FINITE TRAVELING WAVE SOLUTIONS IN A DEGENERATE CROSS-DIFFUSION MODEL FOR BACTERIAL COLONY. Peng Feng. ZhengFang Zhou. (Communicated by Congming Li)

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1 COMMUNICATIONS ON Website: PURE AND APPLIED ANALYSIS Volme 6, Nmber 4, December 27 pp FINITE TRAVELING WAVE SOLUTIONS IN A DEGENERATE CROSS-DIFFUSION MODEL FOR BACTERIAL COLONY Peng Feng Department of Physical Sciences and Mathematics Florida Glf Coast University Fort Myers, FL , USA ZhengFang Zho Department of Mathematics Michigan State University East Lansing, MI 48824, USA (Commnicated by Congming Li) Abstract. In this paper we stdy the existence of finite traveling wave soltions in a degenerate cross-diffsion system modeling the growth of bacteria colony. The importance of establishing the existence lies in the fact that the analysis of the stability of the wave front provides partial answers to the intriging spatial patterns of the colony. There have been very few reslts on the finite traveling wave soltions of degenerate parabolic system. One reason is that the traditional method often leads to phase plane analysis on higher dimension which is sally a difficlt task. Or method in this paper is based on Schader fixed point theorem and shooting argments.. Introdction. Bacteria grown on the srface of thin agar plates often develops colonies of varios spatial patterns, sch as fractal morphogenesis, dense-branching pattern. Recently Kawasaki et al. [9] proposed a degenerate parabolic system with cross diffsion that captres the qalitative featres of the growth patterns. The model is b = D b {nb b} + nb, () t n = D n 2 n nb (2) t with initial conditions b(x, y, ) = b (x), n(x, y, ) = n. Here b represents the bacteria density and n represents the ntrient density. b (x) is sally a compactly spported fnction and n is a constant. D b and D n are positive constants that represent the diffsivity of bacteria and ntrient respectively. Recently, Maini et al. [2] considered the one-dimensional version of this model with D n =. They stdied the existence and niqeness of traveling wave soltions. They fond that sch soltions exist only for speeds greater than some threshold speed and the wave with the minimm speed has a sharp profile. For speeds greater 2 Mathematics Sbject Classification. Primary: 34B6, 35K65; Secondary: 35K4. Key words and phrases. Degenerate, cross-diffsion, finite traveling wave. 45

2 46 PENG FENG AND ZHENGFANG ZHOU than this minimm speed the waves are smooth. By considering the special case D n =, the athors were able to redce the problem to a phase-space analysis in R 2. In this paper, we consider a more general model and we will not assme D n =. The model we will stdy takes the form b = D b {n p b b} + n q b l, (3) t n = D n 2 n n q b l. (4) t Or method is based on Schader-fixed point theorem. By fixing n in a properly chosen space V, we investigate the existence of traveling wave soltion b. For sch a traveling wave b, we can find a traveling wave soltion ñ which lies in a compact sbset of V. In this way, we may define a continos mapping T : V V with T n = ñ. Invoking Schader-fixed point theorem, we conclde the existence of a traveling wave soltion pair (b, n). A key part of the paper is to investigate the existence of finite traveling wave soltion for a degenerate cross-diffsion eqation. Or method is inspired by a very recent stdy of a similar problem by Malagti []. We will transform the existence of traveling wave soltion b to the solvability of a first-order singlar bondary vale problem which can be done by typical shooting and comparison argment. For more information on singlar ODE, see [, 6, 8,, ]. Or main goal is to prove the following theorem. Theorem.. For q >, p, l >, there exists a constant velocity v sch that the system admits a traveling wave soltion (b(ξ), n(ξ)) where b is a monotone finite traveling wave soltion and n is a monotone classical traveling wave soltion. Here ξ = x vt is the sal wave coordinate and by finite traveling wave we mean ξ = sp{ξ : b(ξ) > } <. In section 2, we simlate the problem in one-dimensional case with special initial data and observe the long time behavior of the soltions. In section 3 we will derive some properties of traveling wave soltions. In section 4 we investigate the existence of finite traveling wave b for a given n. For this part, the reslt is general and we place no restriction on k. In section 5, we derive the eqivalent problem and stdy the existence of a traveling wave n for the determined finite traveling wave soltion b. In section 6 we apply Schader fixed point theorem to dedce the existence of traveling wave soltions (b, n) where b is of finite type. 2. Nmerical simlation to D problem. In this section we consider the one dimensional problem on [, ]: b t = b (nb ) + nb, x x (5) n t = 2 n nb, x2 (6) with Nemann bondary condition and initial data is chosen to be b(x, ) = (x.25)(x.75), n(x, ) =. We nmerically simlate the problem on time interval [,.2] and we have the reslts shown in Figre.The nmerical reslts show that b stays compact spported which

3 DEGENERATE CROSS-DIFFUSION 47 is not nsal in degenerate parabolic eqations. We calclate the density fnctions p to t = 9 to see the long time behaviors. The reslts are shown in Figre. We can see that b tends to a niform state and n tends to asymptotically. For more information on long time behavior of soltions to sch degenerate system, we refer the readers to [4]. b(x,t) n(x,t) Distance x Time t Time t Distance x.2 b(x,t) n(x,t) Distance x 2 4 Time t Distance x 2 4 Time t 6 8 Figre. Bacteria and ntrient density and their long term behaviors. 3. Traveling wave soltion. For the rest of the paper we consider the following system b t = D x (np b b x ) + nq b l, (7) n t = 2 n x 2 nq b l, (8) where (x, t) R R + and D is the rescaled non-dimensionalised diffsion coefficient of bacteria.

4 48 PENG FENG AND ZHENGFANG ZHOU We denote the spatially niform steady states by (b, n) = (b s, ), (, n s ), where n s and b s are some constants. We may assme that initially the ntrient is niformly distribted in the plate and there is no bacteria seed. Hence a proper initial steady state is given by b =, n =, for all < x <. We shall also assme that b x, n as x ±, x for all t >. (9) Let ξ = x vt be the wave coordinates in which b and n solve D(n p bb ) + vb + n q b l =, () n + vn n q b l =, () where denotes the derivative with respect to wave coordinate ξ. Eqations ()- () are to be solved sbject to the following bondary conditions: ahead of the wave and behind the wave n(ξ), b(ξ), < ξ <, b, n as ξ +, (2) b b s, n n s as ξ. (3) 3.. Properties of the traveling wave soltions. We derive several sefl properties of traveling wave soltions in this sbsection. Property. n(ξ) and b(ξ) and n(ξ) > for < ξ <. Proof. Sppose n(ξ), then n = n =. From eqation (), we obtain nb =. Since n, we mst have b. However, this is the trivial soltion and not a traveling wave soltion that we are seeking. Sppose that n(ξ) is a traveling wave soltion and there exists ξ sch that n(ξ ) =. Then since n(ξ) is non-negative, we have that n (ξ ) =. Moreover, for any given b(ξ), eqation () can be regarded as a linear second-order ode for n(ξ), which has no singlar points for any ξ. Ths the initial vale problem has a niqe soltion in (, ). Eqation () together with the above conditions at ξ form an initial vale problem for n, which has the niqe soltion n(ξ). However, we mst have n(ξ) as ξ for a traveling wave soltion. Hence, we conclde that n(ξ) > for all < ξ <. Property 2. n s =, b s = and v >. Proof. To conclde that b s =, we integrate eqation () from to ξ, we have n p bb + v(b b s ) + ξ n q b l =. (4) On the other hand, integrating eqation () from to ξ, we have ξ n q b l = n (ξ) + vn(ξ). (5) Sbstitting expression (5) into eqation (4) and passing ξ to +, we have vb s + v =. (6)

5 DEGENERATE CROSS-DIFFUSION 49 Therefore b s. From eqation (), and letting ξ, we obtain n s b s =, n s = follows from the fact that b s =. Finally, integrating eqation () on the range (, ), we obtain v = n q b l >. Property 3. If n, b are the traveling wave soltions, then n is monotone increasing and b is monotone decreasing if < b <. Proof. The monotonicity of n follows directly from eqation (). In fact, n + vn = n q b l, (7) or eqivalently (n e vξ ). (8) Integrating this ineqality from to ξ, we obtain n e vξ >. Hence n > for all ξ. For a traveling wave soltion b(ξ), we prove that b (ξ) < for < b(ξ) <. Sppose that b (ξ ) = for some ξ, it follows from eqation () that Hence We may now define Dn p bb + D(n p b) b + vb + n q b l =. (9) b (ξ ) = n(ξ ) q b(ξ ) l Dn(ξ ) p <. (2) b(ξ ) ξ = inf{ξ : b (s) > for all s (ξ, ξ ).}. (2) Therefore ξ > implies b (ξ ) = since < b < = b( ) =. On the other hand, from eqation () we conclde that n p bb is positive and strictly decreasing in (ξ, ξ ], which contradicts b (ξ ) = = b (ξ ). Ths if b (ξ) > in some interval (ξ, ξ 2 ) with b (ξ ) =, we necessarily have b(ξ ) = which again contradicts the fact that n p bb is positive and decreasing in (ξ, ξ 2 ). Hence we conclde that b (ξ) < when < b <. 4. Existence of b for a given n. The prpose of this section is to stdy the existence of traveling wave b for any given n. We transform this problem into the solvability of a singlar bondary vale problem and characterize the behavior of b sing the singlar eqation. Let ξ = inf{ξ : b(ξ) < } R { }; ξ 2 = sp{ξ : b(ξ) > } R {+ }. Since b is strictly decreasing on (ξ, ξ 2 ), it follows that the inverse of b(ξ), denoted by ξ = ξ(b) is well defined on (, ) and takes vale in (ξ, ξ 2 ). Therefore we may define n(b) := n(ξ(b)) (22) and (b) := Dn p (b)bb (ξ(b)) < for all b (, ). (23)

6 5 PENG FENG AND ZHENGFANG ZHOU In fact, ξ =. The proof is similar to the proof of property 3 in previos section. Now we are ready to derive the corresponding singlar bondary vale problem. 4.. First-order singlar bondary vale problem. Let n(b) V where V is the closed convex set of the Banach space C ([, ]) defined by V := {n(b) C [, ], n, lim sp b n(b) b L, n(b) ( b)} L where L is a sfficiently large constant and will be chosen later. The reason why we define sch a space will become clear in the paper. Differentiating both sides of eqation (23) with respect to ξ we have (b)b (ξ) = Dn p bb + (Dn p b) b = vb n q b l. (24) Therefore as a fnction of b satisfies (b) = v Dnp+q (b)b +l, b (, ). (25) Or next step is to derive the bondary condition at and. Lemma 4.. ( + ) = ( ) =. Proof. If ξ 2 = +, i.e., b is a classical traveling wave, it follows that ( + ) = lim ξ + Dnp (b)bb (ξ) =. If ξ 2 < +, i.e., b is a finite traveling wave soltion. Owing to the invariant property of a traveling wave soltion nder translation, we may assme that it vanishes at ξ =. This allows b to become singlar near ξ =. As is well known([2]), at sch a reglar singlar point, one expects that as ξ Sbstitting this expression into b(ξ) A( ξ) α. (26) Dn p bb + D(n p b) b + vb + n q b l = where the derivative is with respect to ξ, we obtain Dn p A 2 [α(α ) + α 2 ]( ξ) 2α 2 + Dpn p n ( A 2 α)( ξ) 2α vaα( ξ) α + n q A l ( ξ) αl =. (27) Eqating the dominated terms at ξ = we have 2α 2 = α, Dn p A 2 [α(α ) + α 2 ] vaα = where n = lim b n(b) >. Hence we have From the definition of, we have α =, (28) A = v. Dn p (29) ( + ) = lim b + Dnp (b)bb (ξ) = lim ξ + Dnp A( ξ) ( Aα)( ξ) =. (3) ( ) = follows from the definition.

7 DEGENERATE CROSS-DIFFUSION 5 Therefore solves the following singlar bondary vale problem: (b) = v Dnp+q (b)b +l for b (, ), (3) sbject to ( + ) = ( ) =. (32) We shall now consider the solvability of the following singlar problem. Problem P. Find a pair (v, ) with v > sch that (b) = v Dnp+q (b)b +l for b (, ), (33) ( + ) = ( ) =, (34) < in (, ). (35) Here n(b) V where V is as defined before Existence of a critical velocity. In this section we show that P is solvable with a niqe negative soltion = (b) if and only if v v for some positive v which depends on the choice of n(b). We first prove the following lemma Lemma 4.2. If there exists φ C (, ) sch that φ (b) > v Dnp+q (b)b +l φ(b) for b (, ), (36) sch that φ( + ) = and φ(b) < for b (, ). Then P is solvable and the soltion > (b) > φ(b). Proof. The proof is based on shooting argment and some comparison techniqes. First, for a fixed constant b (, ), note φ(b ) <, let α [φ(b ), ), we consider the following initial vale problem: = v Db+l n p+q (b), < b <, (b ) = α φ(b ), and let represent the niqe soltion of it. We first claim that (b) < on (, b ). Sppose otherwise there is a b (, b ) satisfying (b) < for all b (b, b ] and lim (b) =. b b + Since lim ( v (b,) (b,) Db+l n p+q (b)) = D(b ) +l n p+q (b ) <, hence there exists a constant c > sch that = v Db+l n p+q (b) > c (b) > for all b < b < b + c, which wold imply that > for b < b < b + c, clearly a contradiction. Next we assert if α > φ(b ), then Since φ(b ) < α = (b ), we may define φ(b) < (b) < for < b b. b = inf{b : φ(s) < (s) for s (b, b )}

8 52 PENG FENG AND ZHENGFANG ZHOU Sppose b >, then (b) = φ(b) and (b) = v Db+l n p+q (b) (b) = v Db+l n p+q (b) φ(b) < φ (b), (37) which is clearly absrd. In view of the argments above, we have proved that is well defined on (, b ). Now we may define (, b α ) to be the maximal existence interval for soltion. The aim is to show that b α = for some α [φ(b ), ). Let and 2 be two distinct soltions corresponding to initial vale α and α 2 respectively. Sppose for definiteness that α < α 2, then hence < 2 in (, min{b α, b α2 }), b α b α2. We now claim that if the α is sfficiently small, then b α < and (b α ) =. The trick is to constrct an pper soltion and se a similar comparison argment as in the earlier part of the proof of this lemma. Since lim v (b,) (b Db+l n p+q (b) <,,) there exists a sfficiently small constant M > and λ < 2M sch that v Db+l n p+q (b) > M, for all λ < < and b b < b + λ. Let α > λ and define for b b b + α2 2M We have ψ = α 2 2M(b b ) which solves the following initial problem: ψ = M ψ, ψ(b ) = α. b + α2 2M < b + λ <. Moreover, (b ) >, hence (b) > (b ) = α > λ in a right neighborhood of b. Pt I = [b, min{b + α2 2M, b α}], we dedce that (b) > λ for all b I. We apply a similar comparison argment as before to conclde (b) > ψ(b) for all b (b, b α ) (b, b + α2 2M ). Since φ(b + α2 2M ) =, we have b α b + α2 2M <. Now we let α = inf{α (φ(b ), ) : b α < }, then b α =, therefore the corresponding soltion is defined and negative on (, ) and > φ in (, ) and ( + ) =. This completes the proof.

9 DEGENERATE CROSS-DIFFUSION 53 We are now in position to prove the following solvability reslt for Problem P. Theorem 4.3. There exists v >, sch that for all v > v, Problem P has a niqe negative soltion. Proof. We first show P is solvable for v sfficiently large. To this aim, we let which is well defined. Let Then, for v > 2 c, we have v Db+l n p+q (b) φ(b) Dn p+q (s)s +l c = sp s (,), (38) s φ(b) = cb. < 2 c + Db+l n p+q (b) cb 2 c + c = φ (b) (39) for all b (, ). Hence φ(b) satisfies condition of Lemma 4.2. Therefore P is solvable for every v > 2 c. We now show that P is not solvable for v =. Otherwise, let solves = Db+l n p+q (b) defined on some interval (a, ) with < a < and (b) < for all b (a, ). Integrating the eqation above in [b, b] with a < b < b <, we obtain 2 ( b) = 2 (b) 2 Therefore, if ( ) =, we have (b) = 2 b b which implies ( + ) <, a contradiction. We now let v = inf{v : P is solvable} b Ds +l n p+q (s)ds. (4) Ds +l n p+q (s)ds (4) which is well defined and v > based on the observation above. We prove that for every v > v, P is solvable. Given v > v, take v sch that P is solvable with v < v and the niqe soltion ū for v. Since ū = v Db+l n p+q (b) > v Db+l n p+q (b), ū ū hence ū satisfies condition of Lemma 4.2. Therefore, we conclde the solvability of P for v > v. Finally we prove that P admits at most one soltion. Sppose for contradiction that and 2 are two distinct soltions of P. For definiteness, we assme it follows that (b ) > 2 (b ) for b (, ), (b ) 2(b ) = Db+l n p+q (b ) (b ) + Db+l n p+q (b ) >. (42) 2 (b ) Hence if (b ) > 2 (b ), then (b ) > 2(b ). Therefore, it is impossible that ( ) = 2 ( ) =.

10 54 PENG FENG AND ZHENGFANG ZHOU This completes the proof Finite traveling wave at v. The main goal of this section is to show that b is a finite traveling wave at the minimm wave speed. We will also show that b is a classical traveling wave if v > v. The idea is to characterize the type of b by the vale of ( + ). Lemma 4.4. ( + ) = or ( + ) = v. Proof. We first show that if (b ) = for some sfficiently small b then (b ) >. To this aim, we write it follows that Note that g(b) = Db +l n p+q (b), (b) = g(b) (b) 2 g (b). g (b) = Db l n p+q [2n(b) + (p + q)bn (b)] > for b sfficiently small. Hence if there exists b sfficiently small sch that (b ) =, then (b ) = g (b ) >. Therefore we conclde that there exists < b < b sch that (b) > or (b) < on (, b). Case : (b) > on (, b). In this case ( + ) < exists and it follows from ( + v) b = Dbl n p+q (b) that ( + ) = v. Case 2: (b) < on (, b). In this case, v < ( + ), similarly we obtain ( + ) =. Lemma 4.5. b is a finite traveling wave if and only if ( + ) = v. Proof. If b is finite traveling wave, then from Lemma 4., as ξ, we have b(ξ) Therefore by eqation (24) we have ( + ) = lim ( v nq b l ) = v + lim b + On the other hand, if ( + ) = v, we have b v ( ξ). (43) Dn p ξ + nq ( v ) l ( ξ) l Dn p = v. (44) lim b (b) b (ξ) = lim ξ ξ b + b Dbn p = lim v b + Dn p = v. (45) Dn p 2 This implies that b is a finite traveling wave soltion. Lemma 4.6. For v sfficiently large, b(ξ) > for all ξ R. In other words, b(ξ) is a classical traveling wave soltion.

11 DEGENERATE CROSS-DIFFUSION 55 Proof. Note that for v sfficiently large, there exists λ > sch that v + D λ ( + l)λ. (46) We claim that (b) λb +l, (47) for all b [, ]. Sppose ( b) < λ b +l for some b (, ), then Hence ( b) = v D b +l n p+q ( b) ( b) < v + D b +l n p+q ( b) λ b +l < v + D λ λ( + l) λ( + l) b l. (48) (b) + λb +l < ( b) + λ b +l < for all b > b (49) which contradicts ( ) =. Now for any ξ (, ξ 2 ), let b(ξ ) = b, we have ξ ξ 2 = b ξ (b)db = b Dn p (b)b db (b) λ b Dn p (b) b l db =. (5) Hence ξ 2 = + which implies that b is a classical traveling wave soltion. applied the fact that l here. Now or goal is to show that when v = v, b is a finite traveling wave soltion. In view of Lemma 4.5, we shall show that when v = v, the soltion of = v Db+l n p+q (b), ( + ) = ( ) =, satisfies ( + ) = v. The trick is to constrct a converging seqence of soltions which satisfy this property. To show this, we define a continos fnction g n (b) on [, ] as: g n (b) = It holds that on [, n + 2 We ] [ ], g n Db +l n p+q (b) for b n + 2, n + [ ] and g n (b) = Db +l n p+q (b) on n +,. g n g n+. We consider the following problem: To find a fnction n : (, ) R and v n > sch that n = v n g n(b) n, n ( + ) = n ( ) =. We emphasize here that v n is also an nknown of the problem. We shall prove the following theorem. Theorem 4.7. There exists a niqe soltion n : (, ) R and v n of the problem. is of class C.

12 56 PENG FENG AND ZHENGFANG ZHOU Proof. The problem is eqivalent to n = v n g n n n = v n b on [ ] on n + 2,, (5) [ ],, (52) n + 2 n( n + 2 ) = v n, (53) n ( ) =. (54) Note that the soltion of this eqivalent problem is of C since ( n+2 ) = v n. We apply the shooting argment and comparison techniqes on [ n+2, ] similar as in Section 3 to the problem: n = v n g n on [, ], n n + 2 (55) n ( n + 2 ) = v n n + 2, (56) n( n + 2 ) = v n. (57) We can show that there exists a niqe v n sch that n ( ) =. We now examine several properties of n. Some of them will be applied in the proof of the next lemma. The proof is all the similar shooting and comparison argment. We shall briefly show the proof of the second one. Lemma 4.8. n satisfies: p n (b) = v n b on [, n+2 ]. p2 n (b) v b for all b [, ]. p3 n n+ for all b [, ]. Proof. p2 Since n (b) = v n (b) > v b on (, n+2 ], hence if b sch that n ( b) < v b for some n, b mst be in ( n+2, ]. Frthermore, b ( n+2, b) sch that n ( b) = v b and n( b) < v. However, which is a contradiction. n( b) = v n g n( b) n ( b) > v g n( b) v b > v, Now we will prove that v n determined in the theorem above has the following properties. Lemma 4.9. {v n } is monotone increasing and lim n + v n = v. Proof. First we show that {v n } is monotone increasing. Sppose v n > v n+ for some n. Then, n ( n + 2 ) = v n n + 2 < v n+ n + 2 < n+( n + 2 ). We also have n( n + 2 ) = v n < v n+ g n+( n+2 ) n+ ( n+2 ) = n+( n + 2 )

13 DEGENERATE CROSS-DIFFUSION 57 since g n+ ( n + 2 ) >. Hence n (b) < n+ (b) in a right neighborhood of n+2, therefore in this neighborhood n(b) = v n g n < v n+ g n+ = n n+(b). n+ Hence n (b) n+ (b) < n ( n + 2 ) n+( n + 2 ) < for all b [ n+2, ]. This is a contradiction with n ( ) = n+ ( ) =. Hence v n v n+. We now prove that v n v. Sppose v n > v, then n ( n + 2 ) = v n n + 2 < v n + 2 < ( n + 2 ), where we have applied the fact that > v for all b (, ). We also have n( n + 2 ) = v n < v < ( n + 2 ). Hence n (b) < (b) < in a right neighborhood of n+2. For every b in this neighborhood, we have n(b) = v n g n < v g n n v Db+l n p+q (b) = (b). Hence we conclde that [ ] n (b) < (b) for all b n + 2, which contradicts with n ( ) = ( ) =. Finally, we show lim n + v n = v. Let v = lim sp v n v. n + Note that we may define (b) = lim n + n (b), we have (b) ( n + ) = ( ) lim n (b) n ( n + n + ) = b lim v n g b n = n + n+ n Hence solves = lim n + b n+ n v Db+l n p+q (b). (58) = v Db+l n p+q (b) on (, ). Moreover, we have ( + ) = and ( ) =. By the definition of v we conclde that v v. Hence v v. Note that (b) = lim n + n(b) v b for b [, ). Applying monotone convergence theorem to { n } we can show that solves = v Db+l n p+q (b) in (, ),

14 58 PENG FENG AND ZHENGFANG ZHOU ( + ) =, ( ) =. Hence v b ( ] n b = v n for b,. n + 2 Therefore () = v. This shows that the traveling wave at minimm wave speed v is of finite type. We conclde this section with a few properties of the negative soltion. Lemma 4.. Let = (b) be a negative soltion of Problem P. It holds that ( ) =. (b) (b) Proof. Let (b) be a soltion of P. Let M := lim sp and m := lim inf b b b b. There are three possible cases: either m = M > or M > m and M = m =. We show the first two are impossible. In fact, if m = M >, then lim Dn p+q b +l b (b) = v lim = v <, b b b (b) n(b) a contradiction. Here we applied the fact that lim sp b b L. If M > m. Then for any given m (m, M) there exists a seqence {b n } which converges to sch that ( ) (b n ) (b) b n = m and. b b=b n Hence, lim n + (b n ) = lim v Dn p+q (b n )b +l n = v <, n + m b n therefore we may assme (b n ) < for every n. On the other hand, ( ) (b) = [ (b n ) (b ] n) = b b=b n b n b n b n [ (b n ) m ] >. This contradicts to previos statement that, i.e., there exists ( ) =. Using this lemma, we can frther show that ( (b) b ) b=b n. Therefore, M = m = Lemma 4.. For the soltion of Problem P, there exists C <, C 2 < sch that C 2 ( b) p+q (b) C ( b) p+q for b sfficiently close to. Proof. We only need to show that it is impossible to find negative constants C, C 2 sch that (b) C ( b) p+q+γ or (b) C 2 ( b) p+q γ for any γ > as b. Otherwise, we either have (b) = v Dnp+q b +l (b) v Dnp+q b +l C ( b) p+q+γ v D(/L)p+q ( b) p+q b +l C ( b) p+q+γ > (59)

15 DEGENERATE CROSS-DIFFUSION 59 vb (b) C 2 ( p) p+q b Figre 2. Sketch of for b sfficiently close to. A contradiction to Lemma 4.. Or we have (b) v DLp+q ( b) p+q b +l v C ( b) p+q γ as b. A contradiction to Lemma 4. again. In fact, we can improve this reslt in the next lemma. Lemma 4.2. There exists C 2 < sch that (b) C 2 ( b) p+q for all b [, ]. Proof. Sppose that there exists b [, ] sch that ( b) < C 2 ( b) p+q. Then we have ( b) = v Dnp+q ( b) b +l v DLp+q +l ( b) p+q b ( b) C 2 ( b) p+q < C 2 (p + q)( for C 2 sfficiently large so that DLp+q b +l b) p+q v < C 2 (p + q)( C b) p+q. 2 Ths (b) C 2 ( b) p+q < ( b) C 2 ( b) p+q < for every b > b. A contradiction to ( ) =. This ends the proof. This reslt is illstrated in Figre The eqivalent problem. We have shown that for any n V, there exists a v which depends on the choice of n sch that b is a finite traveling wave, we may assme that b(ξ) = for ξ. (6) We shall simplify the problem by redcing it to a system in the interval ξ < only. Note that n satisfies n + vn = for ξ. (6) We have the following lemma:

16 6 PENG FENG AND ZHENGFANG ZHOU Lemma 5.. The problem n + vn = in (, + ), (62) n() = n with < n <, (63) n () = v( n ), (64) has a niqe bonded soltion that satisfies lim ξ + n(ξ) =. Proof. We consider the following ODE system: n = p, p = vp. A phase plane analysis shows that every trajectory can intersect the n axis at most once. Hence p changes sign at most once, and conseqently n(+ ) exists. Let n(+ ) = c, a direct integration shows that hence Therefore + n = + vn, v( n ) = vc + vn. lim n(ξ) = c =. ξ + In view of this lemma, we may reformlate the problem into: Problem P 2. Find (v, b, n) with v > sch that D(n p b) + vb + n q b l =, ξ <, (65) n + vn n q b l =, ξ <, (66) n () = v( n()), n( ) =, (67) b( ) =, b() =. (68) We have shown that as ξ varies from to, b = b(ξ) decreases monotonically from to. We can therefore define ξ = ξ(b) as the inverse fnction of b(ξ) where b varies from to and ξ takes vale in (, ). As before, we define (b) = Dn p bb (ξ(b)) < for all b (, ) (69) and n(b) = n(ξ(b)). (7) Since d dξ = d b (ξ) db = (b) Dn p b d db, we can transform problem P 2 into the following eqivalent problem: Problem P 3. Find (v,, n) that solves = v Db+l n p+q (b), (7) Dbn p ( Dbn p n ) + v Dbn p n n q b l =, (72) ( + ) = ( ) =, (73)

17 DEGENERATE CROSS-DIFFUSION 6 n ( + Dbn p ) = v( n()) lim b + (b) = ( n())dn()p, n( ) =, (74) we shall also need < in (, ) (75) Dbn p db =. (76) (b) Remark. The first eqality in eqation (74) is eqivalent to the first eqality of eqation (67). 6. A Fixed Point. Given n(b) V, let (v, ) be the niqe soltion of Problem P sch that ( + ) = v, i.e., the corresponding b is a finite traveling wave. Consider the following: Problem P 4. Find ñ(b) sch that ( Dbn p ñ ) + vñ Dnp ñ q b l+ = in (, ), (77) ñ + D( ñ)n p = at b =, (78) ñ() =. (79) We shall show that Problem P 4 admits a niqe soltion that is also in V. To prove this fact, we begin with the following local existence reslt. Lemma 6.. ( Dbn p ñ ) + vñ Dnp ñ q b l+ = in (, ), (8) ñ() = n, < n <, (8) ñ () = D(n )n p, (82) ñ(b) > for b > and close to. (83) has a local soltion. Here n = lim b n(b). Proof. Take E = {ñ(b) ñ(b) C([, b ]), ñ n for b b, and some small b > to be determined later}. Then E is a closed convex set in C([, b ]). Before proceeding to the next step, it is helpfl to rewrite eqation (8) as an eqivalent integral eqation. Assming that n is a smooth soltion on the interval (, b ] with b, we integrate once to get Since (s) Dbn p ñ lim (s)ñ s + Dsn p + vñ(b) vn = lim s + (s) Dsn p (s)ñ = we have Dbn p ñ + vñ(b) v = A second integration yields ñ(b) = n + b Dsn p (s) v( ñ(s))ds + (s) b v Dn p D(n )n p, b b Dn p ñ q s +l. (84) (s) Dn p ñ q s +l. (85) (s) Dsn p (s) s Dn p ñ q τ +l dτ. (86) (τ)

18 62 PENG FENG AND ZHENGFANG ZHOU For small b and b b, we have ñ n < +, ñ (b) = Dbnp (b) v( ñ(b)) + Dbnp Dn p ñ q τ +l dτ C < +. (b) (b) (τ) This shows the operator defined above is a compact operator. Therefore it has a fixed point in E, i.e., it has a local soltion. To establish the existence and niqeness of soltion to P 4, we shall also need the following lemmas: Lemma 6.2. If ñ() = n is sfficiently close to, then ñ(). If ñ() = n is sfficiently close to, then ñ(b) = for some < b <. Proof. We prove this lemma by contradiction. We start with the following eqation: [ ] b ñ + vdbnp Dn p ñ q s l+ Dbn p ñ = v + (s). (87) Let h(b) = Dbnp, which is clearly nonpositive. The above eqation can be rewritten as [ ] b (e b vh(s)ds ñ) = e b vh(s)ds h(b) v + b h(s)ñ q s l ds. (88) Let H(b) = e b vh(s)ds, and J(b) = v + b h(s)ñq s l ds. Integrating both sides, we have b H(b)ñ ñ() = v H (s)j(s)ds = v H(b)J(b) v H()J() b = v H(b)J(b) b v H(s)h(s)ñq (s)s l ds v H(s)h(s)ñq (s)s l ds. (89) Sppose that n is sfficiently close to bt ñ(b) = for some < b <. Then the left side of the above eqation is sfficiently close to -. Bt since H(b) >, J(b) = Dbn ñ +vñ(b) > and b p v H(s)h(s)ñq (s)s l ds <. Ths the eqality above is impossible. Hence we conclde that ñ() for n sfficiently close to. To prove the second statement, we note that for l >. Since Dbn p ñ + v () = lim = lim Dbl n p+q b b b [ ( ) = Dbn p [ = Dbn p ] + v ñ + Dnp ñ q b +l Db 2 n p pnp n Dbn 2p + v and lim b Dbn p Db 2 n p = () =. 2Dn p Ths if p =, l >, we have ñ () <. = ] ñ + Dnp ñ q b +l, (9)

19 DEGENERATE CROSS-DIFFUSION 63 Note that there exists δ >, sch that (v + Db ) Db 2 > v and 2 We claim that ñ < on{b [, δ] : ñ > }. If not, δ (, δ) sch that Ths if we let we have I(δ ) =. On the other hand, v 2 < < 2v on [, δ]. b ñ (δ ) =, ñ (b) < on (, δ ). I(b) = Dbñ [ = Db ] Db 2 + v ñ + Dñq b +l, This ends the proof of the lemma. I > v 2 ñ () Dñq () v/2 δl > for ñ() <<. Lemma 6.3. Let ñ, ñ 2 be the soltions corresponding two different initial vale n and n 2 with n > n 2, then n > n 2 on [, min{t (n ), T (n 2 )}] where T (n i ) represent the corresponding maximal existence interval in the sense that ñ(t ) =. Proof. Sppose otherwise there is a b [, min{t (n ), T (n 2 )}] sch that ñ (b ) = ñ 2 (b ) and ñ > ñ 2 on [, b ), then ñ (b ) < ñ 2(b ). We shold have while (b ) )ñ Db n p (b (b ) + vñ (b ) v > (b ) )ñ Db n p (b 2(b ) + vñ 2 (b ) v, (9) b Dn p ñ q s+l (s) which is clearly a contradiction to (85). < b Dn p ñ q 2 s+l, (92) (s) In view of the above lemmas, we have established the following theorem Theorem 6.4. Problem P 4 admits a niqe soltion.the soltion ñ satisfies the following property: ñ, ñ ñ ( b), lim sp L b b L, M ñ. Proof. Since eqation (77) is only degenerate at b =. The bondedness of ñ on [, ) follows from standard ODE theory. We shall stdy the bondedness at b =. To this prpose, we let n(b) ( b) α with α as b. Let ñ(b) ( b) β as b. In view of eqation (77), we have Ths as b, ñ satisfies Dnp+q v ( nq as b. v ñ ) + vñ + v ñq =. (93) nq

20 64 PENG FENG AND ZHENGFANG ZHOU Sppose that β <, matching the leading singlar terms in the eqation above, we have or β = q(β α), β = αq q. This is clearly impossible since α. Ths we conclde β, or eqivalently, ñ () exits. In fact, by matching the coefficients, one can show that lim sp b ñ b L. Let W = Dbn ñ and differentiate (72), we get p ( Dbn p W ) + vw qñ q ñ b l lñ q b l =. (94) Also W () >, W () =. The maximm principle yields W > in (, ), i.e., ñ < in (, ). Note that to dedce W () =, we have applied the fact that ñ () is bonded. By Lemma 6.2, we have the bond on ñ(). The niqeness follows from Lemma 6.3. This ends the proof of this theorem. We shall now combine Lemma 6.3 with Theorem 2.4. For every n V, we define (v, ) to be the soltion of P and ñ by Theorem 2.4, and introdce the mapping T by T n = ñ. Clearly, T maps V into itself, and its image lies in a compact sbset of V (since M ñ ). By the niqeness part in Lemma 2.5, it also follows that T is continos. Invoking the Schader fixed point theorem, we conclde that there exists at least one fixed point for T. We shall denote it by (v,, n). To show that the corresponding (v, b, n) is a soltion to problem P 2, we shall only need to show eqality (76). In fact, we do have the following conclsion. Lemma 6.5. Dbn p db =. (b) Proof. We shall start with the following facts. There exists a constant L > sch that n(b) /L( b) and a constant C 2 < sch that (b) C 2 ( b) p+q where (, n) is the pair in the fixed point stated above. The first fact is trivial. The second fact follows from Lemma 4.2. Ths Dbn p (b) db Remark 2. This holds for any q. Db(/L) p ( b) p C 2 ( b) p+q =. Acknowledgement. The athors wish to thank the referee for many valable sggestions which improved the presentation of this paper.

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