Blowup of solutions to generalized Keller Segel model

Transcription

1 J. Evol. Equ. 1 (21, Birkhäuser Verlag Basel/Switzerlan /1/ , publishe onlinenovember 6, 29 DOI 1.17/s Journal of Evolution Equations Blowup of solutions to generalize Keller Segel moel Piotr Biler an Grzegorz Karch Abstract. The existence an nonexistence of global in time solutions is stuie for a class of equations generalizing the chemotaxis moel of Keller an Segel. These equations involve Lévy iffusion operators an general potential type nonlinear terms. 1. Introuction We consier in this paper the following nonlinear nonlocal evolution equation generalizing the well known Keller Segel moel of chemotaxis t u + ( α/2 u + (ub(u =, (1.1 for (x, t R R +, where the anomalous iffusion is moele by a fractional power of the Laplacian, α (1, 2, an the linear (vector operator B is efine (formally as B(u = (( β/2 u. (1.2 For β (1, ], an 2, one may express the nonlocal nonlinearity in (1.1 using convolution operators since x y B(u(x = s,β u(y y, (1.3 R x y β+2 with some s,β >, an the assumption β>1isneee for the convergence of this integral. Of course, the choice α = 2, β = 2in(1.1 (1.3 correspons to the usual Keller Segel system stuie mainly in space imensions = 1, 2, 3. It is well known that if β = 2, the one-imensional system (1.1 (1.3 possesses global in time solutions not only in the case of classical Brownian iffusion α = 2 but also in the fractional iffusion case 1 <α<2aswas shown in Escuero [17]. On the other han, there are many results on the nonexistence of global in time solutions with large initial ata if 2 Mathematics Subject Classification (2: 35Q, 35K55, 35B4 Keywors: Nonlocal parabolic equations, Blowup of solutions, Lévy iffusion, Chemotaxis, Moment metho.

2 248 P. Biler an G. Karch J. Evol. Equ. an α = 2, β = 2, see, e.g., [1,2,12,19,22]. Even if 2, α = 2 an 1 <β, there are results on the blowup of solutions with suitably chosen initial ata, see [1] (caution: the notation in [1, Prop. 4.2] iffers from that in the present paper. Let us finally recall that, in the limit case α = 2, β =, massm = R u (x x is the critical parameter for the blowup, see [1, Prop. 4.1] an [14]. The usual metho of proving the nonexistence of global in time nonnegative an nontrivial solutions, use in the abovementione papers, consists in the stuy the evolution of the secon moment of a solution w 2 (t = R x 2 u(x, t x an to show (via suitable ifferential inequalities that w 2 (t vanishes for some t >. The secon moment of a typical solution to an evolution equation with fractional Laplacian cannot be finite, see, e.g., [13]. Hence, our goal in this paper is to generalize the classical virial metho an to show the blowup of solution system (1.1 (1.3 by stuying moments of lower orer γ (1, 2 w γ = x γ u(x x. (1.4 R We show the extinction of a moment w γ for some T >, which implies lim t T u(t p = for every p (1, ], seeremark2.8. Thus, we mean this phenomenon as blowup for (1.1. After this paper was complete, we iscovere a recent preprint [24] where the authors show the blowup of solution to system (1.1 (1.3 with fractional iffusion in the particular case = 2 an β = 2. Our argument is ifferent than that in [24], shorter, seems to be more irect, an applies in more general situations. Moreover, we are able to formulate a simple conition on the initial ata which leas to the blowup in a finite time of the corresponing solution. Notation. The L p -norm of a Lebesgue measurable, real-value function v efine on R is enote by v p. The constants (always inepenent of x, t will be enote by the same letter C, even if they may vary from line to line. Sometimes, we write, e.g., C = C( when we want to emphasize the epenence of C on a parameter. 2. Main results The crucial role in the approach in this paper is playe by the following scaling property of system (1.1 (1.3 u λ (x, t = λ α+β 2 u(λx,λ α t for all λ>, (2.1 in the sense that if u is a solution to (1.1 (1.3, then u λ is so. In particular, in our construction of solutions to (1.1 (1.3 we use the fact that the usual norm of the Lebesgue space L /(α+β 2 (R is invariant uner the transformation u (x λ α+β 2 u (λx for every λ>.

3 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 249 THEOREM 2.1. Assume that 2, α (1, 2], an β (1, ]. Let { } max α + β 2, 2 < p. + β 1 i For every u L p (R there exists T = T ( u p an the unique local in time mil solution u C([, T ], L p (R of system (1.1 (1.3 with u as the initial conition. ii There is ε>such that for every u L /(α+β 2 (R satisfying u /(α+β 2 ε (2.2 there exists a global in time mil solution u C([,, L p (R of system (1.1 (1.3 with u as the initial conition. Moreover, if u (x, then the solution u in either i or ii above is nonnegative. Finally, if u L 1 (R, then the corresponing solution conserves mass u(x, t x = u (x x M. (2.3 R R Recall that α>1isausual assumption ([1, Th. 2.2], [8,9] which permits us to control locally the nonlinearity in (1.1 (1.3 by the linear term. The results state above can be easily generalize for equations with general Lévy iffusion operators consiere in [8,9] but we o not pursue this question here. We refer the reaer to the abovementione papers, as well as [1], for physical motivations to stuy such equations. On the other han, motivations stemming from probability theory (propagation of chaos property for interacting particle systems can be foun in, e.g., [7]. REMARK 2.2. For α (1, 2 an β>1satisfying α + β> + 2, it can be shown that local in time solutions can be continue to the global in time ones, see [1, Th. 3.2]. However, in [1], another approach (via weak solutions is use to construct solutions of system (1.1 (1.3. The proof of Theorem 2.1 on local an global solutions to (1.1 (1.3 follows a more or less stanar reasoning which we sketch in Sect. 3. Our main goal, however, is to prove the finite time blowup of nonnegative solutions to the nonlocal system (1.1 (1.3 butaprioriless regular than those in Theorem 2.1. THEOREM 2.3. Assume that 2. The nonnegative solution of (1.1 (1.3 with a nonnegative an nonzero initial conition u(x, = u (x cannot exist globally in time in each of the following cases: i (large mass for α = 2, β =, u L 1 (R,(1 + x 2 x, an if M = R u (x x > 2 s,β, with the constant s,β efine in (1.3; in particular: s 2,2 = 2π 1 threshol value of M is 8π if = 2; so that the

4 25 P. Biler an G. Karch J. Evol. Equ. ii (high concentration for α (1, 2] an β (1, ] satisfying α + β< + 2, u L 1 (R,(1 + x γ x for some γ (1,α, an if R x γ ( γ u (x x +2 α β c u (x x (2.4 R u (x x R for certain (sufficiently small constant c > inepenent of u. The result state in i is essentially containe in [14]. The conition for blowup in the form (2.4 appeare alreay in [2] an [26], of course, for α = 2 only. Note that i is a limit case of ii. Inee, (2.4 written as ( R x γ u (x x R u (x x +2 α β γ cm, becomes a conition on (sufficiently large mass: 1 cm, when (α + β ( + 2. We have to emphasize that Theorem 2.3(ii contains a result which is new even for the classical parabolic-elliptic Keller Segel moel (i.e., equations (1.1 (1.3 with α = β = 2. Inee, for 3, the conitions from part ii of Theorem 2.3 guarantee the blow up in a finite time if the moment of orer γ of the initial conition is finite for some γ (1, 2. All other known proofs of the blowup require just γ = 2. The following result for = 2 is also an immeiate consequence of Theorem 2.3(ii. COROLLARY 2.4. Assume that α = β = = 2 in equations(1.1 (1.3. Suppose that there exists γ (1, 2 such that u L 1 (R 2,(1 + x γ x. There exists M γ > such that if u (x x > M γ, R 2 then the nonnegative solution of (1.1 (1.3 with the initial conition u(x, = u (x ceases to exist in a finite time. It is conjecture that M 2 = M γ = 8π for every γ (1, 2 an we expect that it can be shown by a careful analysis of constants appearing in inequalities (4.6 (4.1 an in (4.18 (4.2 in the proof of Theorem 2.3. However, the assumptions in Corollary 2.4 are not optimal: nonnegative solutions to equations (1.1 (1.3 with α = β = = 2 blow up in finite time uner the assumption that their mass is larger than 8π an no moment conition impose on the initial ata is necessary, see [22, Th. 1.5] an [4, Prop. 2.2, Th. 3.1] in the raially symmetric case for a etaile presentation. REMARK 2.5. Due to the translation invariance of problem (1.1 (1.3, the conitions on moments can be impose on the quantity inf x R x x γ u (x x = R x x γ u (x x, R where x = ( R xu (x x/( R u (x x is the center of mass of u.

5 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 251 REMARK 2.6. Let us observe that the assumptions (2.2 an (2.4 are in a sense complementary ue to the following elementary inequality involving the L p -norms, mass M = R u(x x, an the moment w γ = R x γ u(x x of a nonnegative function u ( ( M γ 1 1 p u p CM. (2.5 To prove (2.5 observe that w γ R γ R \B R ( u(x x, so that u(x x = M u(x x M R γ w γ. B R ( R \B R ( ( Multiplying both sies of this inequality by R 1 1 p we get with C = ω 1 1 p w γ ( 1 C u p = u p p ( (x x R B R ( 1 1 ( p 1p 1 x R ( ( R 1p 1 u(x x R 1p 1 M R γ w γ. B R ( Taking the optimal R, i.e., R γ = Cw γ /M, we get (2.5. Now, it is clear that if conition (2.4 for blowup is satisfie for some γ (1, 2] an a suitable constant c, then for ( p = /(α + β 2 appearing in Theorem 2.1.ii, we have u p CMM γ +2 α β γ 1 1 p = C, so conition (2.2 for global existence is violate for sufficiently small ε>. Vice versa, if (2.2 is satisfie, (2.4 cannot be true with small constants c. REMARK 2.7. Note that one can prove in a similar way the inequality ( M u M p CM w γ ( γ 1 1 p, (2.6 where the Morrey space norm u M p ( C u p is efine as ( sup R 1p 1 u(x x, R>, x R B R (x see [2, (15] in the particular case γ = 2. The scale of Morrey spaces is relevant to stuy mean fiel type problems relate to the Keller Segel moel, since the Morrey norms are particularly well suite to measure the (local concentration of ensities, see, e.g., [3]. REMARK 2.8. We prove Theorem 2.3 an Corollary 2.4 by showing the extinction in a finite time of the function w(t = R ϕ γ (xu(x, t x where ϕ γ (x is smooth an behaves like x γ with some γ (1,αfor large x,see(4.1 below. By (2.5 an (2.3

6 252 P. Biler an G. Karch J. Evol. Equ. this might be interprete as the blowup of certain norms (in particular, L p -norms for each p (1, ] of nonnegative solutions but we stress on the fact that a nonnegative solution coul cease to exist even before the critical time T < suggesting by (2.5 that lim t T u(t p =. Note that if α<2, we cannot expect the existence of higher orer moments w γ efine in (1.4 with γ α. Inee, even for the linear equation t v +( α/2 v =, the funamental solution p α (x, t behaves like p α (x, t ( t /α + x +α /t 1, an therefore the moment (1.4 with γ α cannot be finite, see [13] an references given there. Thus, we cannot apply the usual reasoning which involves an analysis of the evolution of the secon moment w 2 of the solution because the integral efining w 2 may iverge. We recall in Proposition 3.4 a result showing that the moment (1.4 isfinitefora large class of initial conitions in the case of γ<α. 3. Existence of solutions We are going to construct solutions to system (1.1 (1.3 via the following integral formulation t u(t = S α (tu S α (t τ(u(τbu(τ τ, (3.1 i.e., we consier mil solutions. Here S α (tu = p α (t u is the solution to the linear Cauchy problem t v + ( α/2 v =, v(x, = u, (3.2 an p α (x, t is the funamental solution of (3.2 which can be represente via the Fourier transform p α (ξ, t = e t ξ α. In particular, p α (x, t = t /α P α (xt 1/α, where P α is the inverse Fourier transform of e ξ α,see[18, Ch. 3] an [13]formore etail. It is well known that for every α (, 2 the function P α is smooth, nonnegative, an satisfies the (optimal estimates < P α (x C(1 + x (α+ an P α (x C(1 + x (α++1 (3.3 for a constant C an all x R. Hence, it follows immeiately from the Young inequality for the convolution an from the self-similar form of the kernel p α (x, t that for every 1 q p there exists C = C(p, q,α> such that an S α (tu p Ct α S α (tu p Ct α for every u L q (R an all t >. ( 1q 1 p u q (3.4 ( 1q 1 p 1 α u q (3.5

7 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 253 The construction of solution to (3.1inL p spaces is base on the following abstract result, see, e.g., [23], [25]. LEMMA 3.1. Let (X, X be a Banach space an H : X X X a boune bilinear form satisfying H(x 1, x 2 X η x 1 X x 2 X for all x 1, x 2 X an a constant η>. Then, if <ε<1/(4η an if y X is such that y <ε,the equation u = y + H(u, u has a solution in X such that u X 2ε. This solution is the only one in the ball B(, 2ε. We skip an easy proof of this lemma which is a irect consequence of the Banach fixe point theorem. Sketch of the proof of Theorem 2.1. The proof consists in constructing solutions to the quaratic equation (3.1 using Lemma 3.1 with y = S α (tu an with the bilinear form t H(u,v= S α (t τ(u(τbv(τ τ. (3.6 Local existence of solutions It suffices to obtain estimates require by Lemma 3.1 in the Banach space X p T = C([, T ], L p (R supplemente with the usual norm u p = sup X t [,T ] u(t p. T By inequality (3.4, we immeiately obtain S α ( u p u X p. T Combining the efinition (1.3 of the operator B with the Hary Littlewoo Sobolev inequality we obtain B(u q s,β +β 1 u q C u p (3.7 for every 1 < p < q < satisfying 1 p β 1 the Höler inequality lea to S α (t τ(ub(v p C(t τ 1 α α C(t τ 1 α α = q 1. Moreover, inequality (3.5 an ( 1r 1 p ( 1r 1 p ub(v r u p B(v q. (3.8 where r 1 = 1 p + q 1. Hence, by inequalities (3.7 an (3.8, there exists a constant C such that for all u,v X p T, the bilinear form (3.6 satisfies H(u,v X p T C sup t [,T ] CT 1 1 α α t (t τ 1 α α ( 1r 1 p ( 1r 1 p u(τ p v(τ p τ u X p T v X p T. (3.9 In estimates (3.9, we have use the relations r 1 = 1 p + q 1 = 2 p β 1, an we assume that

8 254 P. Biler an G. Karch J. Evol. Equ. p > /(α + β 2 in orer to have 1/α + (/α(1/r 1/p <1; p > 2/( + β 1 to guarantee that r > 1; p /(β 1 to be sure that r p. Choosing a sufficiently small T > in(3.9 we complete the proof of Theorem 2.1.i by an application of Lemma 3.1. Global in time solutions Here, the reasoning is completely analogous: using inequalities (3.4, (3.5, (3.7 (3.9 we estimate the bilinear form (3.6 in the Banach space X p = C([,, L /(α+β 2 (R { supplemente with the norm u C((,, L p (R α : sup t t> u X p = sup t> α u(t /(α+β 2 + sup t t> ( 1p } α+β 2 u(t p < ( 1p α+β 2 u(t p. We skip further etails of this stanar reasoning. Nonnegativity property To prove that u implies u(t for sufficiently regular solutions, it suffices to stuy the function u (x, t = max{ u(x, t, } an to follow the arguments either from [9, Prop. 3.1] or from [16, Prop. 2] in orer to show that u (x, t. Here, we o not give a etaile presentation because the proof from [24, Lemma 2.7] can be rewritten in this more general case. Thus, the nonnegativity property is natural for solutions of (1.1, an our results show that loss of regularity (blowup an loss of nonnegativity are intimately connecte. Conservation of the integral If u L 1 (R, one shoul repeat the fixe point argument from i (or from ii in the space X p T C([, T ], L1 (R (in X p C([,, L 1 (R, resp. to have u(t L 1 (R for all t [, T ] (t (,, resp.. Next, it suffices to integrate over R the both sies of equation (3.1. Using the following consequences of the Fubini theorem p α (t u (x x = u (x x R R an p α (t v(x x = for every v L 1 (R, R we conclue that R u(x, t x = R u (x x for every t [, T ]. REMARK 3.2. The reasoning from the proof of Theorem 2.1 follows the lines of the usual proof of the local in time existence (as well as the global in time existence for small initial conitions of solutions to system (1.1 (1.3 with α = 2. Since that argument is base on an integral representation analogous to that in (3.1 an on counterparts of ecay estimates from (3.4 to(3.5, it can be easily aopte to

9 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 255 the more general case of α (1, 2]. Examples of such a reasoning applie to various semilinear moels an realize in miscellaneous Banach spaces can be foun in [3,5,6,11,2,21,23,25]. Next, we recall weighte estimates of solutions to the linear Cauchy problem (3.2 which have been prove in, e.g., [5,13]. In the following, we use the weighte L spaces { } L ϑ (R = for a fixe ϑ. v L (R : v L ϑ ess sup x R (1 + x ϑ v(x < LEMMA 3.3. ([13, Lemma 3.1] Assume that v L α+ (R. There exists C > inepenent of v an t such that S α (tv L α+ C(1 + t v L α+, S α (tv L α+ Ct 1/α v L α+ + Ct 1 1/α v 1, With these estimates, we are in a position to construct solutions to system (1.1 (1.3 also in the weighte space L α+ (R L 1 (R L (R. PROPOSITION 3.4. Let α (1, 2] an b (1, ]. Assume that u is the solution of system (1.1 (1.3, constructe in Theorem 2.1, supplemente with the initial conition u L α+ (R. Then u C([, T ], L α+ (R. In particular, for each γ<α we have R x γ u(x, t x <. Sketch of proof. Here, it suffices to moify slightly the argument from [13, Proof of Prop. 3.3.i] written the case of a convection equation with the fractional Laplacian. In that reasoning (as well as in the above proof of Theorem 2.1, we construct solutions to equation (3.1 by applying Lemma 3.1 with the Banach space X T = C([, T ], L α+ (R. Obviously, y = S α ( u X T by Lemma 3.3. Next,weshow the following estimate of the bilinear form from (3.6 H(u,v XT CT 1 1/α u XT v XT for all u,v X T an a constant C inepenent of u,v. Here, in orer to estimate the singular integral operator (1.3 in the spaces L ϑ (R, one shoul follow the reasoning from [5, Sec. 2]. Let us omit other etails of this classical argument. 4. Blowup of solutions The main role in our proof of the blowup of solutions to (1.1 (1.3 is playe by the following smooth nonnegative weight function on R ϕ(x = ϕ γ (x (1 + x 2 γ/2 1 (4.1

10 256 P. Biler an G. Karch J. Evol. Equ. with γ (1, 2]. Since (1 + x 2 γ (1 + x γ 2, we have for each ε>, suitably chosen C(ε >, an for every x R ϕ(x x γ ε + C(εϕ(x. (4.2 Next, let us state two auxiliary results concerning the weight function ϕ which will be use in the proof of Theorem 2.3. Here, for a given ϕ C 2 (R, we enote by D 2 ϕ its Hessian matrix. Moreover, the scalar prouct of vectors x, y R is enote by x y. IfA is either a vector or a matrix, the expression A means its Eucliean norm. LEMMA 4.1. Let α (1, 2, γ (1,α, an ϕ be efine by (4.1. Then Proof. First note that by a irect computation we have ( α/2 ϕ L (R. (4.3 ϕ(x = γ(1 + x 2 γ 2 1 x (4.4 an x j xi ϕ(x = (γ(1 + x 2 δ ij γ(2 γx i x j (1 + x 2 γ 2 2. (4.5 In particular, for every R > there exists C(R,γ > such that for all x R we have ϕ(x C(R,γ x γ 1 an D 2 ϕ(x C(R,γ x γ 2. (4.6 Now, we apply the following Lévy Khintchine integral representation of the fractional Laplacian ( α/2 ϕ(x + y ϕ(x ϕ(x y ϕ(x = C(,α R y +α y (4.7 (with a suitable constant C(,α which is vali for every α (1, 2, see, e.g., [16, Th. 1] for a etaile proof of that version of Lévy Khintchine formula. Using the Taylor expansion an estimates (4.6 one can immeiately show that ( α/2 ϕ(x is well efine for every x R an, moreover, sup x R ( α/2 ϕ(x < for each R >. In orer to obtain an estimate uniform in x R, we assume that x 1 an we shall estimate the integral on the right-han sie of (4.7 for y x /2an y > x /2, separately. If y x /2, by the Taylor formula an the secon inequality in (4.6, we obtain ϕ(x + y ϕ(x ϕ(x y y 2 D 2 ϕ(x + sy s 1 C y 2 x + sy γ 2 s.

11 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 257 Since y x /2 an s [, 1] we can estimate x + sy x s y x y 1 2 x. Consequently, for γ 2 <, we obtain ϕ(x + y ϕ(x ϕ(x y y x /2 y +α y C x γ 2 y y +α 2 = C x γ α (4.8 y x /2 for all x 1 an a constant C > inepenent of x. If y x /2an x 1, we combine the first inequality from (4.6 (remember that γ 1 > with the Taylor expansion to show 1 ( ϕ(x + y ϕ(x y ϕ(x + sy s C y x γ 1 + y γ 1. Hence, ϕ(x + y ϕ(x ϕ(x y y > x /2 y +α y C ( x γ 1 y y +α 1 + y > x /2 y > x /2 y y +α γ = C x γ α (4.9 for all x 1 an a constant C > inepenent of x. Finally, inequalities (4.8 an (4.9 complete the proof because γ<α. REMARK 4.2. Note that above we have, in fact, prove that ( sup 1 + x α γ ( α/2 ϕ(x < for every γ (1,α. x R LEMMA 4.3. For every γ (1, 2], the function ϕ efine in (4.1 is locally uniformly convex on R. Moreover, there exists K = K (γ such that the following inequality hols true for all x, y R. ( ϕ(x ϕ(y (x y K x y x 2 γ + y 2 γ (4.1 Proof. Using the explicit expression for the Hessian matrix of ϕ in (4.5 we obtain ( γ(1 + x 2 y 2 γ(2 γ D 2 i x2 i y2 i + i = j x i x j y i y j ϕ(xy y = (1 + x 2 2 γ/2 (4.11

12 258 P. Biler an G. Karch J. Evol. Equ. for every x, y R. Now, by the elementary inequality x i x j y i y j 2 1 (x2 i y2 j + x2 i y2 j we immeiately obtain x i x j y i y j xi 2 y2 j. i = j i = j Consequently, xi 2 y2 i + x i x j y i y j xi 2 y2 j = x 2 y 2. (4.12 i i = j i, j Since γ (1, 2], applying estimate (4.12 to(4.11 we get the inequality D 2 ϕ(xy y γ(1 + (γ 1 x 2 y 2 (1 + x 2 2 γ/2 which leas irectly to the estimate from below D 2 (γ 1 y 2 ϕ(xy y (1 + x 2. ( γ/2 Finally, it follows from the integration of the secon erivative of ϕ that ( ϕ(x ϕ(y (x y = 1 Hence, using inequality (4.13 an the estimate D 2 ϕ (x + s(y x (x y (x y s. (1 + x + s(y x 2 1 γ/2 C(1 + x 2 γ + y 2 γ, vali for all x, y R, s [, 1] an a constant C > inepenent of x, y, s, one can easily complete the proof of Lemma 4.3. Proof of Theorem2.3. We consier the function w = w(t ϕ(x u(x, t x, R where ϕ is efine in (4.1 an 1 <γ <α. Note that, in view of inequalities (4.2, the quantity w is essentially equivalent to the moment w γ of orer γ of the solution u. Moreover, it satisfies the relation t w = ( α/2 u(x, tϕ(x x u(x, tbu(x, t ϕ(x x R R = ( α/2 ϕ(x u(x, t x R s,β u(x, tu(y, t ( ϕ(x ϕ(y (x y x y ( R R x y β+2 after using the efinition of the form B in (1.3 an the symmetrization of the ouble integral. This computation resembles the usual proof of blowup involving the secon moments, cf. [2,1,14,15].

13 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 259 i For α = 2 = γ (hence for ϕ(x = x 2 an for β =, the equality (4.14 can be rewritten as follows t w(t = 2M s,β M 2. Eviently, for M > 2/s,β, this implies the inequality w(t <forsome < T <, a contraiction with the global existence of nonnegative solutions. Thus, we recover the result in [1, Prop. 4.1] refine in [14,15]. ii For 1 <β an fixe M >, we are going to use the following simple ientity M 2 = u(x, tu(y, t x y R R x y ν = u(x, tu(y, t ( R R 1 + x 2 γ + y 2 γ δ ( 1 + x 2 γ + y 2 γ δ x y ν x y with some ν > an δ >. We apply now the Höler inequality with the exponents p > 1 an p = chosen so that p p 1 νp = β, δp = 1, νp + (2 γδp = γ. (4.15 Of course, such a choice of ν, δ, p is possible whenever β< an γ<2 because we only nee β + 2 γ = γ(p 1. Ifβ =, it suffices to take ν = an p = 2/γ > 1. As a consequence, we get M 2 J(t 1/p ( ( u(x, tu(y, t x y νp 1+ x 2 γ + y 2 γ δp 1/p x y, (4.16 R R where the integral J(t satisfies u(x, tu(y, t x y J(t = R R x y β 1 + x 2 γ + y 2 γ 1 u(x, tu(y, t ( ϕ(x ϕ(y (x y x y (4.17 K R R x y β+2 by Lemma 4.3. It follows from relations (4.15 an inequalities (4.2 that there exists a constant C 1 > such that x y νp ( 1 + x 2 γ + y 2 γ δp C 1 (1 + ϕ(x + ϕ(y. (4.18 Hence, (4.16 implies ( 1/p M 2 C 1/p 1 J(t 1/p M 2 + 2Mw(t. (4.19

14 26 P. Biler an G. Karch J. Evol. Equ. Going back to ientity (4.14 we obtain from Lemma 4.1 an from inequalities (4.17 (4.19 that M 2p t w(t C 2M C 3 (M 2 ( Mw(t p/p with C 2 = ( α/2 ϕ an a suitable constant C 3 >. Now, we fix for a while M = M in (4.2 so large in orer to have M 2p C 2 M C 3 (M 2 <. (4.21 p/p Hence, there exists C 4 = C 4 (M > such that for <w( C 4 we still have M 2p C 2 M C 3 (M 2 + 2M <. w( p/p It is clear that if initially <w( C 4 then, by inequality (4.2 with M = M, the function w(t is ecreasing in time. Moreover, M 2p t w(t C 2M C 3 (M 2 + 2M < w( p/p an, consequently, w(t <forsome< T <. This contraicts the global in time existence of regular nonnegative solutions of (1.1 (1.3. Finally, note that ue to the first inequality in (4.2, it suffices to assume w( x γ u (x x C 4 an u (x x = M, (4.22 R R in orer to obtain the blowup in a finite time of the corresponing solution. Now, assume that R u(x, t x = R u (x x = M = M. Recall that system (1.1 (1.3 is invariant uner the rescaling (2.1. Choosing λ α+β 2 = M /M we obtain R u λ (x, t x = R u λ (x x = M an, by (4.22, the blowup of the solution takes place uner the assumption x γ u λ R (x x C 4. Changing the variables an using the explicit form of λ we obtain the blowup of solutions to (1.1 (1.3 uner the following assumption on the initial conition x γ u (x x C 4 M 1+ R γ α+β 2 ( R u (x x 1+ γ +2 α β.

15 Vol. 1 (21 Blowup of solutions to generalize Keller Segel moel 261 Proof of Corollary 2.4. We follow the proof of Theorem 2.3. In particular, we choose M γ so large that the inequality (4.21 hols true for all M > M γ. This leas to the blowup of the corresponing solution uner the assumption (4.22 impose on the initial ata. To complete the proof, we use the scaling argument again. By (2.1, u λ (x, t = λ 2 u(λx,λ 2 t is a solution for every λ>. Note now that u λ R 2 (x x = u (x x an x γ u λ R 2 R 2 (x x = λ γ x γ u (x x. R 2 Hence, each initial ata u L 1 (R 2,(1 + x γ x satisfying R 2 u (x x = M > M γ leas to the blowup in a finite time of the corresponing solution because the moment conition in (4.22 can be satisfie replacing u by u λ an choosing λ large enough. Acknowlegments The preparation of this paper was partially supporte by the Polish Ministry of Science grant N /92, the POLONIUM project ÉGIDE no SG, an by the European Commission Marie Curie Host Fellowship for the Transfer of Knowlege Harmonic Analysis, Nonlinear Analysis an Probability MTKD-CT The authors are greatly inebte to Tomasz Cieślak for pointing them out the preprint [24] an to Takayoshi Ogawa for the reprint of [22]. REFERENCES [1] P. Biler, Local an global solvability of parabolic systems moelling chemotaxis, Av. Math. Sci. Appl. 8 (1998, [2] P. Biler, Existence an nonexistence of solutions for a moel of gravitational interaction of particles. III, Colloq. Math. 68 (1995, [3] P. Biler, The Cauchy problem an self-similar solutions for a nonlinear parabolic equation, Stuia Math. 114 (1995, [4] P. Biler, Raially symmetric solutions of a chemotaxis moel in the plane the supercritical case, 31 42, in: Parabolic an Navier-Stokes Equations, Banach Center Publications 81, Polish Aca. Sci., Warsaw, 28. [5] P. Biler, L. Branolese, On the parabolic-elliptic limit of the oubly parabolic Keller Segel system moelling chemotaxis, Stuia Mathematica 193 (29, [6] P. Biler, M. Cannone, I. Guerra, G. Karch, Global regular an singular solutions for a moel of gravitating particles, Mathematische Annalen 33 (24, [7] P. Biler, T. Funaki, W. A. Woyczyński, Interacting particle approximation for nonlocal quaratic evolution problems, Probab. Math. Stat. 19 (1999, [8] P. Biler, G. Karch, W. A. Woyczyński, Critical nonlinearity exponent an self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Analyse non Linéaire 18 (21, [9] P. Biler, G. Karch, W. A. Woyczyński, Asymptotics for conservation laws involving Lévy iffusion generators, Stuia Math. 148 (21, [1] P. Biler, W. A. Woyczyński, Global an exploing solutions for nonlocal quaratic evolution problems, SIAMJ.Appl.Math.59 (1998,

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