear equation with nonlocal dispersion (1.7). Inspired by [43] for the classical Fisher-KPP equations and the further developments by [39,40], by ingeniously selecting a weight function which is dependent on the critical wave speed c * , and using the weighted energy method and the Green function method with the comparison principles together, we will further prove that, all noncritical planar traveling waves φ(x · e + ct) are exponentially stable in the form of t − n α e −µτ for some constant µ τ = µ(τ ) such that 0 < µ τ ≤ µ 0 for τ ≥ 0; and all critical planar traveling waves φ(x · e + c * t) are algebraically stable in the form of t − n α . These convergence rates are optimal and the stability results significantly develop the existing studies on the nonlocal dispersion equations. We will also show that the time-delay τ will slow down the convergence of the the solution u(t, x) to the noncritical planar traveling waves φ(x · e + ct) with c > c * , and cause the higher requirement for the initial perturbation around the wavefronts.
The paper is organized as follows. In section 2, we will state the existence of the traveling waves, and their stability. In section 3, we will give the solution formulas to the linearized dispersion equations of (1.7) and (1.8), and derive the optimal decay rates by Fourier transform with energy method together. In section 4, we will prove the global existence of the solution to (1.7) and establish the comparison principle. In section 5, based on the results obtained in sections 3 and 4, by using the weighted energy method, we will further prove the stability of planar traveling waves including the critical and noncritical waves. Finally, in section 6, we will give some particular applications of our stability theory to the classical Fisher-KPP equation with nonlocal dispersion and the Nicholson's blowflies model, and make a concluding remark to a more general case.
Before ending this section, we make some notations. Throughout this paper, C > 0 denotes a generic constant, while C i > 0 and c i > 0 (i = 0, 1, 2, · · · ) represent specific constants. j = (j 1 , j 2 , · · · , j n ) denotes a multi-index with non-negative integers j i ≥ 0 (i = 1, · · · , n), and |j| = j 1 + j 2 + · · · + j n . The derivatives for multi-dimensional function are denoted as For a n-D function f (x), its Fourier transform is defined as and the inverse Fourier transform is given by Let I be an interval, typically I = R n . L p (I) (p ≥ 1) is the Lebesque space of the integrable functions defined on I, W k,p (I) (k ≥ 0, p ≥ 1) is the Sobolev space of the L p -functions f (x) defined on the interval I whose derivatives ∂ j x f with |j| = k also belong to L p (I), and in particular, we denote W k,2 (I) as H k (I). Further, L p w (I) denotes the weighted L p -space for a weight function w(x) > 0 with the norm defined as is the weighted Sobolev space with the norm given by

Traveling Waves and Their Stabilities
As we mentioned before, the existence and uniqueness (up to a shift) of traveling waves for the equation (1.1) were proved in [9,10,11,12], particular, in a recent work by Yagisita [53] for the existence and nonexistence of traveling waves, when the nonlinearity F (u) is monostable. Without any difficulty, these results can be extended to the nonlocal equation (1.7) with time-delay with the help of comparison principle established in Section 4, when d(u) and b(u) satisfy the monostable features (H 1 )-(H 3 ). We state these results as follows without detailed proof. • when c ≥ c * , there exits a monotone traveling wavefront φ(x 1 + ct) of (1.9) connecting u ± exists; • when c < c * , no traveling wave φ(x 1 + ct) exists.
Here (c * , λ * ) with c * > 0 and λ * > 0 is given by Furthermore, it can be verified: • In the case of c > c * , there exist two numbers depending on c: and particularly, • In the case of c = c * , it holds (2.8) • When ξ 1 = x 1 + ct → ±∞, for all c ≥ c * , the traveling wavefronts φ(x 1 + ct) converge to u ± exponentially as follows Here λ − = λ 1 (c) > 0 is given in (2.5), and λ + = λ + (c) > 0 is the unique root determined by the following equation For easily understanding all cases mentioned in the above, we show them in Figure 2.1. Before stating our main stability theorems, let us technically choose a weight function: where λ * = λ * (c * ) > 0 is given in (2.3) and (2.4), and x * > 0 is a sufficiently large number such that, The selection of ). In fact, we have

Theorem 2.2 (Stability of planar traveling waves with time-delay) Under assumptions
, then the solution of (1.7) uniquely exists and satisfies: • When c > c * , the solution u(t, x) converges to the noncritical planar traveling wave φ(

15)
and ε 1 = ε 1 (τ ) such that 0 < ε 1 < 1 for τ > 0, and ε 1 = ε 1 (τ ) → 0 + as τ → +∞; • When c = c * , the solution u(t, x) converges to the critical planar traveling wave φ( However, when the time-delay τ = 0, then we have the following stronger stability for the traveling waves but with a weaker condition on initial perturbation. , then the solution of (1.8) uniquely exists and satisfies: • When c > c * , the solution u(t, x) converges to the noncritical planar traveling wave φ( • When c = c * , the solution u(t, x) converges to the critical planar traveling wave φ( 1. Comparing Theorem 2.2 with time-delay and Theorem 2.3 without time-delay, we realize that, the sufficient condition on the initial perturbation around the wave in the case with time-delay is stronger than the case without time-delay, but the convergence rate to the noncritical waves φ(x 1 + ct) for c > c * in the case with time-delay is weaker than the case without time-delay, see This means, the time-delay τ > 0 effects the stability of traveling waves a lot, not only the higher requirement for the initial perturbation, but also the slower convergence rate for the solution to the noncritical traveling waves.
2. The convergence rates showed both in Theorem 2.2 and Theorem 2.3 are explicit and optimal, particularly, the algebraic decay rates for the solution converging to the critical waves. Actually, all of them are derived from the linearized equations.

Linearized Nonlocal dispersion Equations
In this section, we will derive the solution formulas for the linearized nonlocal dispersion equations with or without time-delay, as well as their optimal decay rates, which will play a key role in the stability proof in section 5. Now let us introduce the solution formula for linear delayed ODEs [28] and the asymptotic behaviors of the solutions [41]. Then
We are going to derive its solution formula as well as the asymptotic behavior of the solution. By taking Fourier transform to (3.11), and noting that, and we have dv dt   (3.19) and its derivatives for k = 0, 1, · · · and j = 1, · · · , n. Now we are going to derive the asymptotic behavior of v(t, x).
For τ = 0, the equation (3.11) is reduced to Taking the inverse Fourier transform, we get the solution formula Then, a similar analysis as showed before can derive the optimal decay of the solution in the case without time-delay as follows. The detail of proof is omitted.

Global Existence and Comparison principle
In this section, we prove the global existence and uniqueness of the solution for the Cauchy problem to the nonlinear equation with nonlocal dispersion (1.7), and then establish the comparison principle in n-D case by a different proof approach to the previous work [5,12].
Proof. Multiplying (1.7) by e η 0 t and integrating it over [0, t] with respect to t, where η 0 > 0 will be technically selected in (4.4) below, we then express (1.7) in the integral form Let us define the solution space as, for any T ∈ [0, ∞], with the norm where Clearly, B is a Banach space. Define an operator P on B by Now we are going to prove that P is a contracting operator from B to B. Firstly, we prove that P : , and d(u + ) = b(u + ), and using the facts R n J(x − y)dy = 1, This plus the continuity of P(u) based on the continuity of u proves P(u) ∈ B, namely, P maps from B to B. Secondly, we prove that P is contracting. In fact, let u 1 , u 2 ∈ B, and v = u 1 − u 2 , then we have So, we have for 0 < ρ := 2η 0 −1 2η 0 < 1, namely, we prove that the mapping P is contracting: Hence, by the Banach fixed-point theorem, P has a unique fixed point u in B, i.e, the integral equation (4.1) has a unique classical solution on [0, T ] for any given T > 0. Differentiating (4.1) with respect to t, we get back to the original equation (1.7), i.e., then we can easily confirm from the right-hand-side of (4.13) that u t ∈ C([0, T ] × R n ). This completes our proof.  7). Although the comparison principle in 1D case were proved in [5,12]. Here we give a comparison principle in n-D case with much weaker restriction on the initial data. The proof is also new and easy to follow. Different from the previous works [5,12], instead of the differential equation (1.7), we will work on the integral equation (4.1), and sufficiently use the property of contracting operator P.
Letū(t, x) be an upper solution to (1.7), namely where its integral form can be written as x − y))dy ds, for t > 0 (4.15) and let u(t, x) be an lower solution to (1.7) satisfying (4.14) or (4.15) conversely. Then we have the following comparison result.

Global Stability of Planar Traveling Waves
The main purpose in this section is to prove Theorems 2.2 for all traveling waves including the critical traveling waves.
For τ = 0, it is easy to prove the corresponding results as follows.
Combing Lemma 5.1-Lemma 5.3, we obtain the decay rates for V (t, x) in L ∞ (R n ).
Lemma 5.4 It holds that: 1. when c > c * , then Since V (t, x) = U + (t, x) − φ(x 1 + ct), Lemma 5.4 give directly the following convergence for the solution in the cases with time-delay. Lemma 5.5 It holds that: 1. when c > c * , then 2. when c = c * , then Step 2. The convergence of U − (t, x) to φ(x 1 + ct) For the traveling wave φ(x 1 + ct) with c ≥ c * , let As in Step 1, we can similarly prove that U − (t, x) converges to φ(x 1 + ct) as follows.
Thanks to Lemmas 5.5 and 5.6, by the squeeze argument, we have the following convergence results.
Then, by a similar calculation, we can prove the existence of the traveling waves φ(x 1 + ct) for c ≥ c * , where c * > 0 is a specified minimal wave speed, and that the noncritical traveling waves with c > c * are exponentially stable and the critical waves with c = c * are algebraically stable.