für Mathematik in den Naturwissenschaften Leipzig

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1 ŠܹÈÐ Ò ¹ÁÒ Ø ØÙØ für Mathematik in den Naturwissenschaften Leipzig Simultaneous Finite Time Blow-up in a Two-Species Model for Chemotaxis by Elio Eduardo Espejo Arenas, Angela Stevens, and Juan J.L. Velazquez Preprint no.: 9 9

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3 Simultaneous Finite Time Blow-up in a Two-Species Model for Chemotaxis E. E. Espejo Arenas, A. Stevens y, J. J. L. Velazquez z In honor of Erhard Heinz, on the occasion of his 85th birthday. Abstract A system of two classical chemotaxis equations, coupled with an elliptic equation for an attractive chemical, is analyzed. Depending on the parameter values for the three respective diusion coecients and the two chemotactic sensitivities in the radial symmetric setting, conditions are given for global existence of solutions and nite time blow-up. A question of interest is, whether there exist parameter regimes, where the two chemotactic species dier in their long time behavior. This questions arises in the context of dierential chemotactic behavior in early aggregates of Dictyostelium discoideum Introduction In this paper we address the question if dierential chemotaxis can serve as the main mechanism for cell sorting in Dictyostelium discoideum during self-organisation under starvation conditions. In laboratory experiments [], cell sorting was observed in the early aggregation stages during mound formation. Cells moving in the top of the aggregates show stronger chemotactic abilities when being observed separately, than the cells moving at the bottom. As a rst mathematical ansatz to approach this question on cell sorting, a model of two chemotactic species is considered, both of which react to the same chemical, camp, but with dierent chemotactic strength. Institute for Cell Dynamics and Biotechnology ICDB, Universidad de Chile, Facultad de Ciencias Fisicas y Matematicas, Beauchef 86-Santiago-Chile (eespejo@ing.uchile.cl) y University of Heidelberg, Applied Mathematics and BioQuant, INF 67, D-69 Heidelberg, Germany (angela.stevens@uni-hd.de) z ICMAT (CSIC-UAM-UC3M-UCM), Facultad de Ciencias Matematicas, Universidad Complutense, Madrid 84, Spain (velazque@mat.ucm.es)

4 So we are interested to understand the qualitative behavior of the following t u = u r (u t u = u r (u rv); (.) = v + u + u ; in B R (): t > ; with ; ; > @ = ; R (); t > ; (.3) u (; x) = u ; (x) ; u (; x) = u ; (x) ; x B R () ; (.4) where B R () is the two-dimensional ball of radius R, which is centered in the origin, u ; ; u ; are not identically zero, and u and u are the density variables for the two cell types with dierent chemotactic sensitivities, and v is the chemo-attractant. Chemotaxis models of Keller-Segel type with several chemotactic populations reacting to several chemicals have be studied in []. By using energy methods sucient conditions for global existence of solutions, especially for the case we are interested in, but with Dirichlet boundary conditions, were derived. In [4] the dynamics of two chemotactic species are considered, which are modeled by two classical chemotaxis equations. They are interacting via two chemicals. One of the chemotactic populations is attracted by one chemical and repelled by the other. For the second population this is the other way around. The chemicals are modeled by elliptic equations. Under a certain symmetry condition for the ratio of the attractive and repulsive eects, the full model is numerically analyzed and solutions for the stationary system are described. The stationary system could be simplied by this symmetry assumption. A question of interest in the context of our paper is, if the two chemotactic species in (.) can separate. One possibility could be that for one species blow-up happens, whereas the solution for the other species is still bounded. We will show that this is not the case. If blow-up happens, it is simultaneous for both chemotactic species. Specically, it will be proved that blow-up for one chemotactic species implies blowup for the other one. The dierence between the species though is the following. For some parameter ranges of the chemotactic sensitivities and the diusion coecients, but probably not for all of them, dirac-mass formation in one of the chemotactic species can be observed, whereas the blow-up asymptotics for the other species grow like an integrable power law. This means that for the rst species mass aggregation happens, whereas for the second species no accumulation of mass at the origin takes place, so we obtain blowup without mass aggregation.

5 The outline of this paper is as follows. First, in Section we present a dimensional analysis of the model. Then, in Section 3, we recall several results on sucient conditions for local and global existence of solutions in multi-component chemotaxis systems. Also we give sucient conditions for nite time blow-up in the radial symmetric situation. In Section 4 we prove that blow-up for one of the chemotactic species implies also blow-up for the other one at the same time. In the last section we derive formal asymptotics of the blow-up proles for a certain range of parameters of our system. Dimensional analysis The original model setup t u = u r(u rv) t u = u r(u rv) t v = Dv + u + u v in B R (); t > (.3) with Neumann boundary conditions. We will restrict our attention to the case ; ;. Moreover, we will assume that the concentrations u ; u are of the same order of magnitude. However, many other interesting limit cases would arise without posing these assumptions. Suppose that D >>, since molecular diusion can be expected to be much faster than cell diusion. The characteristic time scale for diusion of the chemical in B R () then is t chem = R ; and the characteristic time scale for cell diusion is given D by t cell = R. Thus t chem << t cell. We are interested in the regime in which the diusive and chemotactic eects in (.), (.) are of the same order of magnitude. If v denotes the typical spatial variation of the concentration of the chemical, and kuk denotes a typical measure of the order of magnitude of the cell concentrations u i, and if we assume that the initial cell concentrations change in scales of the order of magnitude of R, we then need kuk R kuk v R ; whence v : On the other hand (.3) yields v R kuk D ; so R D kuk : 3

6 Notice that kuk N cells R, where N cells is the number of cells. In the regime in which the Keller-Segel model yields interesting critical dynamics, if the diusive and chemotactic terms are balanced, we must have that N cells is of order one. Therefore, we will consider the limit D : (.4) To non-dimensionalize the system we introduce the following new variables t = t cell t ; x = Rx ; u i = D R u i ; v v base = R D v = v ; D R where v base = R We further assume R ( u + u )dx is the base level of the chemo-attractor density. i = i ; i = i ; i = i ; for i = ; ; with i ; i ; i all being of order one. The original system of equations then becomes Therefore t x u r x (u r x v ) ; x u r x (u r x v ) ; = R x v + ( u + ) D u R (v base + v ) : t chem t = x v + u + ) t chem u v base t chem t degrad v ; where t degrad = with So is the characteristic time for the degradation of the chemical and v base v base = v base t ; chem B () ( u + u )dx : (.7) t chem = x v + ( u + u ) v base t chem t degrad v : (.8) 4

7 By assumption all coecients and functions in (.5) and (.6) are of order one. On the other hand, (.8) might be simplied under the assumptions t chem << t cell and t chem << t degrad : (.9) If the conditions (.4), (.9) are satised, then the original system (.)-(.3) reduces formally to (.5), (.6) and with v base given as in (.7). = x v + ( u + u ) v base (.) Actually this limit is similar to the limit considered by Jager and Luckhaus in [7]. Of course there are other regimes yielding interesting limits, for instance t chem << t cell ; t chem t degrad, but these will not be considered in this paper. Notice that now (.5)-(.7), (.) is solved for x = B () ; t R +. It is possible to reduce (.5)-(.7), (.) to a problem containing a minimal number of parameters. To this end we dene Therefore U i = i u i ; i = ; T = t ; V = v ; X = = XU r X (U r X V ) ; X B () ; = X U r X (U r X V ) ; X B () ; T > = X V + (U + U ) B () (U + U ) dx ; X B () ; T > Renaming ; and as ; ; respectively, and T by t we nally = X U r X (U r X V ) ; X B () ; t = X U r X (U r X V ) ; X B () ; t > = X V + (U + U ) with X B () (U + U ) dx ; X B () ; t > X = ; () ; t > (.4) 5

8 Later we will set the last term on the right hand side of (.3) equal to. In case of radially symmetric solutions it is more convenient to rewrite our problem in terms of the rescaled mass functions r M (t; r) = M (t; r) = r U (t; ) d (.5) U (t; ) d ; r = jxj : (.6) As a consequence = M + M c r r ; (.7) where c = RB() (U (; X) + U (; X)) dx. Now (.)-(.4) cr M + ; cr M + ; (.9) with boundary conditions M (t; ) = U (; ) d =: m ; M (t; ) = In the following we will use u ; u ; v instead of U ; U ; V. U (; ) d =: m (.) 3 Global Solutions and Blow-up Here we formulate some standard solvability conditions for system (.) together with (.), (.4), and boundary conditions. First, notice that the function v is not uniquely dened by the given equations, since a solution (u ; u ; v) automatically implies a solution (u ; u ; v + g(t)) for any time dependent real function g(t). The precise normalization of v(t; x) though is not relevant for the problem we are considering, since all estimates are done in terms of rv, which is unique. More precisely, v does not represent the chemical attractor itself, but the dierence of chemical concentration of the attractor with respect to its basal concentration. Let B() v(t; x)dx = (t) (3.) 6

9 be a normalization condition for this dierence, where (t) can be an arbitrary function of time. Integrating (.8) with respect to x we obtain B() vdx = B() v (; x)dx exp t cell t degrad t Thus the relative size of t cell and t degrad provides several natural choices for. For our system the following local existence result holds. Theorem Let u ; ; u ; C ; (B ()) for some < <, with u ; ; u ; not being identically zero, and RB() (u ;(x) + u ; (x))dx =. Then there exists a unique classical solution of (.) with (.), (.3), and (3.) satisfying u (; x) = u ; (x); u (; x) = u ; (x) for t < T for a T >. Moreover, if ku (t; )k + ku (t; _ )k C in t T, it is possible to extend the solution (u ; u ; v) to an interval t T + for some >. Proof. The proof of this result can be done by standard arguments and was shown in the PhD-thesis by E.E. Espejo Arenas, [3]. Global existence results for more general multi-component chemotaxis systems were considered by Wolansky in []. Writing the notation by Wolansky on the left hand side of the following equations and our notation in the present paper on the right hand side we have, that i = ; is the index for the chemotactic species, and k = is the index for the involved chemical. Then =, = and ; =, ; =. For the chemo-attractant we have = =, ; = ; = and f =. Just to make the cross-check easier, we also mention further introduced notation in [], namely: i;j = i; j; = i;. Thus ; = and ; =. Additionally some constants a j > are dened in [] by a ; = a = a ; = a. So for instance a = and a =. Then Theorem 5 in [] applied to our context and our notation reads Theorem Consider our system (.) on the two-dimensional disc of radius. Assume Dirichlet boundary conditions for the chemo-attractant. Further let 8 N + N (N + N N + N ) > ; with N i = m i for i = ; are the respective total preserved masses of the two chemotactic species. 7 :

10 Then for (u (; ); u (; )) R Y N with Y N := n(u ; u ) : B ()! R + : RB() u i = N i ; B() u i log u i dx < a global in time classical solution. o there exists Remark 3 For one species the above mentioned inequality is the classical condition for global solutions of the classical chemotaxis model. For system (.) with (.), (.3), (.4), so Neumann boundary conditions, existence of global solutions for the radial symmetric case, like in our situation, was proved in [3]. The result basically says the following: If T max is the maximal time of existence for the classical radially symmetric solutions of our problem and if m < min ; ( m ) and m < min ; ( m ) then T max = and sup t fku (t; )k L + ku (t; )k L + kv(t; )k L g <. ; (3.) Now we will derive sucient conditions for blow-up in nite time for the solutions of our system. Theorem 4 For u ; ; u ; assume the conditions in Theorem. Let (t) = RB() jxj u (t; x)dx and m (4 m )+ () <, with m ; m being given as in (.). Then there exists a T <, such that lim t!t ku (t; )k L + ku (t; )k L = : (3.3) Remark 5 From this result one can deduce that if m > blow-up happens in nite time. and is "small" then Proof. of Theorem 4 We will argue by contradiction and use techniques related to [8]. Suppose that (3.3) is bounded for any T. Then Theorem implies that the solution is classical for any time. Now t u = u r (u rv) by jxj and integrating the resulting relation over B () we t B() u jxj dx = B() jxj u dx jxj B() r (u rv)dx : 8

11 From Green's identity we t B() u jxj dx = = 4 4 B() jxj u dx B() u dx jxj r (u rv)dx u dx + u r jxj ds u r jxj ds + B() From (.7) and using the identity x rv B() u (x rv)dx = = u From (3.4) it follows that d dt (t) 4m + rdr M + M r B() r jxj (u rv)dx u (x rv)dx (3.4) we dr + m + (t) B() rdr u M dr + B() u jxj dx u jxj dx = m + (t) = m (4 m ) + (t): Since m (4 m ) + () <, it follows that (t) is strictly decreasing with respect to t. Then for every t we have that So d dt (t) < m (4 m ) + () : (t) < () + [m (4 m ) + ()] t: and (t)! as t! T for some nite T. But then (u ; u ; v) cannot be a classical solution. Hence due to Theorem there exists T T such that lim t!t (ku (t; )k L + ku (t; )k L ) = : 9

12 4 Simultaneous Blow-up In this chapter we investigate if blow-up at two dierent times is possible in all densities variables of our system. Some of the techniques and tools that we use here are taken from [5] and [6]. The proof will be done by contradiction. To obtain this result we will prove that u being bounded will imply a lower bound for M near the blow-up point. This estimate will allow us to construct a sub-solution for (.9) that behaves like M Cr for small values of r at the blow-up time. This is a contradiction to M = R r u d Cr for small values of r. To obtain our result we will rst prove Lemma 6 Suppose that u blows up at t = T and that ku (t; )k L C for t T. Then there exist " > and L >, both depending on ; ;, and C such that lim inf t!(t ) M (t; L p T t) " : Proof. We argue by contradiction. Suppose there exists a sequence (t n ) n such that t n! (T ) and M (t n ; L p T t n ) < ". Dene a sequence of functions m n (s; ) = M (t n + L (T t n )s; L for s < and L p L T. Using (.8) we obtain tn p T t n ) (4.) m n;s = m n; m c n; L (T m n + M t n )m n; + m n; ; n : (4.) By assumption the functions m n blow up at s = L. Our strategy is to construct a super-solution for (4.), that shows that this is not the case. To obtain this contradiction the proof is divided into three steps Step. Construction of an auxiliary super-solution In order to obtain a super-solution for (4.) we are looking for a function M which satises M M M + M c + M L (T t n )M M s (4.3) in the sense of distributions. Since u C it follows that M C L (T t n ). Therefore, if M(t; ) satises M M M + M (C c) + L (T t n )M M s (4.4)

13 in the sense of distributions, then M automatically satises (4.3). First we obtain an auxiliary function which satises M() = M + ; M ; > ; (4.5) M M + M M (C c) + L (T t n )M (4.6) for, with M() < equivalent to, and t T. This is possible, since (4.6) is (C c) ( ) + f(m + ) + L (T t n ) g : And the above inequality holds for any (; ) in, t n T for M ; small enough. Now we construct another auxiliary function f M(s; ). Dene f M () for as an increasing, strictly concave function, satisfying the following conditions fm () = M() f M We then dene f M(s; ) as the solution of where fm s = f M +! d^ ds (s) = () < for < fm () m ; f M C ([; )) for ^(s) < ; with f M(; ) = f M () ; (4.8) fm() ^(s) + ^(s) for s ; with ^() = (4.9) and for > suciently small. (C c) In the following we assume, that L (T t n ). Problem (4.8) can be solved explicitly for short time, by using the method of characteristics. Moreover,

14 for small values of s, i.e. s [; s ], with s > suciently small and independent of L, the following estimates hold (i) f M(x; ^(s)) = M() (4.) (ii) (iii) (s; ) < () f M (s; ) for < (4.) (iv) f M(s; ) C ([; )) (v) (vi) f M(s; ) > m ^(s) w for s s : These estimates can be derived by integration over characteristics. From (4.) it follows that fm f M + f M f M (C c) + L (T t n )f M Ms f (4.3) for s s, ^(s) <, if (C c) L (T t n ). We now dene our super-solution as ^M(s; ) = 8 >< >: M ^(s) ; ^(s) fm(s; ) ; ^(s) (4.4) with s s. Due to (4.6),(4.),(4.), and (4.3), we have that M(s; ) satises (4.3) in the sense of distributions for s s,. Moreover, ^M(s; ) m for, with some >. Finally dene M(s; ) = ^M(s; ) for > suciently large, then M(s; ) satises (4.3) in the sense of distributions for s (; s ),, (T t n) small and M(s; ) m for : (4.5)

15 Step. M is "small" close to the origin for t [t n ; T ]. Suppose that " < M, with M as in (4.5). Then from (4.) it follows, that for L suciently large and L s that By comparison we obtain m n (; ) M(; ) for : (4.6) m n (s; ) M(s; ) for ; s L (4.7) m n (s; ) M < for ; s L (4.8) respectively using (4.5),(4.7),(4.4) M (t; r) ; for r L pt t n ; t n t T : (4.9) Step 3. (4.9) implies that u does not blow up at t = T. We give a comparison argument. To do so, a super-solution for (.8) is constructed. Since M C r due to the boundedness of u, we obtain this super-solution by nding ^M (t; r) such ^M + (C c) + ^M : (4.) We can obtain a solution for (4.) by setting the right hand side equal to zero. By simple ODE arguments one can solve the stationary system to get ^M = ^M (r). We can express ^M (r) = '( r " ) with " << and '() = 4 ( + ) + O(" + ) (4.) Choosing " > suciently small, we can then compare M (t; r) and ^M (r) to obtain and K > xed. M (t; r) Kr for t T ; r ; (4.) 3

16 Finally we provide an estimate for u. We have that u (t; r) : (4.3) Therefore we have to To do this, we dene for small R > the of functions M ;R (; ) = M (t + R ; R) R (4.4) for and t (; T R ). Now (4.3) in particular implies jm ;R (; )j C for 4 ; [; ] : On the other + f ; M ;R ; (4.5) where f is bounded for any R, ( ; ), [; ]. Therefore, classical interior 4 regularity theory for parabolic equations C for ; ; ; : (4.6) Combining (4.4) and (4.6) it follows, that T j Cr for r, t ( ; T Using (4.3) we obtain that u (t; r) is bounded for r, t (; T ), and our Lemma follows. Now we focus on the main result of our paper, which is Theorem 7 Assume that u ; ; u ; fulll the assumptions of Theorem. Also, suppose that lim sup ku (t; )k L (B t!(t ) ()) = ; for some T > (4.7) then lim sup ku (t; )k L (B t!(t ) ()) = : (4.8) 4

17 Proof. We argue by contradiction. Suppose that (4.7) is fullled and (4.8) does not hold. Then there exists a constant C > such that Thus u (t; x) C for x B () ; t T : (4.9) M (t; r) Cr for r ; t T : (4.3) Our strategy is to construct a sub-solution for (.9) behaving like Kr with K >, for r! + with (; ). This then contradicts (4.3). Let I frag be the characteristic function of the set fr ag. Due to Lemma 6 and p the monotonicity of M with respect to r we have that M (t; r) " > for L T t r, with (T t) suciently small. Using (.9) as well as the fact it " p I frl T r Therefore, we can obtain a sub-solution for M by nding a function M M cr " I frl r p : (4.3) We are looking for M in the form M(t; r) = r p T t ; log(t t) : (4.3) Let y = pt r and = log(t t), cy exp( ) The structure of (4.33) suggests to look for solutions of the form where " I : (4.33) (; y) = exp( )Q(y) + exp( ( + ))G(y) ; (4.34) d Q y dy + y " I fylg dq + Q = : (4.35) y dy 5

18 and d G dy y + y " I fylg dg y dy + ( + )G c y dg dy : (4.36) It will later be checked if (; y) in (4.34) yields a sub-solution for y exp( ) for small enough. Before, we study the function Q(y) as dened in (4.35). Step. Analysis of Q(y): Equation (4.35) can be solved explicitly using Kummer functions. Let z = y, then 4 (4.35) transforms into dq z d Q dz z " I fz L 4 g If z L, this equation reduces to 4 dz + Q = : (4.37) zq zz zq z + Q = ; (4.38) and the unique solution of this equation, vanishing at the origin is (cf. []) with Q(z) = AzM( + ; ; z) ; (4.39) M(a; b; z) = X l= (a l ) (b) l z l l! ; where (a) l = a(a + )(a + ):::(a + l ) and (a) =. On the other hand, for z L equation (4.37) becomes 4 z d Q dz z " dq + Q = : (4.4) dz The only solution of this equation that does not grow exponentially for large z is with U(a; b; z) = sin(b) Q(z) = BU( ; " ; z) ; (4.4) M(a; b; z) ( + a b) (b) z b M( + a b; b; z) (a) ( b) 6

19 for b =. In particular, using (4.39), (4.4) one obtains that the solution of (4.35) is given as Q(y) = Ay M( + ; ; y ) for y < L ; 4 4 (4.4) Q(y) = BU( ; " y ; ) for y > L : 4 (4.43) Suppose that A > is a given constant. To obtain a solution of (4.35) with the aid of (4.4), (4.43) we need to impose continuity for Q and dq at y = L. Using dy and 3.4. in [] this requires L AM( + ; ; L ) = BU 4 4 A ( + ) 6 = BL U For the moment suppose ; " ; L 4 L 3 M( + ; 3; L ) + AL M( + ; ; L ) ; + " ; L 4 (4.44) : (4.45) U( ; " ; L 4 ) 6= for (; ) : (4.46) Without loss of generality we can always assume that L is suciently large. Then So (4.45) can be rewritten as 4 ) M( + ; ; B = L L A : 4 U ; " ; L 4 f() := ( )L M( + ; 3; L ) + 4 M( + ; ; L ) (4.47) 4 4 U L M( + ; ; L ) 4 + ; + " U ; " ; L 4 = ; L 4 It can be shown that there exists at least one solution of (4.47) for (; ) and L large enough, if (4.46) holds. Indeed, we have f() = L L M(; 3; ) + 4 M(; ; L ) ; 4 4 7

20 and using (cf in []), it follows that M(a; b; z) (b) (a) ez z a b for z! f() 4 exp( L 4 ) > for L! : On the other hand U(; + " f() = 4 M(; ; L ) L M ; ; ; L L ) U( ; " ; L ) : 4 Since M(; b; z) =, it follows that f() = 4 U(; + " L 4 U( ; " ; L ) 4 ; L ) 4! : So we need to study the sign of g(b; x) := U(; b + ; x) x U( ; b; x) Using 3.4. in [] we obtain U (a; b; x) = au(a + ; b + ; x). Thus g(b; x) = x U ( ; b; x) U( ; b; x) : : Using 3.5. in [] we get U(; b; x) x b + O x for x! ; and g(b; x) x (x b) b x < for x! : With b = " > it follows, that f() <. Therefore there exists a root of (4.47) in the interval (; ). It only remains to prove (4.46). For large L this is a direct consequence of 3.5. in []. 8

21 Step. Analysis of G(y): Now we study the properties of G(y) in (4.34). A solution of the inequality (4.37) can be obtained by nding a solution G + ( + )G(y) = (4.48) for > for y L. The unique solution of (4.48), which is polynomially bounded for y! and bounded quadratically for y!, is given by G(y) = y M( ; ; +y U( ; ; y 4 ) y 4 ) y y Q y ()U( ; ; W () 4 ) d (4.49) Q y ()M( ; ; ) 4 d ; (4.5) W () where W (y) = d dy y M( ; ; y y M( ; ; y 4 ) 4 ) d dy y U( ; ; y ) 4 y U( ; ; y ) 4 : (4.5) Using (4.4) we obtain with 3..8 in [] that y Q(y) B 4 for y! : (4.5) Combining this with (4.49), (4.5), and 3..4, 3..8 in [] we get G(y) = O y (+) for y! (4.53) G(y) = O(y ) for y! : (4.54) Similar asymptotics and estimates for the derivatives can be obtained for Q and G. Step 3. Comparison argument: The function dened in (4.34) satises (4.33) for values of y for exp( ( + @y exp( ) The asymptotics (4.5), (4.53) imply that this inequality is satised for jyj exp( ) with > small enough. 9

22 A comparison argument using (.9) with M = for r = implies that M > for r =, T t T. Choosing B > suciently small, we obtain that the subsolution (; y) in (4.34) is smaller than M (t; r) for r =, t (T ; T ) with > small. The same is true for t = T, r. Overall, due to the self-similar structure of (; y) and (4.5), (4.53) it follows that M ((T ) ; r) lim = r! r But since (; ), this contradicts (4.3), thus the result follows. 5 Formal derivation of the blowup prole In this section we derive the asymptotics of the solutions of system (.8), (.9) near the blow-up time by formal asymptotic expansions Our analysis will be restricted to parameters ; ; for which the resulting asymptotics yields M << M near the blow-up point. It seems possible to obtain other asymptotics yielding mass aggregation for both species at time t = T when the chemotactic strength of both species is comparable. But this case will not be considered here. In this paper we will consider the situation where the chemotactic strength of the second species is much weaker than that of the rst species. For M << M we can approximate (.8), M r M @r + h:o:t: (5.) + h:o:t: : (5.) Therefore the dynamics of M is decoupled from that of M to the leading order. Thus the asymptotics of M can be calculated like proved for the one species case in [9]. So one obtains r M (t; r) M ;s p T t (t) (5.3) with M ;s = 4 r ( + r ) ( + ) and (t) exp exp r j log(t t)j! (5.4) for t! T ;(5.5)

23 where M ;s is the stationary solution of the equation for M and is the classical Euler constant. So the problem reduces to the description of the asymptotics of M in (5.) with M as in (5.3). Arguing like in the asymptotics for (5.3) it is convenient to reformulate (5.) using self-similar variables y = r p T t ; = log(t t) ; M = (; y) ; M = (; y) : (5.6) Then (5.) becomes to leading order @y y y + y y : (5.7) Additionally we will need the equation for M in the so-called inner region y. For this we introduce new variables = y ( ) ; (; y) = Q(; ) ; (; y) = G(; ) ; (5.8) where is given as in (5.5) with some slight abuse of @ @ : (5.9) Due to (5.3) we have G(; )! M ;s () for!. Q(; )! Q (; ) for!, where Thus to the + = : (5.) Then Q (; ) = K( )L() ; (5.) with L() = z exp z M ;s () d dz : Using (5.4) it follows that L() A + A for! : (5.)

24 Here = 4 and A = exp A = A z 4 M ;s () d (M ;s () 4 ) d! "exp (M ;s () 4 ) d z! # ; (5.3) dz : (5.4) Combining (5.) and (5.) we obtain the following matching condition for (; y) in (5.7) (; y) A K() (()) y + A K() for! ; (5.5) with () << y <<. In order to compute the asymptotics of (; y) we introduce a new variable Then = (; y) = (; y) y : yy ( + 4 ) y y y y (5.6) Writing N = + 4 = +, then the right-hand side of (5.6) is self-adjoint in L (R + ; y exp( N y 4 )). On the other hand (5.5) implies (; y) K()A + K()A y N : (5.7) To compute the asymptotics of (; y) we argue as in [9] and write with (; y) = a () + Q(; y) ; (5.8) y N Q(; y) exp( y 4 )dy = : (5.9) Plugging (5.8) into (5.6), multiplying by y exp( N y 4 ), integrating in [";) for " >, we obtain, after some computations d d " y (N ) exp y 4 = " N ' y (; ") exp( " '(; y)dy 4 ) " '(; y)y N exp( y 4 )dy : (5.)

25 Using (5.7) and taking the limit "! we get d d = and using (5.9) we obtain y exp( N y )'(; y)dy 4 '(; y)y N exp( y 4 )dy + K( )(N )A ; (5.) a ; = a + (N )A K( ) : (5.) Here K( ) could be expected to behave like exp( ) up to logarithmic corrections, since ( ) >> exp( ) for any >. Due to (5.7) we can assume where to the leading order (N W yy ) + Q(; y) = K( )W (y) ; W y y W y = for y > ; y N exp y W (y)dy = : 4 On the other hand, (5.7) and (5.8) yield So (5.) reads W (y) A y N for y! ; a ( ) = K( ) A : (5.3) a ; = a + (N )A A a ; R + and since (( ))d < it follows that a ( ) C exp( ) for! and some C >. Using (5.3) we get K( ) C A (( )) exp( ) for! : 3

26 This concludes the computation for K() in (5.) to the leading order. Combining this with (5.8), (5.5) gives the asymptotics for M in the region, where jyj is large. Then M (t; x) Cr ; for (T t) << r << ; where t! T ; where = 4. This implies in particular that u (T; r) C r 4 ; as r! + : This shows that there is a singularity for u but no mass aggregation. 6 Conclusions We analyzed a two chemotactic species model, where the chemotactic species are attracted by the same chemical. In the radial symmetric situation we wanted to see, if chemotaxis can separate the two species. From our results so far this does not seem to be the case. There is simultaneous blow-up for both chemotactic species. Specically it is proved that if there is blow-up in one chemotactic species, then there is also blow-up in the other one. We give conditions for local and global existence. Also the existence of blow-up in this system is proved. The blow-up asymptotics of the solutions are formally calculated for certain parameter regimes. The two species can be dierent in this respect, since one of them shows mass concentration in the blow-up regime, whereas the other one does not. Acknowledgement While working on this paper E.E. Espejo Arenas was supported by the the Max- Planck Institute for Mathematics in the Sciences (MPI MIS) in Leipzig and by the University of Heidelberg. A. Stevens' work was partially supported by MPI MIS. J.J.L. Velazquez was supported by the Humboldt Foundation, by MPI MIS, by the International Graduate College 7, Heidelberg, and by DGES Grant MTM J.J.L. Velazquez also thanks the Universidad Complutense for its hospitality. References [] Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions, Dover Publications (97). 4

27 [] Childress, S. Chemotactic collapse in two dimensions. Lecture Notes in Biomath. 55, Springer (984), 6{68. [3] Espejo Arenas, E. E. Global solutions and nite time blow up in a two species model for chemotaxis. PhD-thesis, University of Leipzig (8). [4] Fasano, A.; Mancini, A.; Primicerio, M. Equilibrium of two populations subject to chemotaxis, Mathematical Models and Methods in Applied Sciences. Vol. 4 (4), no. 4, 53{533. [5] Herrero, M. A.; Velazquez, J. J. L. Singularity patterns in a chemotaxis model. Math. Ann. 36 (996), 583{63. [6] Herrero, M. A.; Velazquez, J. J. L. A blow-up mechanism for a chemotaxis problem. Annali Scuola Normale Sup. di Pisa, Serie IV (997), 633{683. [7] Jager, W.; Luckhaus, S. On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 39 (99), 89{84. [8] Nagai, T. Blow-up of Radially Symmetric Solutions to a Chemotaxis System, Advances in Mathematical Sciences and Applications, Gakkotosho, Tokyo, Vol. 5 (995), no., 58{6. [9] Velazquez, J. J. L. Point dynamics for a singular limit of the Keller-Segel model. I. Motion of the concentration regions. SIAM J. Appl. Math. 64 (4), no. 4, 98{3. [] Weijer, C. Dictyostelium morphogenesis. Curr. Opin. Genet. Dev. 4 (4), 39{398. [] Wolansky, G. Multi-components chemotactic system in the absence of conicts. European J. Appl. Math. 3 (), no. 6, 64{66. 5