On traveling wave solutions in full parabolic Keller-Segel chemotaxis systems with logistic source
aa r X i v : . [ m a t h . A P ] J a n On traveling wave solutions in full parabolic Keller-Segelchemotaxis systems with logistic source
Rachidi B. SalakoDepartment of MathematicsThe Ohio State UniversityColumbus OH, 43210-1174andWenxian Shen ∗ Department of Mathematics and StatisticsAuburn UniversityAuburn University, AL 36849
Abstract
This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, ( u t = ∆ u − χ ∇ · ( u ∇ v ) + u ( a − bu ) , x ∈ R N τ v t = ∆ v − λv + µu, x ∈ R N , (0.1)where χ, µ, λ, a, and b are positive numbers, and τ ≥
0. Among others, it is proved that if b > χµ and τ ≥ (1 − λa ) + , then for every c ≥ √ a , (0.1) has a traveling wave solution( u, v )( t, x ) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) ( ∀ ξ ∈ R N ) connecting the two constant steadystates (0 ,
0) and ( ab , µλ ab ), and there is no such solutions with speed c less than 2 √ a , whichimproves considerably the results established in [30], and shows that (0.1) has a minimal wavespeed c ∗ = 2 √ a , which is independent of the chemotaxis. Key words.
Full parabolic chemotaxis system, logistic source, traveling wave solution, minimalwave speed ∗ Partially supported by the NSF grant DMS–1645673 Introduction
This work is concerned with traveling wave solutions of the following full parabolic chemotaxissystem ( u t = ∆ u − χ ∇ · ( u ∇ v ) + u ( a − bu ) , x ∈ R N τ v t = ∆ v − λv + µu, x ∈ R N , (1.1)where χ, µ, λ, a, and b are positive real numbers, τ is a nonnegative real number, and u ( t, x )and v ( t, x ) denote the concentration functions of some mobile species and chemical substance,respectively. Biologically, the positive constant χ measures the sensitivity effect on the mobilespecies by the chemical substance which is produced overtime by the mobile species; the reaction u ( a − bu ) in the first equation of (1.1) describes the local dynamics of the mobile species; λ represents the degradation rate of the chemical substance; and µ is the rate at which the mobilespecies produces the chemical substance. The constant τ in the case τ > τ = 0 is supposed to model the situation whenthe chemical substance diffuses very quickly.System (1.1) is a simplified version of the chemotaxis system proposed by Keller and Segelin their works [18, 19]. Chemotaxis models describe the oriented movements of biological cellsand organisms in response to certain chemical substances. These mathematical models play veryimportant roles in a wide range of biological phenomena and accordingly a considerable literatureis concerned with its mathematical analysis. The reader is referred to [11, 12] for some detailedintroduction into the mathematics of Keller-Segel models.One of the central problems about (1.1) is whether a positive solution blows up at a finitetime. This problem has been studied in many papers in the case that a = b = 0 (see [11, 14,16, 17, 25, 38, 39, 40]). It is shown that finite time blow-up may occur if either N = 2 andthe total initial population mass is large enough, or N ≥
3. It is also shown that some radialsolutions to (1.1) in plane collapse into a persistent Dirac-type singularity in the sense that aglobally defined measure-valued solution exists which has a singular part beyond some finite timeand asymptotically approaches a Dirac measure (see [23, 34]). We refer the reader to [2, 13] andthe references therein for more insights in the studies of chemotaxis models.When the constant a and b are positive, the finite time blow-up phenomena in (1.1) may besuppressed to some extent. In fact in this case, it is known that when the space dimension isequal to one or two, solutions to (1.1) on bounded domains with Neumann boundary conditionsand initial functions in a space of certain integrable functions are defined for all time. Andit is enough for the self limitation coefficient b to be big enough comparing to the chemotaxissensitivity coefficient to prevent finite time blow-up, see [15, 31, 35].Traveling wave solutions constitute another class of important solutions of (1.1). Observe that,when χ = 0, the chemotaxis system (1.1) reduces to u t = ∆ u + u ( a − bu ) , x ∈ R N . (1.2)Due to the pioneering works of Fisher [7] and Kolmogorov, Petrowsky, Piskunov [20] on travelingwave solutions and take-over properties of (1.2), (1.2) is also referred to as the Fisher-KPPequation. The following results are well known about traveling wave solutions of (1.2). Equation(1.2) has traveling wave solutions of the form u ( t, x ) = φ ( x · ξ − ct ) ( ξ ∈ S N − ) connecting 0and ab ( φ ( −∞ ) = ab , φ ( ∞ ) = 0) of all speeds c ≥ √ a and has no such traveling wave solutions2f slower speed. c ∗ = 2 √ a is therefore the minimal wave speed of traveling wave solutions of(1.2) connecting 0 and ab . Since the pioneering works by Fisher [7] and Kolmogorov, Petrowsky,Piscunov [20], a huge amount research has been carried out toward the front propagation dynamicsof reaction diffusion equations of the form, u t = ∆ u + uf ( t, x, u ) , x ∈ R N , (1.3)where f ( t, x, u ) < u ≫ ∂ u f ( t, x, u ) < u ≥ ,
0) and ( ab , µλ ab ). Roughly, it is proved in [30]that, when the chemotaxis sensitivity χ is small relative to the logistic damping b , (1.1) hastraveling wave solutions connecting (0 ,
0) and ( ab , µλ ab ) with speed c , which is bounded below bysome constant c ∗ > c ∗ = 2 √ a and is bounded above by some constant c ∗∗ < ∞ . But manyfundamental questions remain open, for example, whether (1.1) has traveling wave solutionsconnecting (0 ,
0) and ( ab , µλ ab ) with speed c ≫
1; whether there is a minimal wave speed oftraveling wave solutions of (1.1) connecting (0 ,
0) and ( ab , µλ ab ), and if so, how the chemotaxisaffects the minimal wave speed.The objective of the current paper is to investigate those fundamental open questions. Tostate the main results of the current paper, we first introduce the definition of traveling wavesolutions of (1.1) and the induced problems to be studied. An entire solution of (1.1) is a classical solution ( u ( t, x ) , v ( t, x )) of (1.1) which is defined for all x ∈ R N and t ∈ R . Note that the constant solutions ( u ( t, x ) , v ( t, x )) = (0 ,
0) and ( u ( t, x ) , v ( t, x )) =( ab , µaλb ) are clearly two particular entire solutions of (1.1). An entire solution of (1.1) of theform ( u ( t, x ) , v ( t, x )) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) for some unit vector ξ ∈ S N − andsome constant c ∈ R is called a traveling wave solution with speed c . A traveling wave solution( u ( t, x ) , v ( t, x )) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) ( ξ ∈ S N − ) of (1.1) with speed c is said toconnect (0 ,
0) and ( ab , µaλb ) iflim inf x →−∞ U τ,c ( x ) = ab and lim sup x →∞ U τ,c ( x ) = 0 . (1.4)We say that a traveling wave solution ( u ( t, x ) , v ( t, x )) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) of (1.1)is nontrivial and connects (0 ,
0) at one end iflim inf x →−∞ U τ,c ( x ) > x →∞ U τ,c ( x ) = 0 . (1.5)Observe that for given c ∈ R , a traveling solution ( u ( t, x ) , v ( t, x )) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) ( ξ ∈ S N − ) of (1.1) with speed c connecting the states (0 ,
0) and ( ab , µaλb ) gives rise to astationary solution ( u, v ) = ( U τ,c ( x ) , V τ,c ( x )) of the parabolic-elliptic system ( u t = u xx + (( c − χv x ) u ) x + ( a − bu ) u, x ∈ R v xx + τ cv x − λv + µu, x ∈ R . (1.6)connecting the states (0 ,
0) and ( ab , µaλb ). 3onversely, if ( u, v ) = ( U τ,c ( x ) , V τ,c ( x )) is a stationary solution of (1.6) connecting the states(0 ,
0) and ( ab , µaλb ), then ( u ( t, x ) , v ( t, x )) = ( U τ,c ( x · ξ − ct ) , V τ,c ( x · ξ − ct )) is a traveling wavesolution of (1.1) with speed c connecting the states (0 ,
0) and ( ab , µaλb ) for any ξ ∈ S N − .To study traveling wave solutions of (1.1) with speed c connecting the states (0 ,
0) and ( ab , µaλb )is then equivalent to study stationary solutions of (1.6) connecting the states (0 ,
0) and ( ab , µaλb ).It is clear that (1.6) is equivalent to ( u t = u xx + ( c − χv x ) u x + ( a − χv xx − bu ) u, x ∈ R v xx + τ cv x − λv + µu, x ∈ R . (1.7)Hence, to study traveling wave solutions of (1.1) connecting the states (0 ,
0) and ( ab , µaλb ) we shallstudy steady states solutions of (1.7) connecting the states (0 ,
0) and ( ab , µaλb ).Before stating the main results of the current paper, we next recall some existing results on theexistence of solutions of (1.7) with given initial functions and existence of steady states solutionsof (1.7) or traveling wave solutions of (1.1) connecting the states (0 ,
0) and ( ab , µaλb ). Let C b unif ( R ) = { u ∈ C ( R ) | u ( x ) is uniformly continuous in x ∈ R and sup x ∈ R | u ( x ) | < ∞} equipped with the norm k u k ∞ = sup x ∈ R | u ( x ) | . Proposition 1.1 (Local existence) . For every nonnegative initial function u ∈ C b unif ( R ) and c ∈ R , there is a unique maximal time T max ( u ) , such that (1.7) has a unique classical solution ( u ( t, x ; u , c ) , v ( t, x ; u , c )) defined for every x ∈ R and ≤ t < T max ( u ) with u (0 , x ; u , c ) = u ( x ) . Moreover if T max ( u ) < ∞ then lim t → T max ( u ) − k u ( t, · ; u , c ) k ∞ = ∞ . The above proposition can proved by similar arguments as those in [31, Theorem 1.1]. Thefollowing proposition follows from the arguments of [30, Theorems A and B] (it is proved in [30,Theorems A and B] for the case that λ = µ = 1). Proposition 1.2 (Global existence) . Consider (1.7) .(1) Assume that ≤ χµτc √ λ , lim t →∞ h k u ( t, · ; u , c ) − ab k ∞ + k v ( · , t ; u , c ) − µλ ab k ∞ i = 0 . roposition 1.3. (1) For every τ > , there is < χ ∗ τ < b µ such that for every < χ < χ ∗ τ ,there exist two positive numbers < c ∗ ( χ, τ ) < c ∗∗ ( χ, τ ) satisfying that for every c ∈ ( c ∗ ( χ, τ ) , c ∗∗ ( χ, τ )) , (1.1) has a traveling wave solution ( u, v ) = ( U ( x · ξ − ct ) , V ( x · ξ − ct ))( ∀ ξ ∈ S N − ) connecting the constant solutions (0 , and ( ab , µλ ab ) . Moreover, lim χ → c ∗∗ ( χ, τ ) = ∞ , lim χ → c ∗ ( χ, τ ) = ( √ a if < a ≤ λ + τa (1 − τ ) + λ + τa (1 − τ ) + + a (1 − τ ) + λ + τa if a ≥ λ + τa (1 − τ ) + , and lim x →∞ U ( x ; τ ) e − kx = 1 , where k is the only solution of the equation k + ak = c in the interval (0 , min {√ a, q λ + τa (1 − τ ) + } ) .(2) For any given τ ≥ and χ ≥ , (1.1) has no traveling wave solutions ( u, v ) = ( U ( x · ξ − ct ) , V ( x · ξ − ct )) ( ∀ x ∈ S N − ) with ( U ( −∞ ) , V ( −∞ )) = ( ab , µλ ab ) , ( U ( ∞ ) , V ( ∞ )) = (0 , ,and c < √ a . As mentioned before, in the absence of chemotaxis (i.e. χ = 0), c ∗ = 2 √ a is the minimal wavespeed of the Fisher-KPP equation (1.2). Both biologically and mathematically, it is interestingto know whether the results stated in Proposition 1.3(1) can be improved to the following: forany c ≥ c ∗ , (1.1) has a traveling wave solution ( u ( t, x ) , v ( t, x )) = ( U ( x · ξ − ct ) , V ( x · ξ − ct ))( ∀ ξ ∈ S N − ) connecting ( ab , µλ ab ) and (0 , c ∗ = 2 √ a is the minimal wave speed of (1.1). The precise statements of the main results arestated in next subsection. In order to state our main results, we first introduce some notations. For given c ∈ R , let B λ,c,τ = 1 √ λ + τ c ,λ c = ( τ c + √ λ + τ c )2 , λ c = ( √ λ + τ c − τ c )2 , and c κ = a + κ κ ∀ < κ < √ a. Note that λ c and − λ c are the positive and negative roots of the quadratic equations m + τ cm − λ = 0 . λ c λ c = λ, λ c + λ c = 1 B λ,c,τ . (1.8)All the above quantities are defined for any τ ≥ c >
0. This restriction is justified bythe fact that (1.1) does not have a non-trivial traveling wave with speed c ≤ λ c B λ,c,τ λ c + κ (cid:16) κ − λλ c (cid:17) + = λ c ( κ − λ c ) + ( λ c + λ c )( κ + λ c ) < . (1.9)Hence the following quantity is well defined b ∗ τ = sup { λ c κ ( κ − λ c κ ) + ( λ c κ + λ c κ )( κ + λ c κ ) | < κ < √ a } . (1.10)It is clear that b ∗ τ is defined for all τ ≥ b ∗ τ ≤ τ ≥
0, and b ∗ = 1 + ( √ a −√ λ ) + √ a + √ λ ) .For the sake of simplicity in the statements of our results, let us introduce the followingstanding hypotheses. (H1) b > χµ . (H2) b > b ∗ τ χµ . (H3) b > χµ . (H4) τ ≥ (cid:0) − λa (cid:1) + .Observe that (H3) implies (H2) , and (H2) implies (H1) .The following results about the global existence of bounded classical solutions and the stabilityof the positive constant equilibria of (1.7) will be of great use in our arguments. Theorem 1.1.
For any τ ≥ and c > , the following hold. (i) If ( H1 ) holds, then for every u ∈ C b unif ( R ) , with u ≥ , (1.7) has a unique global classicalsolution ( u ( t, x ; u , c ) , v ( t, x ; u , c )) on (0 , ∞ ) × R satisfying lim t → + k u (0 , · ; u , c ) − u ( · ) k ∞ =0 . Moreover it holds that k u ( t, · ; u , c ) ∞ k ≤ max n k u k ∞ , ab − χµ o , t ≥ . (1.11) (ii) If ( H3 ) holds, then for every u ∈ C b unif ( R ) , with inf x ∈ R u ( x ) > , we have that lim t →∞ (cid:16) k u ( t, · ; u , c ) − ab k ∞ + k v ( t, · ; u , c ) − aµbλ k ∞ (cid:17) = 0 . (1.12) Remark 1.1.
When τ = 0 , we recover [31, Theorems 1.5 & τ > , Theorem 1.1improves the results stated in Proposition 1.2. , √ a ) ∋ κ λ c κ − κ is strictly decreasing. Hence the quantity κ ∗ τ := sup { < κ < √ a | λ c κ − κ ≥ } (1.13)is well defined. It holds that λ c κ − κ > < κ < κ ∗ τ . Note also that κ ∗ τ = min ( √ a, s λ + τ a (1 − τ ) + ) . (1.14)Indeed, it holds that λ c √ a > √ a for every τ ≥
1. On the other hand, for 0 ≤ τ <
1, if λ c κ = κ forsome 0 < κ ≤ √ a , then it holds that λ + κτ c κ − κ = 0 ⇔ λ + τ a = (1 − τ ) κ κ = r λ + τ a − τ . Hence (1.14) holds.Let c ∗ ( τ ) = κ ∗ τ + aκ ∗ τ . (1.15)Note that κ ∗ τ and c ∗ ( τ ) are defined for all τ ≥
0, and κ ∗ = min {√ λ, √ a } , c ∗ (0) = min {√ λ, √ a } + a min {√ λ, √ a } . We have the following theorem on the existence of traveling wave solutions of (1.1).
Theorem 1.2.
For any τ ≥ , the following hold.(1) If (H2) holds, then for any c > c ∗ ( τ ) , (1.1) has a nontrivial traveling wave solution ( u, v )( t, x ) = ( U ( x · ξ − c κ t ) , V ( x · ξ − c κ t )) ( ∀ ξ ∈ S N − ) satisfying (1.5) , where κ ∈ (0 , κ ∗ τ ) is such that c κ = c . Furthermore, it holds that lim x →∞ U ( x ) e − κx = 1 . (1.16) If in addition, ( H3 ) holds, then lim x →−∞ | U ( x ) − ab | = 0 . (1.17) (2) If (H2) and (H4) hold, then κ ∗ τ = √ a and c ∗ ( τ ) = 2 √ a . Hence for any c > √ a , theresults in (1) hold true.(3) Suppose that (H3) holds. Then (1.1) has a traveling wave solution ( u, v )( t, x ) = ( U τ,c ( x · ξ − ct, V τ,c ( x · ξ − ct )) ( ∀ ξ ∈ S N − ) with speed c ∗ ( τ ) connecting (0 , and ( ab , aµbλ ) . emark 1.2. (1) Note that the conditions in Proposition 1.3 are χ < χ ∗ τ and b > χµ , whichimply both (H2) and (H3) . Hence the assumptions in Theorem 1.2(1) are weaker than thosein Proposition 1.3 for the existence of traveling wave solutions. Note also that, by Theorem1.2(1), the lower bound c ∗ ( τ ) for the wave speed is independent of χ , and the upper bound is ∞ . By the proof of [30, Theorem C], κ ∗ τ = min {√ a, λ + τa (1 − τ ) + } is an upper bound found for thedecay rate of traveling wave solutions found in [30]. Hence c ∗ ( χ, τ ) ≥ c κ ∗ τ = c ∗ ( τ ) , that is,the lower bound provided in Theorem 1.2 for the wave speed of traveling wave solutions of (1.1) is not larger than that provided in Proposition 1.3. Moreover, under the assumptions(H2) and (H4), c ∗ ( τ ) = 2 √ a < c ∗ ( χ, τ ) . Therefore Theorem 1.2 improves considerablyProposition 1.3.(2) Recall that b ∗ = 1 + ( √ a −√ λ ) + a + √ λ ) , κ ∗ = min {√ a, √ λ } , and c ∗ (0) = min {√ a, √ λ } + a min {√ a, √ λ } .Hence Theorem 1.2 in the case τ = 0 recovers [28, Theorem 1.4].(3) When λ ≥ a , c ∗ ( τ ) = c ∗ = 2 √ a for any τ ≥ . Hence if λ ≥ a and < χµ < b hold, byTheorem 1.2 for every τ ≥ and c ≥ √ a , (1.1) has a traveling wave solution ( u, v )( t, x ) =( U τ,c , V τ,c )( x − ct ) with speed c connecting (0 , and ( ab , aµbλ ) . Whence, if λ ≥ a and < χ < b µ , Theorem 1.2 implies that c ∗ = 2 √ a is the minimal wave speed of traveling wave solutionsof (1.1) connecting (0 , and ( ab , aµbλ ) , and that the chemotaxis does not affect the magnitudeof the minimal wave speed of (1.1) . Biologically, λ ≥ a means that the degradation rate λ of the chemical substance is greater than the intrinsic growth rate a of the mobile species,and < χµ < b indicates that the product of the chemotaxis sensitivity χ and the rate µ at which the mobile species produces the chemical substance is less than half of the logisticdamping b .(4) When λ < a , c ∗ ( τ ) = c ∗ = 2 √ a for τ > (cid:0) − λa (cid:1) . Hence if λ < a and < χµ < b hold,by Theorem 1.2 for every τ > (1 − λa ) and c ≥ √ a , (1.1) has a traveling wave solution ( u, v )( t, x ) = ( U τ,c , V τ,c )( x − ct ) with speed c connecting (0 , and ( ab , aµbλ ) . Thus in thiscase, Theorem 1.2 also implies that c ∗ = 2 √ a is the minimal wave speed of traveling wavesolutions of (1.1) connecting (0 , and ( ab , aµbλ ) , and that the chemotaxis does not affect themagnitude of the minimal wave speed of (1.1) . Biologically, τ > (cid:0) − λa (cid:1) indicates thatdiffusion rate of the chemical substance is not big.(5) By Theorem 1.2 it holds that c ∗ ( τ ) = 2 √ a whenever τ ≥ and (1.1) has a minimal wavespeed, which is c ∗ ( τ ) . When λ < a and ≤ τ < , it remains open whether (1.1) has aminimal wave speed, and if so, whether the minimal wave speed equals √ a . It would beinteresting to study the stability of the traveling wave solutions of (1.1) . When τ = 0 , thespreading speeds of solutions of (1.1) with compactly supported initial functions are studiedin [28]. It would be also interesting to study these spreading results when τ > , which weplan to carry out in our future work. The rest of the paper is organized as follow. In Section 2, we prove some preliminaries resultsto use in the subsequent sections. Section 3 is devoted to the proof of Theorem 1.1, while theproof of Theorem 1.2 will be presented in Section 4.8
Preliminary lemmas
In this section, we prove some lemmas to be used in the proofs of the main results in the latersections. Throughout of this section, we assume τ ≥ u ∈ C b unif ( R ) and c ∈ R , letΨ( x ; u, c, τ ) = µ Z ∞ Z R e − λs e − | x + τcs − y | s √ πs u ( y ) dyds. (2.1)It is well known that Ψ( x ; u, c, τ ) ∈ C ( R ) and solves the elliptic equation d dx Ψ( x ; u, c, τ ) + τ c ddx Ψ( x ; u, c, τ ) − λ Ψ( x ; u, c, τ ) + µu = 0 . Lemma 2.1.
It holds that Ψ( x ; u, c, τ ) = µ √ λ + τ c Z R e − √ λ + τ c | x − y |− τc ( x − y )2 u ( y ) dy = µB χ,c,τ (cid:16) e − λ c x Z x −∞ e λ y u ( y ) dy + e λ x Z ∞ x e − λ c y u ( y ) dy (cid:17) (2.2) and ddx Ψ( x ; u, c, τ ) = µB χ,c,τ (cid:16) − λ c e − λ c x Z x −∞ e λ c y u ( y ) dy + λ c e λ c x Z ∞ x e − λ c y u ( y ) dy (cid:17) . (2.3) Proof.
For the case that τ = 0, the lemma is proved in [28, Lemma 2.1].In the following, we prove the case that τ >
0. Observe that it is enough to prove the resultfor τ = 1. The general case follows by replacing c by τ c . So, without loss of generality, we set τ = 1. First, observe that the following identity holds, Z ∞ e − β s − s √ πs ds = e − β , ∀ β > . (2.4)Next using Fubini’s Theorem, one can exchange the order of integration in (2.1) to obtainΨ( x ; u, c,
1) = µ Z ∞ Z R e − λs e − | x + cs − y | s [4 πs ] u ( y ) dyds = µ Z R h Z ∞ e − | x + cs − y | s − λs √ πs ds i u ( y ) dy = Z R e − c ( x − y )2 h Z ∞ e − (cid:2) ( x − y )24 s + (4 λ + c s (cid:3) √ πs ds i u ( y ) dy (2.5)By the change of variable z = (4 λ + c ) s and taking β = √ λ + c | x − y | , it follows from (2.4) that Z ∞ e − (cid:2) ( x − y )24 s + (4 λ + c s (cid:3) √ πs ds = 2 √ λ + c Z ∞ e − β z − z √ πz dz = 1 √ λ + c e − √ λ + c | x − y | . x ; u, c,
1) = µ √ λ + c Z R e − √ λ + c | x − y |− c ( x − y )2 u ( y ) dy. Thus (2.2) holds. Note that (2.3) then follows from a direction calculation.
Lemma 2.2.
For every u ∈ C b unif ( R ) , u ( x ) ≥ , it holds that | ddx Ψ( x ; u, c, τ ) | ≤ λ c Ψ( x ; u, c, τ ) , ∀ x ∈ R , c ∈ R . (2.6) Furthermore, it holds that χκ Ψ x ( · ; u, c, τ ) − χ Ψ xx ( · ; u, c, τ ) ≤ χµ (cid:16) B λ,c,τ (( τ c + κ ) λ − λ ) + ( λ + κ ) + 1 (cid:17) M e − κx (2.7) whenever ≤ u ( x ) ≤ M e − κx for some κ ≥ and M > .In particular, if χµ (cid:16) B λ,c,τ (cid:0) ( τ c + κ ) λ − λ (cid:1) + ( λ + κ ) + 1 (cid:17) ≤ b, (2.8) holds, then χκ Ψ x ( x ; u, c, τ ) − χ Ψ xx ( x ; u, c, τ ) − be − κx ≤ , ∀ x ∈ R , (2.9) whenever ≤ u ( x ) ≤ e − κx for some positive real numbers κ > and M > .Proof. For the case that τ = 0, the lemma is proved in [28, Lemma 2.2]. In the following, weprove the lemma for any τ ≥ | ddx Ψ( x ; u, c, τ ) | ≤ √ λ + τ c + τ c x ; u, c, τ ) . This implies (2.6).Next, we prove (2.9). It follows from (2.1) and (2.3) that χκ Ψ x ( x ; u, c, τ ) − χ Ψ xx ( x ; u, c, τ ) = χκ Ψ x ( x ; u, c, τ ) − χ ( λ Ψ( x ; u, c, τ ) − τ c Ψ x ( x ; u, c, τ ) − µu )= χ ( τ c + κ )Ψ x ( x ; u, c, τ ) − χλ Ψ( x ; u, c, τ ) + χµu = − χµB λ,c,τ (( τ c + κ ) λ c + λ ) e − λ c x Z x −∞ e λ c y u ( y ) dy + χµB λ,c,τ (( τ c + κ ) λ c − λ ) e λ c x Z ∞ x e − λ y u ( y ) dy + χµu. (2.10)Hence, since 0 ≤ u ≤ M e − κx , it follows that χ ( κ Ψ x ( x ; u, c, τ ) − Ψ xx ( x ; u, c, τ )) ≤ χµB λ,c,τ (( τ c + κ ) λ c − λ ) + M e λ c x Z ∞ x e − λ c y e − κy dy + χµMe κx = χµ (cid:16) B λ,c,τ (( τ c + κ ) λ c − λ ) + ( λ c + κ ) + 1 (cid:17) M e − κx Hence, (2.7) follows. 10 emark 2.1.
Observe that τ cλ c − λ = τ c (cid:16)p λ + τ c − τ c (cid:17) − λ = 2 λτ c √ λ + τ c + τ c − λ = − λλ c λ c < . (2.11) Hence B λ,c,τ λ c ( τ cλ c − λ ) + = 0 , and B λ,c,τ λ c + κ (cid:16) ( τ c + κ ) λ c − λ (cid:17) + = λ c B λ,c,τ λ c + κ (cid:16) κ − λλ c (cid:17) + . We also note from (1.8) that B λ,c,τ (cid:16) λλ c + λλ c (cid:17) = 1 . (2.12) These identities will be frequently used later.
For every 0 < κ < ˜ κ < √ a with ˜ κ < κ and M, D ≥
1, consider the functions ϕ κ ( x ), U κ,D ( x ),and U κ,D ( x ) given by ϕ κ ( x ) = e − κx ,U − D ( x ) = ϕ κ ( x ) − Dϕ ˜ κ ( x ) , x ∈ R , (2.13) U κ,M ( x ) = min { M, ϕ κ ( x ) } , (2.14)and U κ,D ( x ) = ( ϕ κ ( x ) − Dϕ ˜ κ ( x ) , x ≥ x κ,D ϕ κ ( x κ,D ) − Dϕ ˜ κ ( x κ,D ) , x ≤ x κ,D , (2.15)where x κ,D satisfies max { ϕ κ ( x ) − Dϕ ˜ κ ( x ) | x ∈ R } = ϕ κ ( x κ,D ) − Dϕ ˜ κ ( x κ,D ) . (2.16)Letting x κ,D := ln( D )˜ κ − κ , there holds that U − D ( x ) ( > x > x κ,D ,< , if x < x κ,D . For every u ∈ C b unif ( R ), let A u,c ( U ) = U xx + ( c − χ Ψ x ( · ; u, c, τ )) U x + ( a − χ Ψ xx ( · ; u, c, τ ) − bU ) U. (2.17) Lemma 2.3.
For given τ ≥ , assume that (H2) holds and κ < κ ∗ τ . Then there is D ∗ > suchthat for every D ≥ D ∗ , M > , and u ∈ ˜ E := { u ∈ C b unif ( R ) | max { U − D ( x ) , } ≤ u ( x ) ≤ min { M, ϕ κ ( x ) } ∀ x ∈ R } it holds that A u,c κ ( U − D ) ≥ ∀ x ∈ ( x κ,D , ∞ ) . (2.18)11 roof. We first note that (H2) implies (2.8), and κ < κ ∗ τ implies λ c κ > κ. (2.19)Let u ∈ ˜ E be given and U − ( x ) = U − D ( x ). Then A u,c κ ( U − ) = U − xx + ( c κ − χ Ψ x ( · ; u, c κ )) U − x + ( a − χ Ψ xx − bU − ) U − = (cid:0) κ e − κx − ˜ κ De − ˜ κx (cid:1) + ( c κ − χ Ψ x )( − κe − κx + ˜ κDe − ˜ κx ) + a ( e − κx − De − ˜ κx ) − ( χ Ψ xx + bU − ) U − = D (˜ κc κ − ˜ κ − a ) e − ˜ kx − χ Ψ x ( − κe − κx + ˜ κDe − ˜ κx ) − ( χ ( λ Ψ − µu − τ c κ Ψ x ) + bU − ) U − = DA κ e − ˜ kx − χ Ψ x ( − κe − κx + ˜ κDe − ˜ κx ) − ( χλ Ψ − χµu − τ c κ χ Ψ x + bU − ) U − ≥ DA κ e − ˜ kx + χ Ψ x ( κe − κx − ˜ κDe − ˜ κx ) | {z } I + ( − χλ Ψ + τ c κ χ Ψ x − ( b − χµ ) U − ) U − | {z } I . where A κ := ˜ κc κ − ˜ κ − a . Next, observe that since λ c κ > κ , it holds that I = µB λ,c κ ,τ (cid:16) − λ c κ e − λ cκ x Z x −∞ e λ cκ y u ( y ) dy + λ c κ e λ cκ x Z ∞ x e − λ cκ x u ( y ) (cid:17) ( κe − κx − ˜ κDe − ˜ κx ) ≥ − µB λ,c κ ,τ (cid:16) κλ c κ e − ( λ cκ + κ ) x Z x −∞ e λ cκ y u ( y ) dy + ˜ κDλ c κ e ( λ cκ − ˜ κ ) x Z ∞ x e − λ cκ y u ( y ) (cid:17) ≥ − µB λ,c κ ,τ (cid:16) κλ c κ e − ( λ cκ + κ ) x Z x −∞ e λ cκ y e − κy dy + ˜ κDλ c κ e ( λ cκ − ˜ κ ) x Z ∞ x e − λ cκ y e − κy (cid:17) = − µB λ,c κ ,τ (cid:16) κλ c κ λ c κ − κ e − (2 κ − ˜ κ ) x + ˜ κDλ c κ λ c κ + κ e − κx (cid:17) e − ˜ κx and I = χµB λ,c κ ,τ (cid:16) − ( τ c κ + λ ) λ c κ e − λ cκ x Z x −∞ e λ cκ y u ( y ) + ( τ c κ − λ ) λ c κ e λ cκ x Z ∞ x e − λ cκ y u ( y ) dy (cid:17) U − − ( b − χµ )( e − κx − DU − ( x ) e − ˜ κx − De − (˜ κ + κ ) x ) ≥ − χµB λ,c κ ,τ (cid:16) ( τ c κ + λ ) λ c κ e ( λ cκ + κ ) x Z x −∞ e λ cκ y u ( y ) dy + ( τ c κ − λ ) − λ c κ U − ( x ) e λ cκ x Z ∞ x e − λ cκ y u ( y ) dy (cid:17) − ( b − χµ )( e − κx − De − (˜ κ + κ ) x ) ≥ − χµB λ,c κ ,τ (cid:16) ( τ c κ + λ ) λ c κ e ( λ cκ + κ ) x Z x −∞ e λ cκ y u ( y ) dy + ( τ c κ − λ ) − λ c e ( λ c − κ ) x Z ∞ x e − λ c y u ( y ) dy (cid:17) − ( b − χµ )( e − κx − De − (˜ κ + κ ) x ) ≥ − χµB λ,c κ ,τ (cid:16) ( τ c κ + λ ) λ c κ e ( λ c + κ ) x Z x −∞ e λ c y e − κy dy + ( τ c κ − λ ) − λ c κ e ( λ cκ − κ ) x Z ∞ x e − λ cκ y e − κy dy (cid:17) − ( b − χµ )( e − κx − De − (˜ κ + κ ) x )= − χµB λ,c κ ,τ (cid:16) ( τ c κ + λ ) λ c κ λ c κ − κ + ( τ c κ − λ ) − λ c κ λ c κ + κ (cid:17) e − κx − ( b − χµ )( e − κx − De − (˜ κ + κ ) x ) . Thus, with
D >
1, 0 < κ := 2 κ − ˜ κ < κ , and x > x κ,D >
0, it holds that A ( U − ) e − ˜ κx ≥ (cid:16) DA κ − h χµB λ,c,τ (cid:0) ( κ + ( τ c κ + λ )) λ c κ λ c − κ + (˜ κD + ( τ c κ − λ ) + ) λ c κ λ c κ + κ (cid:1) + ( b − χµ ) i e − κ x κ,D (cid:17) . κ = κ + η , we have A κ > e − κ x κ,D = e − ( κ − η ) η ln( D ) = 1 D κ − ηη . Therefore, for 0 < η < min { κ , √ a − κ } , it holds that κ < ˜ κ = κ + η < min { κ, √ a } ,κ − ηη > , andlim D →∞ (cid:16) DA κ − h χµB λ,c,τ (cid:0) ( κ + D ( c + λ )) λ c λ c − κ + (˜ κD + ( c − λ ) + ) λ c λ c + κ (cid:1) + ( b − χµ ) i e − κ x κ,D (cid:17) = ∞ . Therefore, there is D ∗ > D ≥ D ∗ and u ∈ ˜ E . In this section, we prove Theorem 1.1.
Proof of Theorem 1.1. (1) Let ( u ( t, x ; u , c ) , v ( t, x ; u , c )) be defined on [0 , T max ). Note by Propo-sition 1.1 that in order to show that T max = ∞ , it is enough the prove that (1.11) holds. Forevery T ∈ (0 , T max ) let M T := sup ≤ t ≤ T k u ( t, · ; u , c ) k ∞ . With κ = 0 and M = M T , it followsfrom (2.7) that u t ≤ u xx + ( c − χv x ) u x + (cid:16) a + χµ (cid:0) B λ,c,τ ( τ cλ c − λ ) + λ c + 1 (cid:1) M T − bu (cid:17) u, < t < T Hence, by comparison principle for parabolic equations, it holds that k u ( t, · ; u , c ) k ∞ ≤ max n k u k ∞ , a + χµ (cid:16) B λ,c,τ ( τcλ c − λ ) + λ c + 1 (cid:17) M T b o , ∀ t ∈ [0 , T ] . Hence, if M T > k u k ∞ , we must have M T ≤ a + χµ (cid:16) B λ,c,τ ( τcλ c − λ ) + λ c + 1 (cid:17) M T b . By (2.11), ( τ cλ c − λ ) + = 0. Hence M T ≤ ab − χµ . Thus, it holds that M T ≤ max n k u k ∞ , ab − χµ o , < T < T max . Which yield that T max = ∞ , and by Remark 2.1 we conclude that (1.11) holds.(2) We show that (1.12) holds. We follow the ideas of the proof of [31, Theorem 1.8].13et u = lim sup t →∞ k u ( t, · ; u , c ) k ∞ and u := lim inf t →∞ inf x ∈ R u ( t, x ; u , c ) . Since inf x ∈ R u ( x ) >
0, it follows from the arguments of [29, Theorem 1.2 (i) ] that0 < u ≤ u < ∞ . It suffices to prove that u = ¯ u = ab . (3.1)To this end, for every T >
0, let u T := sup t ≥ T sup x ∈ R u ( t, x ; u , c ) and u T := inf t ≥ T inf x ∈ R u ( t, x ; u , c ) . Let L ( u ) = u xx + ( c − χv x ) u x . By (2.10) (with κ = 0), for every t ≥ T and x ∈ R , there holds u t ≤L ( u ) + (cid:16) a − χµB λ,c,τ ( τ cλ c + λ ) e − λ c x Z x −∞ e λ c y u T dy − ( b − χµ ) u (cid:17) u + χµB λ,c,τ (cid:16) ( τ cλ c − λ ) + e λ c x Z ∞ x e − λ c y u T dy − ( τ cλ c − λ ) − e λ c x Z ∞ x e − λ c y u T dy (cid:17) u = L ( u ) + (cid:16) a + χµB λ,c,τ (cid:0) − ( τ c + λλ c ) u T + ( τ c − λλ c ) + u T − ( τ c − λλ c ) − u T (cid:1) − ( b − χµ ) u (cid:17) u. Hence, by comparison principle for parabolic equations, it holds that( b − χµ ) u ≤ a + χµB λ,c,τ (cid:16) − ( τ c + λλ c ) u T + ( τ c − λλ c ) + u T − ( τ c − λλ c ) − u T (cid:17) . Letting T → ∞ , we obtain( b − χµ ) u ≤ a + χµB λ,c,τ (cid:16) − ( τ cλ c + λ ) λ c u + ( τ cλ c − λ ) + λ c u − ( τ cλ c − λ ) − λ c u (cid:17) . (3.2)Similarly, from (2.10) (with κ = 0) it follows for every t ≥ T and x ∈ R that u t ≥L ( u ) + (cid:16) a − χµB λ,c,τ ( τ cλ c + λ ) e − λ c x Z x −∞ e λ c y u T dy − ( b − χµ ) u (cid:17) u + χµB λ,c,τ (cid:16) ( τ cλ c − λ ) + e λ c x Z ∞ x e − λ c y u T dy − ( τ cλ c − λ ) − e λ c x Z ∞ x e − λ c y u T dy (cid:17) u = L ( u ) + (cid:16) a + χµB λ,c,τ (cid:0) − ( τ c + λλ c ) u T + ( τ c − λλ c ) + u T − ( τ c − λλ c ) − u T (cid:1) − ( b − χµ ) u (cid:17) u. Hence, by comparison principle for parabolic equations, it holds that( b − χµ ) u ≥ a + χµB λ,c,τ (cid:16) − ( τ c + λλ c ) u T + ( τ c − λλ c ) + u T − ( τ c − λλ c ) − u T (cid:17) . T → ∞ , we obtain that( b − χµ ) u ≥ a + χµB λ,c,τ (cid:16) − ( τ cλ c + λ ) λ c u + ( τ cλ c − λ ) + λ c u − ( τ cλ c − λ ) − λ c u (cid:17) . (3.3)Since ( τ cλ c − λ ) + = 0 by (2.11), by adding side-by-side inequalities (2.10) and (3.2), we obtain( b − χµ )( u − u ) ≤ χµB λ,c,τ (cid:16) τ cλ c + λλ c + ( λ − τ cλ c ) λ c (cid:17) ( u − u )= χµB λ,c,τ (cid:16) λλ c + λλ c (cid:17) ( u − u ) . By (2.12), B λ,c,τ (cid:16) λλ c + λλ c (cid:17) = 1. Thus, since (H3) holds, we conclude that u = u . By (2.11),(3.2), and (3.3), ( b − χµ ) u = a + χµB λ,c,τ (cid:16) − ( τ cλ c + λ ) λ c u + τ cλ c − λλ c u (cid:17) = a − χµu. This implies (3.1) and (2) thus follows.
In this section, following the techniques developed in [30], we present the proof of Theorem 1.2.Without loss of generality, we assume that N = 1 in (1.1).Through this section we suppose that ( H2 ) holds and 0 < κ < κ ∗ τ . We choose 0 < η < min { κ, √ a − κ } and set ˜ κ = κ + η and M = ab − χµ . We fix a constant D ≥ D ∗ , where D ∗ is givenby Lemma 2.15. Define E := { u ∈ C b ( R ) : U κ,D ≤ u ≤ U κ,M } where U κ,M and U κ,D are given by (2.14) and (2.18) respectively. For every u ∈ E , we let U ( t, x ; u )denote the solution of the parabolic equation ( U t = A u,c κ ( U ) , x ∈ R , t > U (0 , x ) = U κ,M , x ∈ R . (4.1)The following result holds. Lemma 4.1. (i) For every u ∈ ˜ E , the function U ( t, x ) ≡ M satisfies A u,c κ ( U ) ≤ on R × R .(ii) For every u ∈ ˜ E , the function U ( t, x ) = e − κx satisfies A u,c κ ( U ) ≤ on R × R .(iii) For every u ∈ ˜ E , the function U ( t, x ) = U − D , where U − D is given by (2.13) , satisfies A u,c κ ( U ) ≥ on R × ( x κ,D , ∞ ) .(iv) Suppose that (H3) holds. There < δ ≪ such that for every u ∈ ˜ E , the function U ( t, x ) = δ satisfies A u,c κ ( U ) ≥ on R × R .Proof. Using Lemmas 2.2 and 2.3, the results follow.15 roof of Theorem 1.2. (1) Thanks to Lemma 4.1, for D ≫ D ∗ , it follows by comparison principlefor parabolic equations that U ( t , x ; u ) < U ( t , x ; u ) , ∀ x ∈ R , ≤ t < t , ∀ u ∈ ˜ E . Hence the function U ( x ; u ) = lim t →∞ U ( t, x ; u, c κ ) , u ∈ ˜ E is well defined. Moreover, by estimates for parabolic equations, it follows that U xx + ( c κ − Ψ x ( · ; u, c κ )) U x + ( a − χ Ψ xx ( · ; u, c κ ) − bU ) U = 0 , x ∈ R , and U ( · ; u, c κ ) ∈ ˜ E ∀ u ∈ ˜ E . Next we endow ˜ E with the compact open topology. From this point, it follows from the argumentsof the proof of [30, Theorem 4.1] that the function˜ E ∋ u U ( · ; u, c κ ) ∈ ˜ E is compact and continuous. Hence, by the Schauder’s fixed point theorem, it has a fixed point,say u ∗ . Clearly, ( u, v )( t, x ) = ( u ∗ , Ψ( · ; u ∗ , c κ ))( x − c κ t ) is a nontrivial traveling wave solution of(1.1) satisfying (1.16). The proof that lim inf x →−∞ u ∗ ( x ) > (H3) holds, it follows from Lemma 4.1 (iv) that for D ≫ D ∗ , it holds that E ∋ u U ( · ; u, c κ ) ∈ E . Hence lim inf x →−∞ u ∗ ( x ) > . Therefore, by the stability of the positive constant equilibrium established in Theorem 1.1, itfollows that lim x →−∞ u ∗ ( x ) = ab . This completes the proof of Theorem 1.2 (1).(2) Observe that c ∗ ( τ ) = c κ ∗ τ , and, by (1.14), κ ∗ τ = min ( √ a, s λ + τ a (1 − τ ) + ) . This implies that, if λ ≥ a or τ ≥ κ ∗ τ = √ a and then c ∗ ( τ ) = 2 √ a . In the case λ < a and τ < (H4) implies that τ ≥ (cid:0) − λa (cid:1) . This implies that 2 τ a ≥ a − λ a ≤ λ + τ a − τ . Hence we also have κ ∗ τ = √ a and c ∗ ( τ ) = 2 √ a . (2) then follows from (1).(3) Let { c n } n ≥ be a sequence of real numbers satisfying c n > c ∗ ( τ ) and c n → c ∗ ( τ ) as n → ∞ . For each n ≥
1, let ( U c n ,τ ( x ) , V c n ,τ ( x )) denote a traveling wave solution of (1.1) withspeed c n connecting (0 ,
0) and ( ab , aµbλ ) given by Theorem 1.2 (1). For each n ≥
1, since the set { x ∈ R : U c n ,τ ( x ) = a b } is bounded and closed, hence compact, then it has a minimal element,say x n . Next, consider the sequence { U n ( x ) , V n ( x ) } n ≥ defined by( U n ( x ) , V n ( x )) = ( U c n ,τ ( x + x n ) , V c n ,τ ( x + x n )) , ∀ x ∈ R , n ≥ . Then, for every n ≥
1, ( u ( t, x ) , v ( t, x )) = ( U n ( x − c n t ) , V n ( x − c n t )) it a traveling wave solutionof (1.1) with speed c n satisfying U n ( −∞ ) = ab , U n ( ∞ ) = 0 , U n (0) = a b , and U n ( x ) ≥ a b for every x ≤ . Note that k U n k ∞ = k U c n ,τ k ∞ ≤ ab − χµ , ∀ n ≥ . Hence by estimates for parabolic equations, without loss of generality, we may suppose that( U n , V n ) → ( U ∗ , V ∗ ) locally uniformly in C ( R ). Moreover, the function ( U ∗ , V ∗ ) satisfies ( U ∗ xx + ( c ∗ ( τ ) − χV ∗ x ) U ∗ x + ( a − χV ∗ xx − bU ∗ ) U ∗ , x ∈ R V ∗ xx + τ c ∗ ( τ ) V ∗ x − λV ∗ + µU ∗ , x ∈ R , (4.2) k U ∗ k ∞ ≤ ab − χµ , U ∗ (0) = a b , U ∗ ( x ) ≥ a b ∀ x ≤ , and U ∗ ( x ) > , ∀ x ∈ R . (4.3)Hence, since (H3) holds, it follows by the stability of the positive constant equilibrium giving byTheorem 1.1 (2) that lim x →−∞ U ∗ ( x ) = ab . So, in order to complete this proof, it remains to show thatlim sup x →∞ U ∗ ( x ) = 0 . (4.4)Suppose by contradiction that (4.4) does not hold. Whence, there is a sequence { y n } n ≥ with y = 0, y n < y n +1 , y n → ∞ as n → ∞ , andlim n →∞ U ∗ ( y n ) = lim sup x →∞ U ∗ ( x ) > . (4.5)Consider a sequence { z n } n ≥ given by U ∗ ( z n ) = min { U ∗ ( z ) | y n ≤ z ≤ y n +1 } , ∀ n ≥ . Thus lim n →∞ U ∗ ( z n ) = inf x ∈ R U ∗ ( x ) . x ∈ R U ∗ ( x ) = 0, otherwise since (H3) holds, we would have from Theorem 1.1 (2)that U ∗ ( x ) ≡ ab , which contradicts to (4.3). Thus, there is some n ≫ z n is a localminimum point for every n ≥ n , and hence U ∗ xx ( z n ) ≥ U ∗ x ( z n ) = 0 , ∀ n ≥ n . (4.6)By (4.3), k U ∗ k ∞ ≤ ab − χµ , then it follows from the first equation of (4.2), from (2.10) with κ = 0 and M = ab − χµ , that0 ≥ U ∗ xx + ( c ∗ ( τ ) − χV ∗ x ) U ∗ x + (cid:16) a − χµB λ,c ∗ ( τ ) ,τ (cid:0) λλ c ∗ ( τ )1 + λλ c ∗ ( τ )2 (cid:1) ab − χµ − ( b − χµ ) U ∗ (cid:17) U ∗ . Which combined with (2.12) yields,0 ≥ U ∗ xx + ( c ∗ ( τ ) − χV ∗ x ) U ∗ x + (cid:18) a ( b − χµ ) b − χµ − ( b − χµ ) U ∗ (cid:19) U ∗ . (4.7)But lim n →∞ U ∗ ( z n ) = 0 and (4.6) imply that there is n ≫ n such that U ∗ xx ( z n ) ≥ , U ∗ x ( z n ) = 0 , a ( b − χµ ) b − χµ − U ∗ ( z n ) > . This contradicts to (4.7), since U ∗ ( z n ) >
0. Therefore, (4.4) holds.
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