Finite time blow-up is shown to occur for radially symmetric solutions to a
critical quasilinear Smoluchowski-Poisson system provided that the mass of the
initial condition exceeds an explicit threshold. In the supercritical case,
blow-up is shown to take place for any positive mass. The proof relies on a
novel identity of virial type

Cieślak, Tomasz

Laurençot, Philippe

English

arXiv

Tomasz Cieślak

Philippe Laurençot

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242Partial Differential EquationsFinite time blow-up for radially symmetric solutions to a criticalquasilinear Smoluchowski–Poisson systemTomasz Cieślak a, Philippe Laurençot ba Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Polandb Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 118, route de Narbonne, 31062 Toulouse cedex 9, FranceReceived 15 October 2008; accepted after revision 6 January 2009Available online 10 February 2009Presented by Haïm BrezisAbstractFinite time blow-up is shown to occur for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson systemprovided that the mass of the initial condition exceeds an explicit threshold. In the supercritical case, blow-up is shown to takeplace for any positive mass. The proof relies on a novel identity of virial type. To cite this article: T. Cieślak, P. Laurençot, C. R.Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméExplosion en temps fini des solutions à symétrie radiale d’un système de Smoluchowski–Poisson quasilinéaire critique.L’explosion en temps fini est établie pour des solutions à symétrie radiale d’un système de Smoluchowski–Poisson quasilinéaire cri-tique dès que la masse de la donnée initiale dépasse un certain seuil. Dans le cas surcritique, l’explosion peut se produire pour toutemasse positive. L’argument principal de la démonstration est une nouvelle identité de type viriel. Pour citer cet article : T. Cieślak,P. Laurençot, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeL’objet de cette Note est l’étude de l’explosion en temps fini des solutions à symétrie radiale du système deSmoluchowski–Poisson généralisé (1)–(4), où B(0,1) désigne la boule unité de Rn, n  2, et M est la moyenne(spatiale) de u0. Le coefficient de diffusion a ∈ C 2([0,∞)) est supposé positif. Une propriété fondamentale des so-lutions de (1)–(4) est la positivité de u et l’invariance de sa moyenne (5). Lorsque a ≡ 1, il existe des solutions de(1)–(4) à symétrie radiale qui explosent en temps fini et ce phénomène se produit pour tout M > 0 si n  3 maisseulement pour M > 8π si n  2 [5,7]. Plus récemment, il a été établi dans [4] que deux comportements radicalementdifférents sont observés en fonction de la diffusion : d’une part, si a(z)  c(1 + z)α et α > (n − 2)/n, les solutions deE-mail addresses: T.Cieslak@impan.gov.pl (T. Cieślak), laurenco@math.univ-toulouse.fr (P. Laurençot).1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.01.016238 T. Cieślak, P. Laurençot / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242(1)–(4) sont globales. D’autre part, si a(z)  c (1 + z)α et α < (n − 2)/n, des solutions de (1)–(4) à symétrie radialeexplosant en temps fini sont construites dans [4].Le cas critique α = (n − 2)/n n’est pas inclus dans les travaux précédemment mentionnés si n  3 et l’objet decette Note est de montrer qu’il y a aussi des solutions de (1)–(4) à symétrie radiale explosant en temps fini dans ce casdès que M est suffisamment grand. Nous montrons de plus que, dans le cas surcritique α < (n − 2)/n, ce phénomèned’explosion peut se produire pour tout M > 0. Notre approche est différente de celle utilisée dans [4,5] et utilise uneinégalité différentielle qui se révèle être incompatible avec l’existence globale. Lorsque a ≡ 1, cette technique estemployée dans [7] : le moment d’ordre n de u vérifie une inégalité différentielle qui est en fait incompatible avecla positivité (5) de u pour des grands temps. Dans le cas d’une diffusion non linéaire, cette inégalité ne semble paspermettre de conclure et nous développons de nouvelles identités de ce type (cf. Lemma 2.2) pour démontrer le résultatsuivant :Théorème 0.1. Supposons qu’il existe α ∈ [0, (n − 2)/n], c1 > 0 et c2 > 0 telle que 0 < a(z)  c1 zα + c2, z  0.Soit M > 0 et u0 ∈ L∞(B(0,1)) une fonction positive à symétrie radiale telle que ‖u0‖1 = M . S’il existe p > 1 telleque EM,p(m̄p(u0)) < 0, les fonctions m̄p et EM,p étant définies par (7) et (9), alors le système (1)–(4) a une uniquesolution classique (u, v) dont le temps maximal d’existence Tmax ∈ (0,∞) est fini et telle que ‖u(t)‖∞ → ∞ lorsquet → Tmax.Il existe effectivement des conditions initiales u0 pour lesquelles EM,p(m̄p(u0)) < 0 quelle que soit la valeur deM > 0 si α ∈ [0, (n − 2)/n) ou pour M est assez grand si α = (n − 2)/n. En effet, il suffit de choisir une fonction u0dont le support est inclus dans B(0, δ) telle que u0 = Mδ−n 1B(0,δ) pour δ > 0 suffisamment petit. La démonstrationdu Théorème 0.1 repose sur l’identité (10) et le contrôle du terme Rp(u) (cf. Lemma 2.2).1. IntroductionWe study the occurrence of blow-up in finite time for radially symmetric solutions to a generalized Smoluchowski–Poisson system∂tu = div(a(u)∇u − u∇v) in (0,∞) × B(0,1), (1)0 = v + u − M in (0,∞) × B(0,1), (2)∂νu = ∂νv = 0 on (0,∞) × ∂B(0,1), (3)u(0) = u0  0 in B(0,1),∫B(0,1)v(t, x)dx = 0 for any t ∈ (0,∞), (4)where B(0,1) denotes the unit ball of Rn, n  2, and M the mean value of u0. The diffusion coefficient a belongsto C 2([0,∞)) and is assumed to be positive for simplicity (see Remark 2 below). The system (1)–(4) arises in as-trophysics as a model of self-gravitating Langevin particles [2] and in biology [6] where it is also known as theparabolic-elliptic Keller–Segel chemotaxis model.A fundamental property of solutions to (1)–(4) is thatu(t)  0,∫B(0,1)u(t, x)dx = M∣∣B(0,1)∣∣, and ∫B(0,1)v(t, x)dx = 0 (5)for t > 0, which readily follows from (1), (3), the comparison principle, the non-negativity of u0, and the definitionof v. It is by now well-known that, if a ≡ 1, there are radially symmetric initial data u0 for which the correspondingsolution to (1)–(4) blows up in finite time and this singular phenomenon may happen for any M > 0 if n  3 butonly for M > 8π if n = 2 [5,7]. More recently, it was shown in [4] that there is a critical exponent for the non-linear diffusion coefficient a which separates two different behaviours for the solutions to (1)–(4): on the one hand, ifa(z)  c(1 + z)α and α > (n − 2)/n, there is a unique global classical solution to (1)–(4) for any non-negative initialcondition u0 ∈ L∞(B(0,1)) (and this is actually true for a general smooth bounded domain of Rn, n  1). On theother hand, if a(z)  c(1 + z)α and α < (n − 2)/n, radially symmetric solutions to (1)–(4) blowing up in finite timeare constructed in [4].T. Cieślak, P. Laurençot / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242 239Except for n = 2, the critical case α = (n−2)/n is not covered in [4] and the purpose of this Note is to fill this gap:indeed, the main outcome of our analysis is that, if a(z)  c (1 + z)(n−2)/n, there are solutions to (1)–(4) blowing upin finite time when M exceeds an explicit threshold. As a by-product of our analysis, we also establish an alternativeand simpler proof of the blow-up result in [4] for α ∈ [0, (n − 2)/n). Indeed, the construction of solutions to (1)–(4)blowing up in finite time performed in [4] relies on the possibility of reducing (1)–(4) to a single parabolic equation.The approach used in this paper is completely different and relies on the derivation of a differential inequality ofvirial type which cannot hold true for all times. When a ≡ 1, this technique is used in [7] where it is shown that themoment of order n of u satisfies a differential inequality which contradicts the non-negativity of u after a finite time.Seemingly, the moment of order n of u does not give valuable information when the diffusion is non-linear and weintroduce non-linear functions of u to be able to handle this case. We finally point out that the above results are onlyvalid for n  2: the situation is qualitatively different in the one-dimensional case n = 1 and will be considered in aseparate paper [3].2. Finite time blow-upWe first introduce some notation: for (α,p) ∈ [0,∞) × (1,∞), we defineκp(α) := (p − 1)(α + 1)(p + α)(2(n − 1)p − 1)α+1(np)(n−2−αn)/n. (6)We also define m̄p(f ) for p  1 and f ∈ L∞(0,1) bym̄p(f ) := 1p1∫0( 1∫rf (ρ)ρn−1 dρ)prn−1 dr. (7)Our main result then reads as follows:Theorem 2.1. Assume that there are α ∈ [0, (n − 2)/n], and positive real numbers c1 > 0 and c2 > 0 such that0 < a(z)  c1zα + c2 for z  0. (8)Let M > 0 and consider a non-negative radially symmetric function u0 ∈ L∞(B(0,1)) such that ‖u0‖1 = M . Assumefurther that EM,p(m̄p(u0)) < 0 for some p > 1, whereEM,p(z) := c1κp(α)(Mn)(2p+nα(p+1))/nz(n−2−αn)/n + c2κp(0)(Mn)(2p)/nz(n−2)/n+ Mz − 1p(p + 1)(Mn)p+1, z  0. (9)Then the system (1)–(4) has a unique maximal classical solution (u, v) with finite maximal existence time Tmax ∈(0,∞) and ‖u(t)‖∞ → ∞ as t → Tmax.There are initial data u0 for which EM,p(m̄p(u0)) < 0 for some p > 1. Indeed, observe that EM,p(0) < 0 for allM > 0 if α ∈ [0, (n − 2)/n) and n  3, and for M > μn,p := n(c1p(p + 1)κp((n − 2)/n))n/2 if α = (n − 2)/n andn  2. It is then sufficient to take u0 sufficiently concentrated near x = 0 so that m̄p(u0) is close to zero (for instance,u0 = Mδ−n1B(0,δ) for δ > 0 sufficiently small). Consequently, if n  2, α = (n−2)/n, and M > μn := infp>1 {μn,p},there is at least an initial condition u0 such that ‖u0‖1 = M and the corresponding solution to (1)–(4) blows up infinite time according to Theorem 2.1. A further question is whether the threshold mass μn thus found is sharp. Ifn = 2 (α = 0) and a ≡ 1, we have μ2,p = 4(p + 1) so that μ2 = 8 and we recover the well-known threshold condition‖u0‖1 > 8π for finite time blow-up to occur in the parabolic-elliptic Keller–Segel system [7]. The situation is not soclear in higher space dimensions (n  3 and α = (n − 2)/n) as the computation of μn is more complicated (observethat μn,p → ∞ if p → 1 and p → ∞) and no explicit critical value of the mass guaranteeing global existence seemsto be available in the literature. The sharpness of μn is then an open question for n  3. Let us mention at this pointthat the Cauchy problem for the critical Smoluchowski–Poisson equation is better understood and a critical mass is240 T. Cieślak, P. Laurençot / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242identified in [1]. However, it is defined as the optimal constant in a functional inequality and thus not explicit, makingdifficult the comparison with μn.Proof of Theorem 2.1. By [4, Theorem 1.3], there exist a maximal existence time Tmax ∈ (0,∞] and a uniqueradially symmetric classical solution (u, v) ∈ C([0, Tmax);L2(B(0,1);R2))∩ C 1,2((0, Tmax)×B(0,1);R2) to (1)–(4)satisfying (5) for t ∈ [0, Tmax). Moreover, if Tmax < ∞ then ‖u(t)‖∞ → ∞ as t → Tmax.We introduceU(t, r) := 1n|B(0,1)|∫B(0,r)u(t, x)dx and mp(t) := 1p1∫0(Mn− U(t, r))prn−1 drfor (t, r) ∈ [0, Tmax) × [0,1] and derive the following identity of virial type.Lemma 2.2. Let A be defined by A′ = a and A(0) = 0. Thendmpdt= Mmp − 1p(p + 1)(Mn)p+1+ Rp(u) (10)withRp(u) :=1∫0r2n−3(Mn− U)p−2(2(n − 1)(Mn− U)− (p − 1)rnu)A(u)dr.Proof. Integrating (1) gives that U solves∂tU = rn−1∂rA(u) + uU − Mnrnu with U(t,0) = Mn− U(t,1) = 0.Consequently,dmpdt=1∫0(Mn− U)p−1[Mn(rn − 1)∂rU +(Mn− U)∂rU − r2(n−1)∂rA(u)]dr= − 1p(Mn)p+1+ Mmp + 1p + 1(Mn)p+1+1∫0[2(n − 1)r2n−3(Mn− U)p−1− (p − 1)r2(n−1)(Mn− U)p−2∂rU]A(u)dr,and hence (10). The next step is to estimate Rp(u) in terms of mp . To this end, we notice that the assumption (8) on a warrantsthat0  A(z)  c11 + α z1+α + c2z, z  0. (11)In view of (11), we have[2(n − 1)(Mn− U)− (p − 1)rnu]A(u) max{2(n − 1)(Mn− U)− (p − 1)rnu,0}(c11 + αu1+α + c2u) max{2(n − 1)(M − U)− (p − 1)rnu,0}(c1[2(n − 1)n(M − U)]α+ c2)un 1 + α (p − 1)r nT. Cieślak, P. Laurençot / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242 241 c1(p − 1)1 + α(2(n − 1)p − 1)1+α(Mn− U)1+αr−nαu + 2(n − 1)c2(Mn− U)u,and thusRp(u) c1(p − 1)1 + α(2(n − 1)p − 1)1+α 1∫0rn−2−αn(Mn− U)p+α−1∂rU dr+ 2(n − 1)c21∫0rn−2(Mn− U)p−1∂rU dr. (12)Since α ∈ [0, (n − 2)/n], the function r → r(n−2−αn)/n is concave and we infer from the Jensen inequality (withmeasure ((M/n) − U)p+α−1∂rU dr) that1∫0rn−2−αn(Mn− U)p+α−1∂rU dr(1p + α(Mn)p+α)(2+αn)/n( 1∫0rn(Mn− U)p+α−1∂rU dr)(n−2−αn)/n(1p + α(Mn)p+α)(2+αn)/n(np + α1∫0rn−1(Mn− U)p+αdr)(n−2−αn)/n (np)(n−2−αn)/np + α(Mn)(2p+αn(p+1))/nm(n−2−αn)/np .Arguing in a similar way to estimate the second integral in the right-hand side of (12), we deduce from (12) thatRp(u)  EM,p(mp) − Mmp + 1p(p + 1)(Mn)p+1.Inserting this estimate in (10) we arrive atdmp(t)dt EM,p(mp(t))for t ∈ [0, Tmax). (13)Assume now for contradiction that Tmax = ∞. Since z → EM,p(z) is an increasing function and mp(0) = m̄p(u0),we realize that, as soon as EM,p(mp(0)) < 0, we have mp(t0) = 0 for some t0 ∈ (0,∞). Thus, U(t0, r) = M/n for allr ∈ [0,1] which contradicts the fact that U(t0,0) = 0. Consequently, Tmax < ∞ and the proof is complete. Remark 1. If a ≡ 1 (i.e. α = 0), Lemma 2.2 is also valid for p = 1 and m1 coincides with the moment of order n usedin [7].Remark 2. The requirement a > 0 is only used to have classical solutions to (1)–(4) but does not play any role in theblow-up condition and the identity of virial type (Lemma 2.2). Thus, Theorem 2.1 remains valid if the diffusion isdegenerate (for instance, a(r) = mrm−1 with m ∈ [1,2(n − 1)/n]) provided an appropriate notion of weak solutionsis available.AcknowledgementsThis Note was prepared during T. Cieślak’s one-month visit at the Institut de Mathématiques de Toulouse, Univer-sité Paul Sabatier. T. Cieślak would like to express his gratitude for the invitation, support, and hospitality.242 T. Cieślak, P. Laurençot / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 237–242References[1] A. Blanchet, J.A. Carrillo, Ph. Laurençot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc.Var. Partial Differential Equations, in press.[2] P.-H. Chavanis, Generalized thermodynamics and Fokker–Planck equations. Applications to stellar dynamics and two-dimensional turbulence,Phys. Rev. E 68 (2003) 036108.[3] T. Cieślak, Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski–Poisson system, in preparation.[4] T. Cieślak, M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008) 1057–1076.[5] W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math.Soc. 329 (1992) 819–824.[6] E.F. Keller, L.A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399–415.[7] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581–601.

Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system

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Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system

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