Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay

This paper is concerned with a class of advection hyperbolic-parabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The influence of advection on the propagation speed is also considered.

They studied the globally exponential stability of traveling wave solutions of (1), see also Volpert et. al. [42]. Schaaf [36] considered the following delayed system ∂u(x, t) ∂t = d ∂ 2 u(x, t) ∂x 2 + g(u(x, t), u(x, t − τ )), x ∈ R, t > 0, τ > 0 (2) for the so-called Huxley nonlinearity as well as Fisher nonlinearity. He proved the existence of traveling wave solutions of (2) by using the phase-plane technique, the maximum principle for parabolic functional differential equations and the general theory for ordinary functional differential equations. By first establishing the existence and comparison theorem of solutions for (2), where they applied the theory of abstract functional differential equations developed by Martin and Smith [32], and then using the elementary sub-and supersolutions comparison and the squeezing technique developed by Chen [13], Smith and Zhao [39] studied the global asymptotic stability, Lyapunov stability and uniqueness of traveling wave solutions of (2) under the bistable assumption.
The nonlocal system was also studied by many researchers. When τ = 0, Chen [13] considered the existence, uniqueness and global asymptotic stability of traveling wave solutions of (3) by the squeezing technique. For more similar results related to this technique, one can refer to Alikakos et al. [1], Berestycki and Nirenberg [6], Chen [12], Chen and Guo [14,15], Ermentrout and McLeod [17], Evans et. al. [18], Fife and McLeod [19], Ma and Zou [27,28] and Shen [37,38]. When τ > 0, g(u, v) = −αu + v, by using similar method in Smith and Zhao [39] to establish the existence and comparison theorem of solutions for (3), Ma and Wu [26] considered the uniqueness and global asymptotic stability of traveling wave solutions of (3) with under bistable assumption by means of the moving plane technique and the squeezing technique. In fact, Ma and Wu [26] studied the existence of traveling wave solutions with the help of a nonlocal system without time delay, see also Chen [13] for a similar technique, and then passing to (3). For more similar results related to this technique, one can refer to Chen [11] for a neural network model and Ou and Wu [33] for a delayed hyperbolic-parabolic model. For some special cases of the following system ∂u(x, t) ∂t = D∆u(x, t) − d(u(x, t)) f (s)e −γs b(u(y, t − s))dyds, the existence of traveling wave solutions has been extensively studied by many researchers, one can refer to Gourley and Kuang [23] and Al-Omari and Gourley [3] for d(u(x, t)) = αu 2 (x, t) and b(u(x, t)) = βu(x, t), So et. al. [41] for d(u(x, t)) = αu(x, t), b(u(x, t)) = βu(x, t)e −au(x,t) . It is easy to see that (4) with discrete delay is more general equation than (3). In 2005, Ou and Wu [33] considered a delayed hyperbolic-parabolic model

TRAVELING WAVE SOLUTIONS IN ADVECTION HYPERBOLIC-PARABOLIC SYST. 2093
where f (x) = 1 √ 4πα e −x 2 /4α , r > 0 was a time lag that the density of adult population u(x, t) moves randomly in space. For more about the usual structured population model including (5), one can refer to [8,9,34,40,45,46]. When r = 0, So et al. [41] considered the existence of traveling wave front of (5) with monostable case by using the standard techniques involving sub-and supersolutions. The existence, uniqueness and asymptotic stability of a traveling wavefront to (5) were recently studied in [26] by using the comparison and squeezing techniques under bistable case. When r > 0, under bistable case, Ou and Wu [33] considered the existence and uniqueness of traveling wave fronts by considering the same problem for a properly parameterized parabolic system, and then by considering the continuous dependence of the wave speed on the parameter involved.
In [25], Liang and Wu derived a reaction advection diffusion equation with nonlocal delay where J α (x) = 1 √ 4πα e −x 2 /4α , τ > 0 is the time delay, ε reflects the impact of the death rate of the immature, α represents the effect of the dispersal rate of the immature on the growth rate of the matured population, and B is the velocity of the spatial transport field. By choosing three different birth functions b(u), they established the existence of traveling wave fronts of (6). We note that they only considered (6) with monostable nonlinearity.
In fact, some reaction-diffusion processes taking place in moving media such as fluids can be described by reaction advection diffusion equations, for example, combustion, atmospheric chemistry, and plankton distributions in the sea, see [5,10,20] and the references therein. We note that the advection terms on the propagation of traveling wave solutions play an important role, see [5,20,29,30,31]. However, the above results can not applied to the equations with time delay and nonlocal effect.
In [44], Wang, Li and Ruan considered the reaction advection diffusion equation with nonlocal delay where and the convolution is defined by Under bistable assumption, they proved existence, uniqueness and globally asymptotic stability of traveling wave fronts. Their method is similar to Chen [13] and Smith and Zhao [39]. As applications, they also studied (4) and (6) under bistable case.
If we take into account the velocity of the spatial transport field in (5) or a time lag that the density of adult population u(x, t) moves randomly in space in (6), then the model (5) or (6) becomes the following form where J α (x) = 1 √ 4πα e −x 2 /4α , τ > 0, α is the probability that a new born at time t − τ and location 0 moves to the location x after maturation time τ , ε ∈ (0, 1] is the survival rate during the maturation period. r > 0 is a time lag that the density of adult population u(x, t) at time t and spatial location x ∈ R of a given single species population with two age classes (the immature and mature with maturation time τ > 0 being a constant) moves randomly in space, D m and d m > 0 are constant diffusion and death rates of the adult at time t and location x, B is the velocity of the spatial transport field, b(u(x, t)) is the birth function.
The rest of this paper is organized as follows. In Section 2, we introduce associated parameterized advection parabolic system. In Section 3, by constructing a pair of sup-and subsolutions and comparison principle, which are similar to [13,39,44], we consider the uniqueness of traveling wave solutions of associated system. In Section 4, by using the method in [13,33,44], the existence of traveling wave solutions for a class of reaction-diffusion equation without delay is established. Our key point is to investigate the continuity of wave speed C(c) of associated system and C(c) = c has a solution, then the existence of traveling wave solutions of (9) is obtained. In Section 5, we apply our results to (8). 2. Preliminaries. A traveling wave solution of (9) is a translation invariant solution of the special form u(x, t) = U (x+ct), where c is a given positive constant, U ∈ C 2 (R, R) is the profiles of the wave that propagates through the one-dimensional spatial domain at a constant velocity c > 0. Substituting u(x, t) = U (x + ct) into (9), denoting x + ct by t, and denoting x + Bτ − y by z, then the corresponding wave equation is Let Therefore, we obtain where D(c) = D − rc 2 . We will consider the existence of traveling wave solutions for (2.2) and (2.3). It is equivalent to consider the following associated parabolic system and where c ∈ R satisfies D(c) = D − rc 2 > 0. (15) We shall find the existence of traveling wave solution w( (13) and (14) satisfying where K > 0 is the maximal positive solution of equation g(u, b(u)) = 0, see Lemma 3.1 and Remark 1 for V max . For any c, r and D satisfying (15) and another technical conditions (24) and (33) given in the next section, there exists a wave speed C(c), denoting x + C(c)t by t, such that U (t) and V (t) satisfying and as well as the asymptotic boundary condition (16). If there exists c such that c = C(c), then for such c > 0, we find a solution for (11) and (12).
3. Uniqueness of traveling wave solutions for associated system. For x ∈ R, we begin with the following more general system with the following initial data In this section, we shall prove that (19) has at most one traveling wave solution (up to translation) w(x, t) = V (x + C(c)t), ϕ(x, t) = U (x + C(c)t) with the wave speed C(c) depending on c. In particular, when τ 1 = 0 system (19) reduces to (13) and (14).
Let X = BU C(R, R) be the Banach space of bounded and uniformly continuous functions from R into R with the usual supremum norm. Let It is easy to see that X + is a closed cone of X and X is a Banach lattice under the partial ordering induced by X + . By Theorem 1.5 in [16], it then follows that the X-realization D∆X of D∆ generates a strongly continuous analytic semigroup T (t) on X and T (t)X + ⊂ X + , t ≥ 0. Moreover, by the explicit expression of solutions of the heat equation we have Consider the following equation The relation between (21) and (22) is as follows.
is a solution of (21). So the existence and uniqueness of solutions of (22) follows from the existence and uniqueness of solutions of (21). In particular, Define bounded linear operator S(t) : X → X, t ≥ 0, by It is easy to show that S(t) is a strongly continuous semigroup on X. Obviously, By the continuity of f 0 and (H2), it easily follows that there exist To show the existence and positiveness of solutions to (19) and (20) when φ ∈ C + , we need some order preserving property for the second equation in (19). For any v ∈ X, we consider an operator : X → X defined by We also assume that

TRAVELING WAVE SOLUTIONS IN ADVECTION HYPERBOLIC-PARABOLIC SYST. 2099
The following lemma is similar to Lemma 3.1 in [33] and we omit the proof here.
Lemma 3.1. Assume that r is sufficiently small so that Then (i) for every v ∈ X, the equation has one and only one solution Remark 1. We can see from Lemma 3.
Remark 2. For any fixed t in system (19), we can solve the second equation by Lemma 3.1 to obtain ϕ(x, t) = F (w)(x, t). Thus system (19) with initial data (20) can be transformed into Now we return to system (19).
in what follows, all the similar definitions will not be given. By the global Lipschitz continuity of for all 0 ≤ t 0 < t < a. We call (w, F (w)) as a mild solution of (3.1) if it is both a supersolution and a subsolution on [0, a).

Remark 3. Assume that there is a bounded and continuous function pair
Now we give the following existence of solution and comparison theorem.
Proof. We know that ϕ(x, t) = F (w)(x, t). It follows from [32] that a mild solution (w, F (w)) of (19) and (20) is a solution of the following integral equation ) are supersolution and subsolution of (19) and (20), respectively. Notice that F 1 : [−δ 0 , K + δ 0 ] C is globally Lipschitz continuous. It also satisfies the quasi-monotone condition in the sense that , Therefore, by (H1) and (H3) and the above results, we have Now we choose r sufficiently small such that for all ξ ∈ R. Therefore and hence, for any h > 0 with L 1 h < 1, which implies that (32) holds. Hence, it follows that the existence and uniqueness of w(x, t, φ) by Corollary 5 in [32] with S(t, s) = T (t, s) = T (t − s) for t ≥ s ≥ 0 and B(t, φ) = F 1 (φ). Moreover, by using a similar argument to Theorem 1 in [32], it follows that (w, F (w)) is a classical solution for t ≥ τ 1 .
Now we estimate the derivative for the traveling wave solutions.

Lemma 3.4. Let (V (x+C(c)t), F (V )(x+C(c)t)) be a nondecreasing traveling wave solution of (19). Then
Proof. By Lemma 3.3, We have that for ξ = x + C(c)t and every h > 0, which implies that Then It follows from the dominant convergence theorem that lim |ξ|→∞ V (ξ) = 0. The proof is completed.
By the definition ofδ, we have by the choice of c 1 , we have Therefore, by the choice of m 0 and σ 0 , we have Therefore, by (40) and (44), it follows that for sufficiently small r > 0 satisfying (15), (24) and (33) ∂w Case (iii) ξ(x, t) ≤ −M 0 . The proof is similar to that in case (ii) and hence is omitted. Now we prove that (30) holds for w + . Letw + (x, t) = w + (x − Bt, t). We only need to Prove thatw + (x, t) is a supersolution of (3.20), namely,

Furthermore, integration by parts gives
Hence, it follows that it follows that for 0 ≤ t 0 < t, Then (45) holds since H(w + )(y, s) ≤ 0, which implies that w + (x, t) is a supersolution of (19). This completes the proof.
Theorem 3.6. Assume that (19) has a nondecreasing traveling wave solution (V (x+ C(c)t), F (V )(x+C(c)t)). Then for any traveling wave solution Proof. We extend the standard proof in [13,26,39] to our case. Since V andV have the same limit as ξ → ±∞, there existξ ∈ R and large enough h > 0 such that for every s ∈ [−τ 1 , 0] and x ∈ R, <V (x +C(c)s) where β 0 , σ 0 andδ are constants given in Lemma 3.5. Noting that the operator F (v)(·) defined in Lemma 3.1 is non-decreasing if v is nondecreasing, we can still use the comparison result to obtain that for all t ≥ 0 and x ∈ R, Keeping ξ = x +C(c)t fixed and letting t → ∞, we have from the first inequality that C(c) ≤C(c) and from the second inequality that C(c) ≥C(c). This yields C(c) =C(c). Moreover, we get Define ξ * = inf{ξ :V (·) ≤ V (· + ξ)}, ξ * = sup{ξ :V (·) ≥ V (· + ξ)}.

4.
Existence of traveling wave solutions. As discussed in in Section 2, in order to the existence of traveling wave solution for (9), we consider system (19) with τ 1 = 0, that is, where ξ = x + Bτ − cτ − y and c ∈ R is a parameter.
In this section, we consider the case u + = u − , namely, f 0 (u) = g(u, b(u)) has only three zeros. If f 0 (u) has more than three zeros, one can refer to [19,42]. Denotē u = u + = u − . In addition, we assume the following condition hold.
(H4) f 0 (u) < 0 for u ∈ (0,ū), f 0 (u) > 0 for u ∈ (ū, K) and f 0 (ū) > 0. By in Lemma 3.3, we have the following fact. If V (x + ct) is a traveling wave solution of (19) with wave speed c, then V (x+(−B +c)t) is a traveling wave solution of (37) with wave speed −B + c. Inversely, if V (x + ct) is a traveling wave solution of (37), then V (x + (B + c)t) is a traveling wave solution of (19) with wave speed B + c. Hence we only need to consider the existence of traveling wave solutions of (37). We first prove the existence of traveling wave solutions of the following equation wherē   (15), (24) and (33). Then there exist two small constants δ > 0, ε 0 > 0 and a large constant C 0 > 0, which are independent of c and τ such that ) are a supersolution of (49) for c ≥ 0 and a subsolution of (49) for c ≤ 0, respectively; ) are a supersolution and a subsolution of (49) for all c satisfying (15), respectively, where ε c = ε 0 /(1 + |c|τ ) and C c = (1 + |c|τ )C 0 .

KUN LI, JIANHUA HUANG AND XIONG LI
By the choice of 0 and , we have . Then for c ≥ 0, For all t ≥ 0, we havē Noting that c ≤ 0 , c C c = 0 C 0 , we can prove that (50) holds for v + c (x, t) in a similar way. The proof is completed.  (15), (24) and (33), there exists a unique strictly monotonic traveling wave solution (V (x+C(c)t), F (V )(x+C(c)t)) for system (49) with speed C(c) being a continuous function of c.
Proof. The existence result is very similar to [13,26,44]. From Lemma 3.3 (or Remark 4), we can get the monotonicity result be obtained. And the uniqueness result can be obtained from Theorem 3.6. Now we prove that C(c) is continuous in c. Assume that (V c (x+C(c)t), F (V c )(x+ C(c)t)) is a traveling wave solution with wave speed C(c). Without loss of generality, we assume that 0 where By a similar argument as that in Lemma 3.4, we have |V c (ξ)| ≤ G 2 D(c)L 1 .
Proof. By (53), we can see that (15) holds for any c satisfying |c| ≤ C 0 . Therefore, by Lemma 50, (49) has a strictly monotonic traveling wave solution (V (x + C(c)t), F (V )(x + C(c)t)). Motivated by the methods in [33,44], now we prove that there exists at least one c * such that C(c * ) = c * and |c * | ≤ C 0 . It only need to show that that the curves y = c and y = C(c) have at least one common point in region |y| ≤ C 0 . For c ≤ 0, let v − 0 (x, t) be the subsolution of (49) described in Lemma 4.1. Then there exists a large constant X > 0 such that V (·) ≥ v − 0 (· − X, 0). So we have by the comparison that V (x + C(c)t) ≥ v − 0 (x − X, t) for all t ≥ 0 and x ∈ R. Hence by the choice of δ (letting δ → 0), it follows that C(c) ≥ −C 0 for c ≤ 0. Similarly, C(c) ≤ C 0 for c ≥ 0. By Lemma 4.2, we can see that C(c) is continuous for any |c| < C 0 < D/r. Therefore, it follows from (53) that there is at least one common point c * such that C(c * ) = c * in the region |c| < C 0 < D/r and |y| ≤ C 0 . Hence, when τ 1 = 0, (V (x + c * t), F (V )(x + c * t)) is also a strictly monotonic traveling wave solution of (37) and (V (x + (B + c * )t), F (V )(x + (B + c * )t)) is also a strictly monotonic traveling wave solution of (19) with wave speed B + c * , furthermore, U (x + c * t) = F (V )(x + c * t) is a strictly monotonic traveling wave solution of (9) with wave speed c * . This completes the proof.

5.
Applications. As mentioned in the introduction, in this section, we apply our results in section 3 and 4 to establish the existence and uniqueness of traveling wave solutions for (8).