Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig

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1 Max-Planck-Institut für Mathematik in en Naturwissenschaften Leipzig Simultaneous Blowup an Mass Separation During Collapse in an Interacting System of Chemotactic Species by Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki Preprint no.:

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3 SIMULTANEOUS BLOWUP AND MASS SEPARATION DURING COLLAPSE IN AN INTERACTING SYSTEM OF CHEMOTACTIC SPECIES Elio Euaro Espejo Department of Systems Innovation, Osaka University 1-3 Machikaneyama-cho, Toyonaka , Japan Angela Stevens Institut für Numerische un Angewante Mathematik Fachbereich für Mathematik un Informatik er Universität Münster Einsteinstr. 62, Münster, Germany Takashi Suzuki Department of Systems Innovation, Osaka University 1-3 Machikaneyama-cho, Toyonaka , Japan (Submitte by: ****) Abstract. We stuy an interacting system of chemotactic species in two-space imension. First, we show that there is a parameter region which ensures simultaneous blowup also for non-raially symmetric solutions. If the existence time of the solution is finite, there is a formation of collapse (possibly egenerate) for each component, total mass quantization, an formation of subcollapses. For raially symmetric solutions we can rigorously prove that the collapse concentrates mass on one component if the total masses of the other components are relatively small. Several relate results are also shown. 1 Introuction The Smoluchowski-Poisson equation or rift-iffusion moels are systems of elliptic-parabolic equations escribing the motion of particle ensities in nonequilibrium statistical mechanics. They have been use in semiconuctor Accepte for publication: Xxxxxxxx 20xx. 1

4 2 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki physics, high-molecular chemistry, an astrophysics (see [33] an the references therein). They arise also in biology, moeling aggregation an selforganization of the cellular slime mol Dictyostelium iscoieum (D) ue to chemotaxis [18, 23, 6]. This moel in biology has a crucial negative rift. The first rigorous proof that chemotaxis can serve as the main mechanism for the onset of self-organization of D was given in [16]. Namely, for the critical space imension two, epening on the L 1 -norm of the initial values an on the strength of the chemotactic sensitivity, this moel exhibits solutions blowing up in finite time or existing global-in-time. Later stuies have clarifie the blowup threshol [20, 21, 1, 11, 27], formation of collapses [12, 26], mass quantization [12, 32, 29], an type II blowup rates [12, 25, 22]. On the other han, [13] gave a family of blowup solutions with precise asymptotics for the full parabolic-parabolic system. Along this line of thoughts, recently a two-component system for chemotaxis has been stuie in [9], to approach the question if cell sorting of D in the moun-stage can be mainly ue to ifferential chemotaxis, i.e., the chemotactically weaker cell type sorts out to the bottom of the moun, whereas the chemotactically stronger cell type sorts out of the top, c.f. [34]. Of course this question has finally to be aresse in a spatially three imensional moel. But as a first mathematical test-problem to this question, in [9] it was analyze if there exists a set of chemotactic sensitivities in the parabolic-elliptic chemotaxis system n the raially symmetric setting, such that the solution for one cell type shows finite time blowup, while the solution for the other cell type still exists at that instant in time, i.e., the first cell type starts to self-organize, while the other cell type oes not yet. In the raially symmetric setting this is not possible. It was rigorously proven in [9] that if the solution for one cell type shows blowup in finite time, then the other cell type oes so too, an at the same time, no matter how weak its chemotactic sensitivity is. Also sufficient conitions for blowup in finite time for this multi-component system were given. Finally, a formal computation in [9] showe that the blowup mechanism for the two cell types can be ifferent, even up to the situation that one species can exhibit blowup without mass aggregation. Explicit formal asymptotics for the blowup profile for special cases were given to confirm this property using rescale mass functions, that is, only one component can form a singularity of elta functions at the blowup time. The consiere system is a special form of the multi-component chemotaxis system introuce in [35], where conitions for the existence of global solutions for such systems were erive. Then its stationary state is stuie by [15]. Further analysis on the critical mass for blowup for a two-species moel for chemotaxis in R 2 have been more recently one [14, 7, 8]). The authors ientifie a curve in the plane of masses such that outsie of this

5 Simultaneous blowup an mass separation 3 curve there is blow-up of solutions an insie of it the solutions are global in time, if the initial masses satisfy a threshol conition. This moel may also escribe a competitive feature of chemotaxis observe in cancer cell biology, especially, in tumor microenvironment at the stage of in-travasation. More precisely, there is an interaction between cancer cells an tumor associate macrophages through chemical substances which causes their localize cell eformations calle invaopoia an poosomes, respectively (see [36, 17]). At the tissue level, this phenomenon may be escribe by a competitive chemotactic feature between two species of cells to chemical substance secrete by themselves. Uner this agreement it will be interesting from both the mathematical an biological points of view, if actually only one component of the solution can form δ-functions at the blowup time. The aim of this paper is to approach such a property of the solution, especially also in the non-raial symmetric situation. We restrict our analysis to the case of two space-imension which is the critical imension of the moel in such a biological setting. Relatives of this 2-imensional moel also arise as a kinetic mean fiel equation of point vortices. See [5] an [4] for single an multi-comonent cases, respectively, an also [31] for a higher-imensional multi-component case. Given a boune omain R 2 with smooth bounary, this moel takes the part of Smoluchowski equations: t u 1 = 1 u 1 χ 1 u 1 v t u 2 = 2 u 2 χ 2 u 2 v in (0, T ), (1) using the chemical graient term v, with the bounary conition u 1 1 ν χ v 1u 1 ν = u 2 2 ν χ v 2u 2 = 0 on (0, T ) (2) ν an the initial conition u 1 t=0 = u 10 (x) 0, u 2 t=0 = u 20 (x) 0 in, (3) where 1, 2, χ 1, an χ 2 are positive constants an ν is the unit normal vector. The initial value (u 10, u 20 ) satisfies 1 u 10 + u 20 x = 1, (4) an then (1), (2), an (3) are couple with the Poisson equation v v = u 1 + u 2 1, ν = 0, v x = 0. (5)

6 4 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki We put the last conition of (5) to normalize an aitive constant of v. This normalization is not essential because only the graient v of v is use in the Smoluchowski equations (1), while conition (4) is necessary for the solvability of (5) at t = 0. Since t u 1 x = t u 2 x = 0 (6) arises for (1) with (2), this compatibility conition for the solvability of (5), that is, 1 u 1 + u 2 x = 1, is kept for all t > 0. System (1), (2), (3) an (5) with (4) is thus a generalization of the parabolic-elliptic system of chemotaxis given in [16] for two chemotactic cell types (u 1, u 2 ). Here we o not aopt the normalization 1 = 1 to make the statements below simpler, particularly for more than two-components systems. We assume that u 10, u 20 0 are in C 2 () an satisfy u 10 1 ν χ v 0 1u 10 ν = u 20 2 ν χ v 0 2u 20 ν = 0 for v 0 = v 0 (x) efine by v 0 = u 10 + u 20 1, v 0 ν = 0, on v 0 x = 0. This assumption guarantees the unique existence of a local-in-time classical solution satisfying u i (, t) > 0 on, 0 < t < T, i = 1, 2 (7) with T = T max (0, + ] staning for the maximum existence time [9]. Since this T is estimate from below by 2 u i0, it hols that T < + lim u i (, t) = + (8) by a stanar argument on the continuation of the solution in time (see, for example, [29]). The solution (u 1, u 2, v) to (1), (2), (3), an (5) with (4) satisfies v = u 1 u x, v ν = 0, v x = 0, u = u 1 + u 2. (9)

7 Simultaneous blowup an mass separation 5 Here, the system compose of (1), (2), (3), an (9) may be calle interacting system of Smoluchowski-Poisson equations, an henceforth is enote by (ISP ) in short. Similarly, we have a local-in-time well-poseness with the properties (7) an (8) for this system. In particular, u 10, u 20 > 0 may be assume on by the strong maximum principle. Then, there is A > 0 such that Au 10 u 20. If 1 = 2 = an χ 1 = χ 2 = χ, we obtain for w = Au 1 u 2, which implies w t = w χ w v in (0, T ) w = 0 ν on (0, T ) w t=0 = Au 10 u 20 0 in Au 1 u 2 on [0, T ). (10) If T < + an there is t k T such that lim inf u 1 (, t) < +, an therefore, by (10). Then it follows that lim sup u 1 (, t k ) < +, k lim sup u 2 (, t k ) < + k lim inf { u 1(, t) + u 2 (, t) } < +, a contraiction to (8). Hence we have an similarly, The simultaneous blowup lim u 1 (, t) = +, lim u 2 (, t) = +. T < + lim u 1 (, t) = lim u 2 (, t) = + (11)

8 6 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki is thus vali in any imensions if 1 = 2 an χ 1 = χ 2. In two-space imensions with u i = u i ( x, t), however, a similar property hols for any pairs of ( i, χ i ), i = 1, 2. More precisely, it hols that T < + lim sup u 1 (, t) = lim sup u 2 (, t) = +, (12) which is not the case of the relate rift-iffusion moel given in [19] (see [9, 10] for the proof). Our first result shows that there is a parameter region where (11) hols even for non-raially symmetric solutions. Henceforth, we put ξ i = i /χ i, u i0 1 = λ i, i = 1, 2, (13) inicating inverse motilities an initial masses of the species. Theorem 1 If then (11) hols. if λ i < 4πξ i, i = 1, 2 (14) Here, conition (14) is consistent with the assumption T < +. In fact, ( ) 2 λ i < 4π ξ i λ i, λ i < 4πξ i, i = 1, 2 then it always hol that T = +, while T < + can occur in case ( ) 2 λ i > 4π ξ i λ i (15) (see Theorems 9 an 10 an the escription on the relation between known results, particlularly [35], in 2). Then the parameter region efine by (14) an (15) in λ 1 λ 2 plane in λ i > 0, i = 1, 2, is not empty. Since property (8) means T < + lim u(, t) = +, (16) recalling u = 2 u i in (9), the blowup set of (u 1, u 2 ) efine by S = {x 0 (x k, t k ) (x 0, T ), u(x k, t k ) + } (17) is not empty. In the case of the Smoluchowski-Poisson system for a single unknown species, the formation of collapses occurs with a quantize mass [32, 33, 29]. Our secon result shows that this property arises also in (ISP ) for each component, with possibly egenerate collapses. Here, we say that the collapse m i (x 0 )δ x0 (x), i = 1, 2, in (18) below is egenerate if m i (x 0 ) = 0.

9 Simultaneous blowup an mass separation 7 Theorem 2 If T < +, the blowup set S efine by (17) is finite. It hols that u i (x, t)x m i (x 0 )δ x0 (x) + f i (x)x, i = 1, 2 (18) x 0 S in M() = C() as t T = T max < +, where m i (x 0 ) 0, i = 1, 2, are constants satisfying (m 1 (x 0 ), m 2 (x 0 )) (0, 0), an 0 < f i = f i (x) L 1 (), i = 1, 2, are smooth functions in \ S. Equality (19) in the following theorem may be calle a total mass quantization because it is an ientity involving all the collapse masses m i (x 0 ), i = 1, 2. Theorem 3 It hols that ( 2 m i (x 0 )) = m (x 0 ) for any x 0 S, where ξ i m i (x 0 ) (19) m (x 0 ) = { 8π, x0 4π, x 0. (20) We can write (1) as t u 1 = 1 u 1 χ 1 u 1 v + χ 1 u 1 (u λ ) t u 2 = 2 u 2 χ 2 u 2 v + χ 2 u 2 (u λ ), using u = u 1 + u 2 an λ = λ 1 + λ 2, where λ i = 1 u i0 x, i = 1, 2. Thus the ODE part of (ISP ) may be efine by u 1 = χ 1 u 1 (u 1 + u 2 a) t u 2 = χ 2 u 2 (u 1 + u 2 a) t u = u i (21)

10 8 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki with a = λ, where u i0 = u i (0), i = 1, 2, are positive constants. Although the solution (u 1, u 2 ) to (21) is constant: u i (t) = u i0, i = 1, 2 if a = we may have a blowup of the solution to (21). In this connection, we mention that there is actually a blowup of the solution to (ISP ) (see Theorems 10 an 11). In (21) it hols that u i0 t (χ 1 1 log u 1 χ 2 2 log u 2 ) = 0, an hence u 2 = cu γ 1 with γ = χ 2/χ 1 an c = u 20 /u γ 10. Therefore, the blowup of the solution (u 1 (t), u 2 (t)) to (21) is simultaneous for u i (t), i = 1, 2, if it actually occurs. Assuming γ 1 without loss of generality, we obtain u 1 t χ 1 u 2 1, which implies u 1 (t) χ 1 (T t) 1 as t T = T max < +. The type (I) blowup rate of (ISP ) thus may be efine by u(, t) (T t) 1, u = u 1 + u 2. The next theorem concerning the formation of subcollapses implies that any blowup rate of (ISP ) is not of this type. To state the result, let (u 1, u 2, v) = (u 1 (x, t), u 2 (x, t), v(x, t)) be a solution to (ISP ) satisfying T = T max < +, take x 0 S, an let z i (y, s) = (T t)u i (x, t), w(y, s) = v(x, t) y = (x x 0 )/(T t) 1/2, s = log(t t) (22) i = 1, 2, be its backwar self-similar transformation. Henceforth, we assume the 0-extensions of z i (y, s), i = 1, 2, where they are not efine. Furthermore, R 2 { } enotes the one-point compactification of R 2, an C 0 (R 2 ) stans for the set of continuous functions on R 2 { } taking the value 0 at, an M 0 (R 2 ) = C 0 (R 2 ). Theorem 4 We have z i (y, s + s )y m i (x 0 )δ 0 (y), i = 1, 2 (23)

11 Simultaneous blowup an mass separation 9 in C (, + ; M 0 (R 2 )) as s +. In particular, it hols that for any b > 0. lim (T t) u(, t) L ( B(x 0,b(T t) 1/2 ) = + (24) Now we approach the property of the collapse mass separation. The first observation is the following theorem. Here we note that the blowup set S coincies with the origin for raially symmetric solutions satisfying T < +. Theorem 5 Let be a isc with center at the origin, u i = u i ( x, t), i = 1, 2, an T < +. Then m i = m i (0) must satisfy besies (19) with x 0 = 0. m i 8πξ i, i = 1, 2 (25) Here we give several remarks. First, inequality (25) arises in the context of a global-in-time continuation of the solution associate with a Truinger- Moser or logarithmic HLS inequality (see 2). Next, (25) is a consequence of (19) if 1/2 ξ i /ξ j 2, i, j = 1, 2. (26) More precisely, if (26) is the case, the curve (a parabola if ξ 1 ξ 2 an a line in the other case) efine by ( ) 2 m i = m ξ i m i, m = m (x 0 ), (27) in the m 1 m 2 -plane in {(m 1, m 2 ) m i > 0, i = 1, 2} oes not cross the lines m i = ξ i m, i = 1, 2. Finally, in the other case of ξ i /ξ j > 2 or ξ i /ξ j < 1/2 for i j, one of (m 1, m 2 ) = (8πξ 1, 0) an (m 1, m 2 ) = (0, 8πξ 2 ) is an isolate point of (27) in the m 1 m 2 -plane with {(m 1, m 2 ) m i 0, i = 1, 2} an (25). Uner such observations we can expect that the above isolate enpoint of (m 1, m 2 ), which we call mass separation, may be actually the case, an is stable uner non-raially symmetric perturbations. In this context we recall that simultaneous blowup (12) is always the case for raially symmetric solutions, regarless of the parameter region inicate by (14). If both simultaneous blowup an mass separation arise, say, m i (x 0 ) = 0 in (18), then it will hol that f i L ( B(x 0, R)) for 0 < R 1, where B(x 0, R) = {x x x 0 < R}. The following theorem shows that the mass separation of raially symmetric solutions, which was formally given in [9], actually occurs if the total mass of one component is relatively small compare with that of the other.

12 10 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki Theorem 6 Uner the assumption of Theorem 5, let ξ i /ξ j > 2 for some i j. Then m i = 0 an hence m j = 8πξ j hols, provie that u i0 1 < 8π(ξ i 2ξ j ). We note that a sufficient conition for T < + in the above theorem is given in [9], that is (see Theorem 11 in 5), u j0 1 > 8πξ j, x 2 u j Theorem 10 in 3 is also available. Theorem 1 is proven by the variational structure of (ISP ) an the logarithmic HLS inequality erive in [30]. Theorem 2 is obtaine by an argument of [26], using an ε-regularity an a monotonicity formula. Then we have the formation of collapses of û(x, t)x as t T, where û = 2 χ 1 i u i. A careful analysis then assures this property component-wisely an also m(x 0 ) m i (x 0 ) > 0. (28) To prove Theorem 3, first we apply an argument evelope for the single component case [32, 33, 29]. We use the backwar self-similar transformation, weak scaling limit, scaling back, an translation limit, to obtain a full-orbit efine on the whole (or the half) space omain. Here establishing a parabolic envelope is essential. Then, an existence criterion of such orbits follows from the metho of local secon moments an scaling, which guarantees an estimate of the total collapse mass from above, that is, ( 2 m i (x 0 )) m (x 0 ) ξ i m i (x 0 ). (29) We use a new argument to erive the reverse inequality ( 2 m i (x 0 )) m (x 0 ) ξ i m i (x 0 ). (30) Namely, we show the bouneness of the total secon moment of the rescale solution an use the scaling limit equation. We have, at the same time, the

13 Simultaneous blowup an mass separation 11 formation of subcollapses inicate by Theorem 4. We note that a weaker estimate of the total collapse mass from below is obtaine similarly to the single component case, that is, either (30) or m i (x 0 ) ξ i m (x 0 ), i = 1, 2 (31) by the logarithmic HLS inequality. Inequality (30), however, is eventually selecte for (19) to be establishe. The proof of Theorem 5 is base on the fact that the interaction between two-components is neglecte in the collapse mass estimate from above for raially symmetric solutions. Then Theorem 6 arises with the total mass conservation of each component of the solution. This paper is compose of five sections an two appenices. In 2 we take preliminaries an prove Theorem 1. We show Theorem 2 in 3 an then Theorems 3 an 4 in 4. Theorems 5 an 6 are proven in 5. In the first appenix, we show that either (30) or (31) hols by the previous argument use in [26]. The secon appenix is evote to some criteria for mass separation an simultaneous blowup. Concluing this section, we note that the proof of the above theorems are vali even for the multi-components case, that is, a system of Smoluchowski equations t u i = i u i χ i u i v in (0, T ) u i i ν χ v iu i = 0 ν on (0, T ) u i t=0 = u i0 (x) 0 in, (32) i = 1, 2,, N, couple with the Poisson equation: v = u 1 u x, v ν = 0, v x = 0, u = N u i. Here, R 2 is a boune omain with smooth bounary, i, χ i, i = 1, 2,, N, are positive constants, an ν is the unit normal vector. Thus we have the local-in-time existence of the solution an blowup criterion (16). Next the simultaneous blowup T < + lim u i (, t) = +, i = 1, 2,, N occurs in the parameter region ( ξ i λ i, J {1, 2,, N}, J = N 1. i J λ i)2 < 4π i J

14 12 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki We have finiteness of the blowup set S efine by (17) an the formation of collapses: u i (x, t)x m i (x 0 )δ x0 (x) + f i (x)x, i = 1, 2,, N (33) x 0 S in M() as t T, where m i (x 0 ) 0, i = 1, 2,, N, are constants satisfying m(x 0 ) N m i (x 0 ) > 0 an 0 f i L 1 (), i = 1, 2,, N, are smooth functions in \ S. Letting ξ i = i /χ i, i = 1, 2,, N, it hols that ( N 2 N m i (x 0 )) = m (x 0 ) ξ i m i (x 0 ) (34) an z i (y, s + s )y m i (x 0 )δ 0 (y), i = 1, 2,, N (35) in C (, + ; M 0 (R 2 )) as s + for any x 0 S, where z i = z i (y, s), i = 1, 2,, N, are self-similar transformations of u i = u i (x, t) efine by (22), which implies lim(t t) u(, t) L ( B(x 0,b(T t) 1/2 )) = +, u = for any b > 0. If u i = u i ( x, t), i = 1, 2,, N, we have ( ) 2 m i (0) 8π ξ i m i (0) i K i K for any K {1, 2,, N}, K, besies (34), x 0 = 0. The solution exists global-in-time, provie that ( ξ i λ i, K {1, 2,, N}, K, i K λ i)2 < 4π i K where λ i = u i0 1, i = 1, 2,, N, while its blowup occurs if ( N 2 N λ i (x 0 )) > m (x 0 ) ξ i λ i (x 0 ) N u i (36) x x 0 2 u i0 L 1 ( B(x 0,2R)) 1, i = 1, 2,, N (37)

15 Simultaneous blowup an mass separation 13 for some x 0, where λ i (x 0 ) = u i0 L 1 ( B(x 0,R)). In the case of u i = u i ( x, t), i = 1, 2,, N, we obtain the following. First, the other blowup criterion than (37) arises, that is, ( ) 2 λ i (0) > 8π ξ i λ i (0) i K i K x x 0 2 u i0 L1 ( B(x 0,2R) 1, i K for some K {1, 2,, N}, K implies T < +. Next, we obtain always the simultaneous blowup lim u i (, t) = +, i = 1, 2,, N. A collapse mass m k = m k (0), k = 1, 2,, N, on the other han, vanishes if ξ i > 2ξ k for any i k an u k0 1 < 8π min i k (ξ i 2ξ k ). Consequently, we have mass separation, for example, provie that an furthermore, m N = 8πξ N, m i = 0, i = 1, 2,, N 1, ξ i > 2ξ j, j = 1, 2,, N 1, i = j + 1,, N, u j0 1 < 8π min (ξ i 2ξ j ), j = 1, 2,, N 1. i=j+1,,n 2 Preliminaries an Proof of Theorem 1 We begin with a escription of the variational structure of (ISP ). Let v = ( ) 1 u stan for (9). First, the total mass conservation follows from (1) an (2) in each component of the solution: Next, we obtain u i (, t) 1 = u i0 1 λ i, i = 1, 2. (38) t u 1 = u 1 ( 1 log u 1 χ 1 v) in (0, T ) u 1 ν ( 1 log u 1 χ 1 v) = 0 on (0, T ) (39)

16 14 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki by (7), which implies 1 u 1 (log u 1 1)x χ 1 v, u 1t = t u 1 ( 1 log u 1 χ 1 v) 2 x, where, enotes the L 2 -inner prouct. Similarly, it hols that 2 u 2 (log u 2 1)x χ 2 v, u 2t = u 2 ( 2 log u 2 χ 2 v) 2 x, t an hence { t = } ξ i u i (log u i 1)x 1 2 ( ) 1 u, u χ 1 i u i ( i log u i χ i v) 2 x by ξ i = i /χ i an v, u t = 1 2 t ( ) 1 u, u. We thus obtain the following lemma using free energy (40) erive in [35]. Lemma 1 Total mass is conserve for each component of the solution, inicate by (38), an also the ecrease of the free energy: in (ISP ), where F ξ1,ξ 2 (u 1, u 2 ) = t F ξ 1,ξ 2 (u 1, u 2 ) 0 for u = u 1 + u 2 (recall (13)). ξ i u i (log u i 1)x 1 2 ( ) 1 u, u (40) Here we prescribe a parameter region of (ξ 1, ξ 2 ) which ensures the logarithmic HLS (or ual Truinger-Moser) inequality First, the conition inf{f ξ1,ξ 2 (u 1, u 2 ) u i 0, u i 1 = λ i, i = 1, 2} >. (41) ( ) 2 λ i 8π ξ i λ i, λ i 8πξ i, i = 1, 2 (42)

17 Simultaneous blowup an mass separation 15 implies where inf{ ˆF ξ1,ξ 2 (u 1, u 2 ) u i 0, u i 1 = λ i, i = 1, 2} >, (43) ˆF ξ1,ξ 2 (u 1, u 2 ) = 1 2 ξ i u i (log u i 1)x 1 2π log 1 x x u u xx for u = u 1 + u 2 an (u u)(x, x ) = u(x)u(x ). This property is proven in [30], by using the logarithmic Hary-Littlewoo-Sobolev inequality 1 2 F (x) log x x G(x ) xx (1 α)f log F + αg log G x + C α, 0 < α < 1, vali to F, G 0, F x = G x = 1, an an inequality erive from linear programming. Secon, we eal with (ISP D), which enotes the system of equations combining (1) an (2) with v = u, v = 0, u = u 1 + u 2. (44) In fact, for this system we also have conservation of total mass (38) an a ecreasing free energy t F ξ1,ξ 2 (u 1, u 2 ) 0, (45) where F ξ1,ξ 2 (u 1, u 2 ) = ξ i u i (log u i 1)x 1 G(x, x )u u xx 2 with u = u 1 + u 2. Here, G = G(x, x ) is the Green s function to (44) which satisfies G(x, x ) 1 2π log 1 x x + C (46) with a constant C = C(). Hence it follows that inf{ F ξ1,ξ 2 (u 1, u 2 ) u i 0, u i 1 = λ i, i = 1, 2} >

18 16 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki uner the assumption of (42). If a slightly stronger conition hols for the total masses of the initial value, that is ( ) 2 λ i < 8π ξ i λ i, λ i < 8πξ i, i = 1, 2 (47) for λ i = u i0 1, i = 1, 2, then we have that ξ i u i (log u i 1) x C (48) by (38) an (45), where (u 1, u 2 ) = (u 1 (, t), u 2 (, t)) is the solution to problem (ISP D), an C > 0 is a constant inepenent of t [0, T ). (See the argument for the proof of Theorem 12.) Now the following theorem is obtaine, similarly to the case of a single component [1, 11, 21], using an iterative scheme, Gagliaro-Nirenberg s inequality, an an inequality ue to [2], that is, (22) (see also Lemma 4.1 of [32]). Theorem 7 ([35]) If (47) is the case, it hols that T = +, lim sup u i (, t) < + (49) for (ISP D). Inequalities (45) an (46) imply ˆF ξ1,ξ 2 (u 1, u 2 ) C, t [0, T ). Here we write 1 u u log x x xx 4π = 1 b i b j ρ i ρ j log x x xx 4π i,j=1 using b i > 0, i = 1, 2, where ρ i = u i /b i. Then we examine the criterion ue to [30] (the main theorem), concerning the bouneness of Ψ(ρ 1, ρ 2 ) = ρ i log ρ i x + a ij i,j=1 ρ i ρ j log x x xx

19 Simultaneous blowup an mass separation 17 efine for ρ i 0 with ρ i 1 = M i, i = 1, 2, where M i = λ i b i, a ij = 1 4π b ib j. In fact, since a ii = b 2 i /(4π) > 0 this property is controlle by Λ J ( M i ) = 2 M i a ij M i M j = 2 ( ) 2 λ i 1 λ i. b i 4π i J i,j J i J i J Thus the above Ψ is boune below if an only if that is, Using Λ J ( M i ) 0, J {1,, N}, J, N = 2, ( ) 2 λ i 8π ρ i log ρ i x = 1 b i we obtain the following lemma. λ i b i, λ i 8π b i, i = 1, 2. (50) u i log u i x λ i b i log b i, Lemma 2 Uner the assumption (50) we have (ξ i 1 )u i log u i x C, t [0, T ) b i for (ISP D). In case λ 1 < 8πξ 1, we can fin b i > 0, i = 1, 2, in (50) by taking Then it follows that δ 0 < b 1 1 ξ 1 1, 0 < b 2 1. u 1 log u 1 x u 2 log u 2 x + δ 1 with 0 < δ 1. The reverse inequality is similarly obtaine if λ 2 < 8πξ 2, while the inequality in (49) hols if an only if sup t [0,T ) (u i log u i )(, t) 1 C with C > 0 inepenent of t (see, for example, [32]). We thus en up the following theorem by (16).

20 18 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki Theorem 8 If then we have (11) for (ISP D). λ i < 8πξ i, i = 1, 2, (51) Here we note again that the case T < + can arise in the parameter region (51) with ( ) 2 λ i > 8π ξ i λ i (see Theorem 10). To hanle the original system (ISP ), we use the fact that the Green s function G = G(x, x ) of the Poisson equation satisfies v = u 1 u x, v ν = 0, v x = 0 G(x, x ) 1 π log 1 x x + C (52) with a constant C = C () etermine by. In fact, each x 0 amits a smooth conformal mapping X = X(x) : B(x 0, 2R) R 2 +, 0 < R 1, where R 2 + stans for the upper half-plane an 0 < R 1. Then the following relation arises G(x, x ) = 1 2π log 1 X(x) X(x ) + 1 2π log 1 X(x) X(x ) + K(x, x ) (53) with K = K(x, x ) C 1+θ,θ ( B(x 0, R) B(x 0, R)) C θ,1+θ ( B(x 0, R) B(x 0, R)) (54) for 0 < θ < 1, where X = (X 1, X 2 ) for X = (X 1, X 2 ) (see [32]). The case x 0 is easier, an it hols that with G(x, x ) = 1 2π log 1 x x + K(x, x ) (55) K = K(x, x ) C 1+θ,θ (B(x 0, R) B(x 0, R)) C θ,1+θ (B(x 0, R) B(x 0, R)). (56) By a stanar covering argument, these relations imply (52). Then we obtain the following lemma.

21 Simultaneous blowup an mass separation 19 Lemma 3 For the free energy F ξ1,ξ 2 (u 1, u 2 ) efine by (40), if ( ) 2 λ i 4π ξ i λ i, λ i 4πξ i, i = 1, 2 it hols that inf{f ξ1,ξ 2 (u 1, u 2 ) u i 0, u i 1 = λ i, i = 1, 2} >. (57) This lemma implies the following theorem an also Theorem 1 similarly to Theorems 7 an 8 erive from (46), respectively. Theorem 9 We have (49) in (ISP ) if ( ) 2 λ i < 4π ξ i λ i, λ i < 4πξ i, i = 1, 2. (58) We procee to a weak form of (ISP ) erive from the symmetry of the Green s function, G(x, x ) = G(x, x). Thus we take φ = φ(x) satisfying φ C 2 φ (), ν = 0, (59) to confirm an hence with t t u 1 φ x 1 u 2 φ x 2 u 1 φ x = χ 1 u 2 φ x = χ 2 [ ] χ 1 i u i φ x t = u v φ x = 1 2 u 1 v φ x u 2 v φ x, [ ] ξ i u i φ x ρ φ u u xx (60) ρ φ (x, x ) = φ(x) x G(x, x ) + φ(x ) x G(x, x ).

22 20 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki The weak form (60) can be use to erive a blowup criterion. The argument is similar to that of [27], using (53) (see also [32]). First, we take a nice cut-off function introuce by [26] (see also [32]), which is enote by φ = φ x0,r(x) for x 0 an R > 0. It is a C -function with support raius R > 0, equal to 1 on B(x 0, R/2), an satisfies 0 φ 1, (59), an φ O(R 1 )φ 2/3, 2 φ O(R 2 )φ 5/6 (61) as R 0. Here, we note the property 1 lim R 0 R 2 x x 0 φ + x x φ x = 0. (62) In fact, if x 0 the above φ = φ x0,r is a scaling of a fixe function with respect to R, an therefore, (62) is easy to see. If x 0 is the case, just the above-escribe conformal mapping X : B(x 0, 2R) R 2 + is involve other than the scaling, in constructing φ = φ x0,r (see [32]). Then, relation (62) is proven similarly. Given x 0, we take φ = x x 0 2 φ x0,r(x) for R > 0. Let B = B(λ 1, λ 2 ) 1 be a constant etermine by λ i, i = 1, 2, an let û(x, t) = χ 1 i u i (x, t) I R (t) = x x 0 2 û(x, t)φ x0,r(x) x MR(t) i = u i (x, t)φ x0,r(x) x, i = 1, 2 J R (t) = 4 Then, we can erive ( 2 ξ i MR(t) i 1 M i 2π R(t)) + 8BR 1 I 4R (t) 1/2. I R t (t) J R(0) + a(r 1 t 1/2 ) + BR 1 I R (t) 1/2 for a(s) = B(s 2 + s), which implies T = T max < + in case J R (0) < 0 an I R (0) 1. Given x 0, on the other han, we take φ = X(x) 2 φ x0,r(x). Then we can argue similarly, an in particular, obtain the following theorem. Theorem 10 We have T < + in (ISP ) if ( 2 λ i (x 0 )) > m (x 0 ) ξ i λ i (x 0 ) (63)

23 an Simultaneous blowup an mass separation 21 B(x 0,2R) x x 0 2 u i0 (x) x 1, i = 1, 2, where x 0 an λ i (x 0 ) = u i0 L1 ( B(x 0,R)), i = 1, 2. 3 Proof of Theorem 2 In this section, we show the finiteness of the blowup set S an the formation of collapses inicate by (18). The first observation is that inequality (58) hols for 0 < λ 1, λ 2 1, an therefore, 0 < λ 1, λ 2 1 implies (49). This property was observe in [16] for the case of a single component. Then, an ε-regularity is proven by the metho of localization [26], that is, taking the above φ = φ x0,r to localize the global-in-time existence result of [16]. This metho is applicable to (ISP ), an we obtain the following lemma. Henceforth, (u 1, u 2 ) = (u 1 (, t), u 2 (, t)) enotes the solution to (ISP ) with T < +. Lemma 4 There is ε 0 > 0 such that implies lim sup lim sup χ 1 i u i (, t) L1 ( B(x 0,R)) < ε 0 (64) u i (, t) L ( B(x 0,R/4)) < +, (65) where x 0 an 0 < R 1. For the proof, first, we erive lim sup B(x 0,R/2) u i log u i x < + (66) from (64) with 0 < ε 0 1. Then, (65) is obtaine by an iteration scheme. In this process of localization, the cut-off function φ = φ x0,r, x 0, R > 0, satisfying (59) an (61) is use. The weak form (60) implies an estimate, which may be calle the monotonicity formula. More precisely, we have ρ φ L ( ) in this formula (see Lemma 5.2 of [32]), an hence [ t χ 1 i u i φ x ] ( ) 2 ξ i λ i φ ρ φ λ i.

24 22 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki The following lemma is thus obtaine. Lemma 5 There is C > 0 such that ûφ x t C φ C 1 () for any φ = φ(x) satisfying (59), where û = χ 1 1 u 1 + χ 1 2 u 2. A irect consequence of Lemma 5 is the existence of an extension of µ(x, t) = û(x, t)x, 0 t < T, as We have, in particular, µ(x, t) C ([0, T ]; M()). µ({x 0 }, T ) = lim lim sup û(, t) L1 ( B(x 0,R)) R 0 = lim lim inf û(, t) L1 ( B(x 0,R)) R 0 for each x 0. Lemma 4, on the other han, implies x 0 S lim lim sup û(, t) L 1 ( B(x 0,R)) ε 0, R 0 recalling (17), an hence µ({x 0 }, T ) ε 0, x 0 S. Since µ(, T ) = λ, we obtain the finiteness of S. Thus the following lemma arises similarly to the case of a single component [26], that is, the formation of collapses of û(x, t)x erive from Lemmas 4 an 5 an the parabolic-elliptic regularity. Lemma 6 If T < + in (ISP ), it hols that S < + an û(x, t)x ˆm(x 0 )δ x0 (x) + ˆf(x)x (67) x 0 S in M() as t T, where ˆm(x 0 ) ε 0 is a constant an 0 ˆf = ˆf(x) L 1 () is a smooth function in \ S. We are reay to prove Theorem 2. First, we take φ satisfying (59) an φ = 0 near S. For such φ = φ(x), it hols that u i φ x = u i φ + u i v φ x t

25 Simultaneous blowup an mass separation 23 an hence u i φ x t C φ, i = 1, 2 with a constant C φ, because (u 1, u 2, v) = (u 1 (x, t), u 2 (x, t), v(x, t)) is smooth in ( [0, T ]) \ (S {T }) from the elliptic-parabolic regularity. Since S < +, the set of such φ = φ(x) is ense in C(). Therefore, like û(x, t)x, the measures u i (x, t)x, i = 1, 2, are extene as µ i (x, t) C ([0, T ], M()), i = 1, 2, using (38). Since ˆµ(x, t) = 2 χ 1 i µ i (x, t), the singular parts of µ i (x, T ), i = 1, 2, are compose of finite sums of elta functions supporte on S. We shall write µ i (x, T ) = m i (x 0 )δ x0 (x) + f i (x)x, i = 1, 2 x 0 S with m i (x 0 ) 0 an 0 f i = f i (x) L 1 (), i = 1, 2. Then, it hols that ˆm(x 0 ) = χ 1 i m i (x 0 ) ε 0, an hence (m 1 (x 0 ), m 2 (x 0 )) (0, 0). Finally, (u 1, u 2, v) = (u 1 (x, t), u 2 (x, t), v(x, t)) satisfies t u i = i u i χ i u i v in ( [0, T ]) \ (S {T }), i = 1, 2. Since (7) hols, therefore, the functions u i (x, t) 0, i = 1, 2, cannot attain 0 in ( \ S) {T } from the maximum principle. Hence it hols that u i (x, T ) = f i (x) > 0, i = 1, 2, in \ S. 4 Proof of Theorems 3 an 4 Theorem 3 is compose of two inequalities, (29) an (30). First, inequality (29) is regare as a localization of the blowup criterion, Theorem 10. For the proof we use the metho of scaling limit evelope in the case of a single component [32, 33, 29]. Henceforth, φ = φ x0,r enotes the cut-off function mentione in the previous section satisfying (59) an (61). Hence we take the backwar self-similar transformations (22), to obtain s z 1 = 1 z 1 χ 1 z 1 (w + y 2 4 ) s z 2 = 2 z 2 χ 2 z 2 (w + y 2 4 ) in s> log T s {s} z 1 ν = z 2 ν = w ν = 0 on s> log T s {s} (68)

26 24 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki an w = z 1 z y, s s w ν = 0, s s w y = 0, (69) where z = z 1 + z 2 an s = (T t) 1/2 ( {x 0 }). As s +, the omain s expans to the whole or the half-spaces enote by L, accoring to x 0 or x 0, respectively. Then, similarly to the prescale case, any s k + amits {s k } {s k} an ˆζ(y, s) such that ẑ(y, s + s k)y ˆζ(y, s) in C (, + : M 0 (R 2 )), (70) where ẑ = χ 1 1 z 1 + χ 1 2 z 2. Here, we recall that the 0-extensions of z(y, s) are taken where it is not efine. Hence this ζ = ζ(y, s) has a support containe on L. We recall also that M 0 (R 2 ) = C 0 (R 2 ), where C 0 (R 2 ) enotes the set of continuous functions on R 2 = R 2 { } vanishing at. In fact, first, we have Next, the Green s function to (69) is given by z i (, s) 1 = λ i, i = 1, 2. (71) G s (y, y ) = G(e s/2 y + x 0, e s/2 y + x 0 ) where (53) is applicable. In the case of x 0, we thus use ẑφ y s C Φ (72) R 2 vali for Φ = Φ(y) C 2 0(L). In the other case of x 0, we take the conformal mapping X : B(x 0, R) R 2 + escribe in 2 an put Y (y, s) = e s/2 X(e s/2 y + x 0 ). Let C0(R 2 2 +) be the set of C 2 -functions on L with compact supports. Given ζ = ζ(y ) C0(R 2 2 ζ +), ν Y = 0, R 2 + we take Φ s = ζ Y (, s), to obtain ẑφ s y s C Φ (73) s (see [32]). Then inequality (71) with (72) an (73) for x 0 an x 0, respectively, implies the esire convergence (70).

27 Simultaneous blowup an mass separation 25 We have also the component-wise convergence of z i (y, s)y, i = 1, 2. In fact, taking subsequences of {s k } enote by the same symbol, we have z i (y, s + s k) ζ i (y, s) in L (, + ; M 0 (R 2 )), i = 1, 2 (74) with the limit measures ζ i (y, s), i = 1, 2, of which supports are containe in L, where L (, + ; M 0 (R 2 )) = L 1 (, + ; C 0 (R 2 )). This convergence guarantees ζ(y, s) = ˆζ(y, s) = ζ(y, s) = ζ i (y, s) χ 1 i ζ i (y, s) ξ i ζ i (y, s) ζ i (L, s) m i (x 0 ), i = 1, 2 (75) for a.e. s, recalling (18). The estimate φ x0,r û(x, t) x t C 7R 2, 0 < R 1, on the other han, is erive from Lemma 5 concerning the prescale solution, which ensures the parabolic envelope similarly to the case of a single component [32]. More precisely, it hols that φ x0,r, û(x, t)x φ x0,r, ˆµ(x, T ) C 7 (T t)r 2 for 0 t < T. Here, putting R = b(t t) 1/2, we make t T an then b +. It follows that an hence for a.e. s by (75) an (76). ˆζ(L, s) = χ 1 i m i (x 0 ), (76) ζ i (L, s) = m i (x 0 ), i = 1, 2 (77)

28 26 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki In case L = R 2 +, we use even extensions of ζ i (y, s), i = 1, 2. Henceforth, we hanle with such ζ i (y, s), i = 1, 2, enote by the same symbols, efine on L = R 2 an put ζ(y, s) = ζ i (y, s) ˆζ(y, s) = ζ(y, s) = χ 1 i ζ i (y, s) ξ i ζ i (y, s). Then it hols that { ζ i (R 2, s) = m m(x0 ), x i (x 0 ) 0 2m(x 0 ), x 0. These measures make up a weak solution to s ẑ = z z (Γ z + y 2 4 ) in R2 (, + ) Γ(y) = 1 2π log 1 y, (78) where z = 2 z i an z = 2 ξ iz i. This property is formulate as follows, using X = C 0 (R 2 R 2 ) [(C 0 (R 2 ) R) R] [R (C 0 (R 2 ) R)]. First, let E be the close linear hull of E 0 = {ρ φ + ψ φ C 2 0(R 2 ), ψ X } L (R 2 R 2 ), where C 0 (R 2 R 2 ) is the set of continuous functions on R 2 R 2 vanishing on R 2 { } { } R 2. Then, the mapping s (, + ) φ, ˆζ(, s) R is locally absolutely continuous for any φ Y. There are, furthermore, ( ) 2 0 K = K(,, s) E, K(,, s) E χ 1 i λ i a.e. s an 0 ζ L (, + ; M 0 (R 2 )), ζ(r 2, s) 0 ζ L (, + ; M 0 (R 2 )), ζ(r 2, s) λ i ξ i λ i

29 Simultaneous blowup an mass separation 27 satisfying for a.e. s, where K(,, s) X = ζ(y, s) ζ(y, s) s φ, ˆζ(y, s) = φ, ζ(y, s) + 1 y φ, ζ(y, s) ρ0 φ, K(,, s) (79) ρ 0 φ = ρ φ (y, y ) = Γ(y y ) ( φ(y) φ(y )). Now, we take the scaling back transformation use in [33]: Then, we obtain a weak solution to where Then it hols that ζ i (y, s) = e s A i (y, s ), i = 1, 2 s = e s/2 y, s = e s/2. s  = à A Γ A in R2 (, 0), A = A i,  = χ 1 i A i, à = ξ i A i. A i (R 2, s ) = ζ i (R 2, s) = m i (x 0 ). Next, we take the translation limits as in [29]. Thus any s k + amits {s k } {s k} such that Â(y, s s k) ˆB(y, s) 0 in C (, + ; M(R 2 )) A i (y, s s k) B i (y, s) 0 in L (, + ; M(R 2 )), i = 1, 2 with the limit measures B i (y, s), i = 1, 2, an ˆB(y, s) = χ 1 i B i (y, s), where M(R 2 ) = [C 0 (R 2 ) R]. Since the space M(R 2 ) envelopes the total masses of A i (y, s), i = 1, 2, it hols that B i (R 2, s) = m i (x 0 ), i = 1, 2.

30 28 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki These measures form a weak solution to s ˆB = B B Γ B in R 2 (, + ), (80) where B = B 1 + B 2 an B = ξ 1 B 1 + ξ 2 B 2. Then, we argue similarly to the case of a single component [19] (see also [32]), using a smooth function c = c(s), s 0, satisfying 0 c (s) 1, 1 c(s) 0, an { s 1, 0 s 1/4 c(s) = 0, s 4. We have C, δ > 0 such that if it hols that ( 2 σ 1 m i (x 0 )) 4 2π ξ i m i (x 0 ) > 0, s c( y 2 ) + 1, ˆB(y, s) C c( y 2 ) + 1, ˆB(y, s) δσ for a.e. s. Therefore, if is the case, we obtain a contraiction. Hence it hols that c( y 2 ) + 1, ˆB(y, 0) < η δσ C c( y 2 ) + 1, ˆB(y, s) < 0, s 1, c( y 2 ) + 1, ˆB(y, 0) η. (81) Equation (80), on the other han, is invariant uner the scaling transformation B µ i (y, s) = µ2 B i (µy, µ 2 s), i = 1, 2, where µ > 0 is a constant. Since C, δ > 0 are absolute constants an σ is invariant uner this transformation, inequality (81) applie for B µ i (y, s), i = 1, 2, takes the form c( y 2 ) + 1, ˆB µ (y, 0) = c(µ 2 y 2 ) + 1, ˆB(y, 0) η, (82) where ˆB µ (y, s) = χ 1 i B µ i (y, s).

31 Simultaneous blowup an mass separation 29 However, since 0 c(µ 2 y 2 ) an lim µ + c(µ 2 y 2 ) + 1 = 0, y R 2 we obtain η 0 by the ominate convergence theorem, a contraiction. Hence it hols that σ 0, that is, (29). For the proof of (30), we use the total secon moment of the scale solution. First, similarly to the scalar case [25, 22] (see also [33]), Lemma 5 applie to φ = x x 0 2 φ x0,r implies the convergence an the uniform bouneness of the global secon moment of ˆζ(y, s): 0 y 2, ˆζ(y, s) = I(s) C (83) with a constant C > 0. If x 0, we moify the above φ = x x 0 2 φ x0,r, using a conformal mapping as in the proof of Theorem 10. Then, we obtain by (79), where I s min,2 χ i I σ(x 0 ), σ(x 0 ) Then it follows that ( 2 4 m i (x 0 )) 4 m (x 0 ) I(s) max,2 χ 1 i σ(x 0 ) from (83). Then, since I(s) 0, we obtain (30). a.e. s (, + ) ξ i m i (x 0 ). Now we turn to the proof of Theorem 4. In fact, having proven Theorem 3, we obtain σ(x 0 ) = 0, an hence I 0 which implies ˆζ(y, s) = ˆζ(R 2, s)δ 0 (y), s R. Then it hols that supp ζ i (y, s) {0}, an hence ζ i (y, s) = m i (x 0 )δ 0 (y), a.e. s, i = 1, 2. (84) The convergence (74) is thus refine as z i (y, s + s k) ζ i (y, s) in C (, + ; M 0 (R 2 )) similarly to the prescale case. Furthermore, the limit measures are prescribe as in (84), an hence we obtain (23) in C (, + ; M 0 (R 2 )) as s +. The latter part of Theorem 4, relation (24), is obvious from this convergence an (28).

32 30 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki 5 Proof of Theorems 5 an 6 Let = B(0, 1) an u i = u i ( x, t), i = 1, 2, in (ISP ). In this case, the blowup criterion (63) of Theorem 10 is reuce to a component-wise conition [9]. First, we confirm this property, using a metho of symmetrization introuce by [24]. In fact, the Poisson equation (9) is reuce to rv r (r, t) = r 0 s ( u(s, t) λ ) s r 0 s ( u 1 (s, t) λ ) s, recalling λ = λ 1 + λ 2 an λ i = u i0 1, i = 1, 2. Then, we obtain x 2 u 1 (x, t) x = x 2 ( 1 u 1 χ 1 u 1 v) x t x <l x <l = x 2 ( 1 u 1r χ 1 u 1 v r ) s 2x ( 1 u 1 χ 1 u 1 v) x = x =l x =l x <l l 2 ( 1 u 1r χ 1 u 1 v r ) 2 1 (x ν)u 1 s l +4πχ 1 r 2 u 1 v r r 0 for 0 < l 1 an r = x with ( 1 u 1r χ 1 u 1r v r ) s = an x =l l 0 = t x <l u 1 x r 2 u 1 v r r l 0 x <l ru 1 (r, t)r { } 2 = 1 l ru 1 (r, t) r + λ r x <l u x ( 1 u 1 χ 1 v) x 0 l 0 ( s u 1 (s, t) λ ) r 3 u 1 (r, t) r. Thus, it follows that x 2 u i (x, t) x l 2 u i (x, t) x + 4 i u i (x, t) x t x <l t x <l x <l { } 2 χ i u i (x, t) x + χ iλ x 2 u i (x, t) x (85) 2π x <l x <l s

33 Simultaneous blowup an mass separation 31 for i = 1. This inequality is also vali to i = 2, an thus we obtain the following theorem. Here, ifferently from Theorem 10, the blowup criterion is taken only for one component. Theorem 11 ([9]) Given a raially symmetric solution, let λ i = u i0 1 an ξ i = i /χ i. If λ i > 8πξ i an x 2 u i0 1 1, then u i = u i ( x, t) cannot exist global-in-time, where i = 1, 2. To prove Theorem 5, we erive [ x 2 u i (x, t) x x <l [4 i + t2 t 1 + χ iλ ] t=t2 t=t 1 [l 2 u i (x, t) x χ i x <l ] x <l x 2 u i (x, t) x u i (x, t) x x <l 2π ( t x <l ] t=t2 t=t 1 u i (x, t) x) 2 from (85), where 0 t 1 < t 2 < T. Using the backwar self-similar transformations (22) with x 0 = 0, this inequality means [e s y 2 z i (y, s) y y <le s/2 [4 i s2 + e s s 1 + χ iλ e s ] s=s2 s=s 1 [l 2 y <le s/2 z i (y, s) y χ i 2π ( y <le s/2 y 2 z i (y, s) y ] s z i (y, s) y y <le s/2 ] s=s2 s=s 1 y <le s/2 z i (y, s) y) 2 for log T s 1 < s 2 < +. Here, we put l = be s, s 1 = s, an s 2 = s +1, to make s +. Then, convergence (23) implies 0 4 i m i χ i 2π m2 i, i = 1, 2, where m i = m i (x 0 ), x 0 = 0. Since S is finite, it hols that S = {0}, an hence we obtain (25). We may assume i = 1 to prove Theorem 6, that is, ξ 1 /ξ 2 > 2 an u 10 1 < 8π(ξ 1 2ξ 2 ). By ξ 1 /ξ 2 > 2, as we have notice, (m 1, m 2 ) = (0, 8πξ 2 ) is the

34 32 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki only point on the curve ( ) 2 m i = 8π ξ i m i satisfying 0 m i 8πξ i, i = 1, 2, m 1 < 8π(ξ 1 2ξ 2 ). Using (38), on the other han, we have m 1 λ 1 < 8π(ξ 1 2ξ 2 ), an hence it hols that (m 1, m 2 ) = (0, 8πξ 2 ). The proof of Theorem 6 is complete. A A total collapse mass estimate from below Here we confirm that the localization of Theorem 9 concerning the existence of a global-in-time solution implies either (30) or (31). This argument is use for the single component case [32, 33, 29], which, however, oes not provie a sharp inequality in the multi-component case. Inequality (30) is thus always selecte. We note, however, that (31) is also true for raially symmetric solutions, which has not been known for the other case. If it is vali, then Theorem 6 hols even for non-raially symmetric solutions. First observation is the following lemma. Lemma 7 Given x 0 S, let φ = φ x0,r, 0 < R 1, an let ũ i = u i φ, i = 1, 2. Then, it hols that lim sup F ξ1,ξ 2 (ũ 1 (, t), ũ 2 (, t)) < +. (86) Proof: From (39) it follows that u 1t ( 1 log u 1 χ 1 v)φ x = u 1 ( 1 log u 1 χ 1 v) [( 1 log u 1 χ 1 v)φ]x = u 1 ( 1 log u 1 χ 1 v) 2 φ u 1 ( 1 log u 1 χ 1 v) ( 1 log u 1 χ 1 v) φ x u 1 ( 1 log u 1 χ 1 v) ( 1 log u 1 χ 1 v) φ x. (87)

35 Simultaneous blowup an mass separation 33 We have S B(x 0, 2R) = {x 0 } for 0 < R 1, an then the right-han sie is estimate from above by a constant inepenent of t, using parabolic-elliptic regularity of (u 1, u 2, v). Henceforth, C i, i = 1, 2,, 6, enote positive constants inepenent of t an R. First, the left-han sie of (87) is equal to u 1t ( 1 log u 1 χ 1 v)φ x = ũ 1t ( 1 log ũ 1 1 log φ χ 1 v) x = ũ 1t ( 1 log ũ 1 χ 1 v) x 1 u 1 (φ log φ) x, t which implies 1 ũ 1 (log ũ 1 1) x χ 1 v, ũ 1t C 1 + t t Similarly, it hols that 2 ũ 2 (log ũ 2 1) x χ 2 v, ũ 2t C 2 + t t an hence t ξ i ũ i (log ũ i 1) x v, ũ t C 3 + t Here, the parabolic-elliptic regularity ensures 1 u 1 (φ log φ) x. 2 u 2 (φ log φ) x [ ] ξ i u i φ log φ x. v, ũ t = φv, φu t + (1 φ)v, φu t = φv, φu t + O(1) an also for h = h(x, t) satisfying (φv) = φu + h, (φv) ν = 0 h L ( (0,T )) + h t L ( (0,T )) C 4. Therefore, from the elliptic regularity it follows that φv = ( ) 1 (φu) + g with g = g(x, t) such that g L ( (0,T )) + g t L ( (0,T )) C 5. (88)

36 34 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki Hence we obtain ũ t, v = ũ t, φv + O(1) = ũ t, ( ) 1 ũ + g(, t) + O(1) = 1 2 t ũ, ( ) 1 ũ + t ũ, g ũ, g t + O(1) = 1 2 t ũ, ( ) 1 ũ + ũ, g + O(1) t by (38) an (88). We thus en up with t F ξ 1,ξ 2 (ũ 1 (, t), ũ 2 (, t)) C 6 + [ ] ξ i u i (, t) φ log φ + ugφ x, t an then (86) follows from (38) an (88). Theorem 12 We have either (30) or (31) for any x 0 S. Proof: If these inequalities are not fulfille, there is σ i > 1, i = 1, 2, such that ( ) 2 m i m (x 0 ) ξ i m i for any 0 < m i < σ i m i (x 0 ), i = 1, 2, where ξ i = i /χ i. Then (18) guarantees the existence of θ i > 1, i = 1, 2, such that θ i ũ i (, t) 1 < σ i m i (x 0 ) for 0 < T t 1 an 0 < R 1. If x 0, we have m (x 0 ) = 4π. Then Lemma 3 is applicable. We have that lim inf F ξ1,ξ 2 (θ 1 ũ 1 (, t), θ 2 ũ 2 (, t)) >. (89) Combining (89) with (86), we obtain lim sup ũ i (log ũ i 1) x < + by θ i > 1, i = 1, 2. This inequality implies (66), an hence x 0 S, a contraiction. In the case of x 0, we use Lemma 8 below in stea of Lemma 3. Actually this lemma is a irect consequence of equality (55) with (54) an inequality (43) vali to (λ 1, λ 2 ) in the parameter region (42). Then, either (30) or (31) is obtaine with m (x 0 ) = 8π. Lemma 8 If λ i, i = 1, 2, are in the parameter region (42), then it hols that inf{f ξ1,ξ 2 (u 1, u 2 ) u i 0, u i 1 = λ i, supp u i, i = 1, 2} >.

37 Simultaneous blowup an mass separation 35 B Criteria for mass separation an simultaneous blowup First, mass separation is the case of raially symmetric solution if an only if the aggregation to the component forming a collapse is slower than that of the other. It is proven by the same ientity use for the proof of Theorem 5. Theorem 13 Let be a isc with center at the origin, u i = u i ( x, t), i = 1, 2, an T < +. Then, there is k {1, 2} such that m i (0) = 0 for i k (an hence m k (0) = 8πξ k by (19)) if an only if for any b > 0. lim x < x <b(t t) 1/2 u k ( x, t)u i ( x, t)xx = 0 (90) Proof: We have only to use a refine form of (85), x 2 u i (x, t) x = l 2 u i (x, t) x + 4 i u i (x, t) x t x <l t x <l x <l { 2 χ l r i u i (x, t) x} 4πξ ru i (r, t)r su j (s, t)s 2π x <l + χ iλ x <l x 2 u i (x, t) x an apply the same argument to the proof of Theorem 11. The final theorem shows that the simultaneous blowup hols even for non-raially symmetric solutions if one can prescribe the rate of convergence in (18). To state the result, let µ j (x, T ) = m j (x 0 )δ x0 (x) + f j (x)x, j = 1, 2 µ(x, T ) = x 0 S µ i (x, T ). It is proven by the total mass quantization, (19), combine with a linear analysis to each component u i (x, t), i = 1, 2. Theorem 14 If (25) an lim inf r 0 0 inf f i > 0 (91) B(x 0,r) 0

38 36 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki in (18), an if as t T with β > 1, it hols that where x 0 S, R > 0, an i = 1, 2. u(, t)x µ(x, T ) M() = o((t t) β ) (92) lim sup u i (, t) L ( B(x 0,R)) = + (93) Proof: The L 1 -estimate to the Poisson part (9) implies lim sup v(, t) W 1,q () < +, 1 q < 2 (94) by u(, t) 1 = λ (see [3]). We recall that this elliptic estimate is erive from the uality argument, using the bouneness of ( ) 1 : W 1,q () = W 1,q 0 () L (), 1 q + 1 q = 1, that is, ( ) 1 f C q f W 1,q, which is erive from Stampacchia s truncation metho. Since we use the triple W 1,q 0 () L 2 () = L 2 () W 1,q (), the space W 1,q 0 () is ense in W 1,q (). We use the Sobolev imbeing W 1,q 0 () L p () an the L p -elliptic regularity with p > 2, which implies the bouneness of ( ) 1 : L p () C(). Since L p () is ense in W 1,q () by the above reason, we obtain ( ) 1 : W 1,q C(), which implies the bouneness of ( ) 1 : W 1,q 0 () M(), 1 < q < 2. Hence (92) implies for any 1 q < 2, where v T (x) = v(, t) v T W 1,q () = o((t t) β ) (95) x 0 S m(x 0 )G(x, x 0 ) + G(x, x )f(x )x with m(x 0 ) = 2 m i(x 0 ) an f(x) = 2 f i(x). Given φ = φ x0,r with x 0 S, 0 < R 1, it hols that [ ] 1 v(, t)φ = γ log x x 0 + g φ + o((t t) β ) in L q (), 1 < q < 2, as t T, where g = g(x) is a smooth function inepenent of 0 < R 1 an { m(x0 )/2π, x γ = 0 m(x 0 )/π, x 0.

39 Simultaneous blowup an mass separation 37 We assume the existence of R 0 > 0 satisfying lim sup u 1 (, t) L ( B(x 0,R 0)) < +. (96) Here we use x x 0 2 φ u 1 (x, t)x t = [ 1 ( x x 0 2 φ) + χ 1 v ( x x 0 2 φ)]u 1 (x, t) x = [( χ 1 (x x 0 ) v)φ (x x 0 ) φ + 1 x x 0 2 φ + χ 1 x x 0 2 v φ]u 1 (x, t) x. First, we have φ = O(R 1 ), an therefore, x x 0 2 [ v φ]u 1 (x, t) x = O(R 3 ) sup t [0,T ) as R 0. Next, we obtain [4(x x 0 ) φ + x x 0 2 φ]u 1 (x, t) x = o(r 2 ) sup t [0,T ) by (62). It also hols that [ χ 1 (x x 0 ) v]φ (4 1 2χ 1 γ)φ + [2(x x 0 ) g]φ q c(t) for 1 < q < 2 with c(t) = o((t t) β ). Here, we have (x x 0 ) g φ x = O(R 3 ), while (19) with (m 1 (x 0 ), m 2 (x 0 )) (0, 0) implies m(x 0 ) m i (x 0 ) min{ξ 1 m (x 0 ), ξ 2 m (x 0 )} by (25) an in particular, γ > 2ξ 1. From the assumption (96), therefore, we have 0 < A(R) = o(r 2 ), δ 1 > 0, an 0 < c(t) = o((t t) β ) such that x x 0 2 φ x0,r u 1 (x, t)x t δ 1 φ u 1 (x, t)x + A(R) + c(t)r 2/q δ 1 φ f 1 (x)x + o((t t) β ) + A(R) + c(t)r 2/q

40 38 Elio Euaro Espejo, Angela Stevens, an Takashi Suzuki by (92) with j = 1. From (91) with i = 1, we have δ 2 > 0 such that φ f 1 (x)x δ 2 R 2 for 0 < R 1. We thus en up with x x 0 2 φ x0,r u 1 (x, t)x δr 2 + o((t t) β ) + c(t)r 2/q, t where δ > 0 is a constant. So we obtain x x 0 2 φ f 1 (x)x δ(t t)r 2 + C 9 R 4 T +o((t t) 1+β ) + R 2/q c(t )t. (97) We set R = b(t t) 1/2 with 0 < b < C 9 /δ. Then T R 2/q c(t )t = o((t t) 1/q +1+β ) = o((t t) 2 ), t hols for 0 < q 2 1 by β > 0. Since β > 1, however, the right-han sie of (97) is negative for 0 < T t 1, which is impossible. Hence (91) with j = 1 implies lim sup u 1 (, t) L ( B(x 0,R)) = +. The proof for the other case, j = 2, is similar. Acknowlegements This research is sponsore by JST project, CREST (Alliance for Breakthrough Between Mathematics an Sciences). t References [1] P. Biler, Local an global solvability of some parabolic systems moelling chemotaxis, Av. Math. Sci. Appl., 8 (1998), [2] P. Biler, W. Hebisch, an T. Nazieja, The Debye system: existence an large time behavior of solutions, Nonlinear Analysis, 23 (1994),