Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N

In this paper, we study the spreading speeds and traveling wave solutions of the PDE $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(1-u),\ \ x\in\mathbb{R}^N 0=\Delta v-v+u, \ \ x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ and $v(x,t)$ represent the population and the chemoattractant densities, respectively, and $\chi$ is the chemotaxis sensitivity. It has been shown in an earlier work by the authors of the current paper that, when $0<\chi<1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x,t;u_0),v(x,t;u_0))$ with initial condition $u(x,0;u_0)=u_0(x)$. Furthermore, if $0<\chi<\frac{1}{2}$, then the constant steady-state solution $(1,1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0<\chi<1$, then there are nonnegative constants $c_{-}^*(\chi)\leq c_+^*(\chi)$ such that for every nonnegative initial function $u_0(\cdot)$ with nonempty compact support $$ \lim_{t\to\infty} \sup_{|x|\le ct} \big[|u(x,t;u_0)-1|+|v(x,t;u_0)-1|\big]=0\ \ \forall\ 0c_{+}^*(\chi).$$ We also show that if $0<\chi<\frac{1}{2}$, there is a positive constant $c^*(\chi)$ such that for every $c\ge c^*(\chi)$, the system has a traveling wave solution $(u(x,t),v(x,t))$ with speed $c$ and connecting $(1,1)$ and $(0,0)$, that is, $(u(x,t),v(x,t))=(U(x-ct),V(x-ct))$ for some functions $U(\cdot)$ and $V(\cdot)$ satisfying $(U(-\infty),V(-\infty))=(1,1)$ and $(U(\infty),V(\infty))=(0,0)$. Moreover, $$ \lim_{\chi\to 0}c^*(\chi)=\lim_{\chi\to 0}c_+^*(\chi)=\lim_{\chi\to 0}c_-^*(\chi)=2. $$ We first give a detailed study in the case $N=1$ and next we extend these results to the case $N\ge 2$.


Introduction
The origin of chemotaxis models was introduced by Keller and Segel (see [20], [21]). The following is a general Keller-Segel model for the time evolution of both the density u(x, t) of a mobile species and the density v(x, t) of a chemoattractant, complemented with certain boundary condition on ∂Ω if Ω is bounded, where Ω ⊂ R N is an open domain, τ ≥ 0 is a non-negative constant linked to the speed of diffusion of the chemical, the function χ(u, v) represents the sensitivity with respect to chemotaxis, and the functions f and g model the growth of the mobile species and the chemoattractant, respectively.
Among the central problems about (1.1) are global existence of classical/weak solutions with given initial functions; finite-time blow-up; pattern formation; existence, uniqueness, and stability of certain special solutions; spatial spreading and front propagation dynamics when the domain is a whole space; etc.
In the present paper, we restrict ourselves to the case that τ = 0, which is supposed to model the situation when the chemoattractant diffuses very quickly. System (1.1) with τ = 0 reads as complemented with certain boundary condition on ∂Ω if Ω is bounded. Global existence and asymptotic behavior of solutions of (1.2) on bounded domain Ω have been extensively studied by many authors. The reader is referred to [3], [9], [16], [39], [41], [45], [46], [47], [48], [49], [50], [51], and references therein for the studies of (1.2) on bounded domain with Neumann or Dirichlet boundary conditions and with f (u, v) being logistic type source function or 0 and m(u), χ(u, v), and g(u, v) being various kinds of functions.
In the current paper, we will study spatial spreading and front propagation dynamics of (1.2) with the following choices, Ω = R N , m(u) = 1, χ(u, v) = χu with χ being a nonnegative constant, f (u, v) = u(1 − u), and g(u, v) = u − v. With such choices, (1.2) becomes (1.3) We will provide a detailed study on the spatial spreading and front propagation dynamics of (1.3) in the case N = 1 and then discuss the extensions of the obtained results for the case N = 1 to N ≥ 2. Here are three main reasons for doing that. First, the study of traveling wave solutions on R N reduces to the study of traveling wave solutions on R. Second, we can get some nicer results in the case N = 1 (compare Theorem B(i) and Theorem D(i)). Third, it is for the simplicity in notations. Consider (1.3) with N = 1, that is, In the very recent work [33], the authors of the current paper studied the global existence of classical solutions with various given initial functions and the asymptotic behavior of global positive solutions of (1.4) (actually, [33] considered a little more general system, namely, (1.3) with u(1 − u) being replaced by u(a − bu)). Let C b unif (R) = {u ∈ C(R) | u(x) is uniformly continuous in x ∈ R and sup x∈R |u(x)| < ∞} (1.5) equipped with the norm u ∞ = sup x∈R |u(x)|. For given 0 < ν < 1 and 0 < θ < 1, let equipped with the norm u C ν unif = sup x∈R |u(x)| + sup x,y∈R,x =y |u(x)−u(y)| |x−y| ν , and C θ ((t 1 , t 2 ), C ν unif (R)) = {u(·) ∈ C((t 1 , t 2 ), C ν unif (R)) | u(t) is locally Hölder continuous with exponent θ}.
Among other, the following are proved in [33].
The limit properties stated in (iii) in the above reflect some spatial spreading feature of the mobile species in (1.4). Note that in [29], the authors studied traveling wave solutions of (1.4) and proved that for any 0 < χ < 1, there is a c * ∈ [2, 2 + χ 1−χ ] such that (1.4) has a traveling wave solution connecting (1, 1) and (0, 0) with speed c * (see [29,Theorem 1.1]). Besides the above mentioned results, up to our best knowledge, there is no other existing results on the spatial spreading and front propagation dynamics of (1.4).
(ii) Let 0 < χ < 1 2 and Theorem A shows that c * min (χ) exists and c * min (χ) ≤ c * (χ). It remains open whether c * min (χ) ≥ 2. It also remains open whether (1.11) has no traveling wave solutions with speed c < c * min (χ) and connecting (1, 1) and (0, 0). These questions reflect the effect of chemotaxis on the wave front dynamics and are very interesting.
(iii) The stability and uniqueness of traveling wave solutions of (1.11) connecting (1, 1) and (0, 0) is also a very interesting problem. We believe that the limit behavior described in (1.14) would play a role in the study of this problem.
(v) Suppose that the logistic source function is replaced by f (u) = u(a − bu) with a > 0 and b > 0. For any given 0 < χ < b 2 , let µ * (χ) be defined by Similarly, we can prove that for any c > c * (χ) (c can also equal c * (χ) when a ≥ 1), (1.11) has a traveling wave solution (u, v) = (U (x − ct), V (x − ct)) with speed c connecting the constant solutions ( a b , a b ) and (0, 0). Moreover, where µ is the only solution of the equation µ + a µ = c in the interval (0 , min{ √ a, 1}).
To state our main results on spreading speeds for (1.4), we first introduce some standing notations.
converges to the single point {2} as χ → 0+, which is the spreading speed of (1.9).
(ii) For any given (iii) When the source function in (1.4) is replaced by f (u) = u(a − bu), similarly, we can prove that if where c * − (χ) and c * + (χ) are such that (iv) Regarding the spatial spreading speeds of (1.4), there are still many interesting problems to be studied. For example, whether c * − (χ) = c * + (χ); whether c * + (χ) = c * (χ) for 0 < χ < 1 2 ; what is the relation between c * − (χ), c * + (χ) and 2 for 0 < χ < 1. These questions are important in the understanding of the spreading feature of (1.4) because they are related to the issue whether the chemotaxis speeds up or slows down the spreading of the species.
We now consider the extensions of Theorems A and B for (1.4) to (1.3). We have the following theorems.
be the spreading speed interval of (1.3) (see Section 5 for the detail).
Because of the lack of comparison principle, the proofs of Theorems A -D are highly non trivial. Our approach to prove Theorem A is based on the construction of a bounded convex non-empty subset of C b unif (R), called E µ (see (2.7)), and a continuous and compact function U : E µ → E µ . Any fixed point of this function, whose existence is guaranteed by the Schauder's fixed theorem, becomes a traveling solution of (1.4). The construction of the set E µ itself is also based on the construction of two special functions. These two special functions are sub-solution and sup-solution of a collection of parabolic equations. At each u ∈ E µ we shall first associate a function which is the solution of a certain parabolic equation, and next define U (·, u) to be the pointwise limit as t goes to infinity of the previous function. One important ingredient in the proof of Theorem B is to prove that for any u 0 ∈ C + c (R), there is M > 0 such that where (u(x, t; u 0 ), v(x, t; u 0 )) is the solution of (1.4) with u(x, 0; u 0 ) = u 0 (x). To do so, for given u 0 ∈ C + c (R) and T > 0, we also construct a bounded convex non-empty subset E T µ (u 0 ) of C b unif (R × [0, T ]) and a continuous and compact functionŪ : E T µ * (χ) (u 0 ) → E T µ * (χ) (u 0 ). Then we prove u(·, ·; u 0 )| R×[0,T ] is a fixed point ofŪ . We use the ideas in the proofs of Theorems A and B and some results in Theorems A and B to prove Theorems C and D.
The rest of this paper is organized as follows. Section 2 is to establish the tools that will be needed in the proof of our main results. It is here that we define the two special functions, which are sub-solution and sup-solution of a collection of parabolic equations, and the non-empty bounded and convex subset E µ . In sections 3 and 4, we prove the main results on the existence of traveling wave solutions and on the spreading speeds for (1.4), respectively. We give the idea of proofs of Theorems C and D in section 5.

Super-and sub-solutions
In this section, we will construct super-and sub-solutions of some related equations of (1.11), which will be used to prove the existence of traveling wave solutions of (1.11) in next section.
It is well known that the function V (x; u) is the solution of the second equation of (1.11 For given open intervals D ⊂ R and I ⊂ R, a function U (·, ·) ∈ C 2,1 (D ×I, R) is called a super-solution or sub-solution of (2.8) on D × I if Theorem 2.1. Suppose that 0 < χ < 1 2 and 0 < µ < 1 satisfy Then for every u ∈ E µ , the following hold.
To prove Theorem 2.1, we first establish some estimates on V (·; u) and V ′ (·; u). It was established in [33] that Combining this with inequality (2.11), we obtain that The next Lemma provide a pointwise estimate for |V (·; u)| whenever u ∈ E µ . Lemma 2.2. For every 0 < µ < 1 and u ∈ E µ , let V (·; u) be defined as in (2.9), then (2.14) Thus, we have Next, we present a pointwise estimate for |V ′ (·; u)| whenever u ∈ E µ .
be the corresponding function satisfying the second equation of (1.11). Then for every x ∈ R and every u ∈ E u .
Proof. Let u ∈ E µ and fix any x ∈ R.

Traveling wave solutions
In this section, we investigate the existence of traveling wave solutions of (1.11) connecting (1, 1) and (0, 0) and prove Theorem A. We first prove the following theorem and then prove Theorem A.
In order to prove Theorem 3.1, we first prove some lemmas. Fix u ∈ E µ . For given u 0 ∈ C b unif (R), let U (x, t; u 0 ) be the solution of (2.8) with U (x, 0; u 0 ) = u 0 (x). By the arguments in the proof of Theorem 1.1 and Theorem 1.5 in [33], we have U (x, t; U + µ ) exists for all t > 0 and U (·, Lemma 3.2. Assume that 0 < µ, χ < 1 satisfy (2.10). Then for every u ∈ E µ , the following hold.
and by (i), U (·, t 2 − t 1 ; U + µ ) ≤ U + µ , (ii) follows from comparison principle for parabolic equations. Let us define U (x; u) to be By the a priori estimates for parabolic equations, the limit in (3.2) is uniform in x in compact subsets of R and U (·; u) ∈ C b unif (R). We shall provide sufficient hypothesis on the choice of d to guarantee that the function U (·; u) constructed above is not identically zero for each u ∈ E µ . Now, we are ready to prove that the function u ∈ E µ → U (·; u) ∈ E µ for d large enough.
Then by comparison principle for parabolic equations, we have that Then by comparison principle for parabolic equations again, The lemma then follows.
Remark 3.4. It follows from Lemmas 3.2 and 3.3 that if the assumptions of these two lemmas hold, then This implies that From now on, we suppose that 0 < µ, χ < 1 are fixed and satisfy inequality (2.10). Next chooseμ such that µ <μ < min{1, 2µ, µ + 1 and take d ≥ d 0 , where d 0 is given by Lemma 3.3. We have the following important result.
Lemma 3.5. Assume that 0 < µ, χ < 1 satisfy (2.10). Then for every u ∈ E µ the associated function U (·; u) satisfied the elliptic equation, Proof. Let {t n } n≥1 be an increasing sequence of positive real numbers converging to ∞. For every n ≥ 1, define U n (x, t) = U (x, t + t n ; u) for every x ∈ R, t ≥ 0. For every n, U n solves the PDE Let {T (t)} t≥0 be the analytic semigroup on C b unif (R) generated by ∆ − I and let X β = Dom((I − ∆) β ) be the fractional power spaces of I − ∆ on C b unif (R) (β ∈ [0, 1]). The variation of constant formula and the fact that . (3.5) Let 0 < β < 1 2 be fixed. We have that Next, using inequality (3.1) in [33] , we have that And
We now prove Theorem 3.1.

Proof of Theorem 3.1. First of all, let us consider the normed linear space
For every u ∈ E µ we have that Hence E µ is a bounded convex subset of E. Furthermore, since the convergence in E implies the pointwise convergence, then E µ is a closed, bounded, and convex subset of E. Furthermore, a sequence of functions in E µ converges with respect to norm · * if and only if it converges locally uniformly convergence on R.
We prove that the mapping E µ ∋ u → U (·; u) has a fixed point. We divide the proof in two steps.
Step 1. In this step, we prove that the mapping E µ ∋ u → U (·; u) is compact. Let {u n } n≥1 be a sequence of elements of E µ . Since U (·; u n ) ∈ E µ for every n ≥ 1 then {U (·; u n )} n≥1 is clearly uniformly bounded by 1 1−χ . Using inequality (3.6), we have that for all n ≥ 1 where M 1 is given by (3.7). Therefore there is 0 < ν ≪ 1 such that for every n ≥ 1 whereM 1 is a constant depending only on M 1 . Since for every n ≥ 1 and every x ∈ R, we have that U (x, t; u n ) → U (x; u n ) as t → ∞, then it follows from (3.21) that for every n ≥ 1. Which implies that the sequence {U (·; u n )} n≥1 is equicontinuous. The Arzela-Ascoli's Theorem implies that there is a subsequence {U (·; u n ′ )} n≥1 of the sequence {U (·; u n )} n≥1 and a function U ∈ C(R) such that {U (·; u n ′ )} n≥1 converges to U locally uniformly on R. Furthermore, the function U satisfies inequality (3.22). Combining this with the fact U − µ (x) ≤ U (x; u n ′ ) ≤ U + µ (x) for every x ∈ R and n ≥ 1, by letting n goes to infinity, we obtain that U ∈ E µ .
Step 2. In this step, we prove that the mapping E µ ∋ u → U (·; u) is continuous.
Let u ∈ E µ and {u n } n≥1 ∈ E N µ such that u n − u * → 0 as n → ∞. Suppose by contradiction that U (·; u n ) − U (·; u) * does not converge to zero. Hence there is δ > 0 and a subsequence {u n1 } n≥1 such that For every n ≥ 1, we have that U (·, u n1 ) satisfies Claim 1. V (·; u n ) − V (·; u) * → 0 as n → ∞. Indeed, for every R > 0, it follows from (2.9) that Thus for every k ∈ N and every R > 1, we have that Now, let ε > 0 be given. Choose R ≫ 1 and k ≫ 1 such that It follows from inequalities (3.26), (3.27) and (3.28) that for every n ≥ N , we have (3.29) Thus, the claim follows.
By Lemma 3.6, U (·) = U (·; u). By (3.23), which is a contradiction. Hence the mapping E µ ∋ u → U (·; u) is continuous. Now by Schauder's Fixed Point Theorem, there is U ∈ E µ such that U (·; U ) = U (·). Then (U (x), V (x; U )) is a stationary solution of (2.1) with c = c µ . It is clear that We claim that if χ < 1 2 , then lim For otherwise, we may assume that there is x n → −∞ such that U (x n ) → a = 1 as n → ∞. Define U n (x) = U (x + x n ) for every x ∈ R and n ≥ 1. By observing that U n = U (·; U n ) for every n, hence it follows from the step 1, that there is a subsequence {U n ′ } n≥1 of {U n } n≥ and a function U * ∈ E µ such that U n ′ − U * * → 0 as n → ∞. Next, it follows from step 2 that (U * , V (·; U * )) is also a stationary solution of (2.1).
Hence, It follows from Remark 3.4 that Letting n goes to infinity in the last inequality, we obtain that U − µ (x δ ) ≤ U * (x) for every x ∈ R. The claim thus follows.
As a direct consequence of Theorem 3.1 we present the proof of Theorem A.

20
In this section, we study the spreading properties of solutions of (1.4) with nonnegative initial functions u 0 which have nonempty and compact supports, and prove Theorem B. Throughout this section, we assume that 0 < χ < 1, unless specified otherwise. One important ingredient in the proof of Theorem B is to prove that for any where (u(x, t; u 0 ), v(x, t; u 0 )) is the solution of (1.4) with u(x, 0; u 0 ) = u 0 (x), µ * is as in (1.12), and c µ * = µ * + 1 µ * . To this end, we first prove some lemmas.
Recall that for every 0 < µ < 1, ϕ µ (x) = e −µx . For every T > 0 and µ ∈ (0, 1), we define For given u ∈ E T µ (u 0 ), let V (x, t; u) be the solution of the second equation in (1.4). Note that In what follows, some of the arguments are similar to those of the previous sections. Hence, some details might be omitted. The next Lemma is an equivalent of Lemmas 2.2 and 2.3, whence it provides pointwise estimates on V (·, t; u) and |∂ x V (·, t; u)| for every u ∈ E T µ (u 0 ).
Proof. In this proof, we put µ = µ * . Consider the normed linear space For every u ∈ E T µ (u 0 ) we have that u * ,T ≤ M e µcuT . Hence E T µ (u 0 ) is a bounded convex subset of E T . Since the convergence in E T implies the pointwise convergence, then E T µ (u 0 ) is a closed, bounded, and convex subset of E T . Furthermore, a sequence of functions in E T µ (u 0 ) converges with respect to norm · * ,T if and only if it converges locally uniformly on R × [0, T ].
Claim 2 : lim t→0 +Ū (·, t) = u 0 in C b unif (R) Indeed, let ε > 0 be fixed. There is 0 < t ε < T such that Thus, by taking β = 0, it follows from inequality (4.16) that for every 0 ≤ t < t ε . Hence, letting n goes to infinity in the last inequality, we obtain that Thus Claim 2 is proved. It is clear thatŪ ∈ E T µ (u 0 ). Thus complete the proof of step 1.

Spreading speeds and traveling waves on R N
In this section, we consider the spatial spreading speeds and traveling wave solutions of (1.3) with N ≥ 1 and prove Theorems C and D. The proofs are based the ideas used in the proofs of Theorems A and B and some results in Theorems A and B. We will skip the details of those arguments which are similar to some arguments in Theorems A and B.
Proof of Theorem C. Assume that 0 < χ < 1 2 and that c * (χ) is as in Theorem A. For given c ≥ c * (χ), let (u, v) = (U (x − ct), V (x − ct)) be the traveling wave solution of (1.4) connecting (1, 1) and (0, 0) with speed c. It is then easy to verify that is a traveling wave solution of (1.3) which connects (1, 1) and (0, 0) and propagates in the direction of ξ ∈ S N −1 with speed c. This proves Theorem C. Observe that M is independent of T ,Ū µ (x, 0) ≥ u 0 (x) for every x ∈ R, and For given u ∈ E T µ (u 0 ), let V (x, t; u) be the solution of the second equation in (1.3). Note that Claim 1. For every 0 < µ < 1 √ N and for every u ∈ E T µ (u 0 ) we have that and The claim can be proved by the arguments similar to those in Lemma 4.1. We provide some indication of the proof in the following.
For every x ∈ R N , t > 0 and i = 1, · · · , N , we have that Combining (5.6) and (5.7), we obtain (5.4). Hence the claim is proved. For every u ∈ E T µ , letŪ (·, ·; u) be the solution of the Initial Value Problem Then for every u ∈ E T µ * N (u 0 ) we have that 0 ≤Ū (·, ·; u) ≤Ū µ * N (·, ·). (5.10) The claim can be proved by the arguments similar to those in Lemma 4.2. In the following, we provide some indication of the proof.
(ii) If 0 < χ < 2 3+ √ N +1 , the proof of the uniform lower bound for c * low (u 0 ) when N = 1 also apply to the general case.