THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC SPACE

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THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC SPACE

Patrick Maheux, Vittoria Pierfelice

To cite this version:

Patrick Maheux, Vittoria Pierfelice. THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC

SPACE. 2018. �hal-01899214�

THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC SPACE

PATRICK MAHEUX & VITTORIA PIERFELICE

Abstract. In this paper, we shall study the parabolic-elliptic Keller-Segel system on the Poincar´e disk model of the 2D-hyperbolic space. We shall investigate how the nega- tive curvature of this Riemannian manifold influences the solutions of this system. As in the 2D-Euclidean case, under the sub-critical conditionχM <8π, we shall prove global well-posedness results with any initialL¹-data. More precisely, by using dispersive and smoothing estimates we shall prove Fujita-Kato type theorems for local well-posedness.

We shall then use the logarithmic Hardy-Littlewood-Sobolev estimates on the hyper- bolic space to prove that the solution cannot blow-up in finite time. For larger mass χM >8π, we shall obtain a blow-up result under an additional condition with respect to the flat case, probably due to the spectral gap of the Laplace-Beltrami operator. Ac- cording to the exponential growth of the hyperbolic space, we find a suitable weighted moment of exponential type on the initial data for blow-up.

1. Introduction

In the last forty years, various models of the Keller-Segel system (also called Patlak- Keller-Segel) for chemotaxis have been widely studied due to important applications in biology. Most of the results of these analytical investigations focus on the fact that the global existence or the blow-up of the solutions of these problems is a space dependent phenomenon. Historically, the key papers for this family of models are the original con- tribution [12] E. F. Keller and L. A. Segel, and a work by C. S. Patlack [16]. The optimal results in the Euclidean space R

are obtained by A. Blanchet, J. Dolbeault and B.

Perthame (see [6],[3]). Moreover, a large series of results, mostly in the bounded domain case has been obtained by T. Nagai, T. Senba and T. Suzuki (see [15],[18],[19],[22],[10]).

The literature on this subject is huge and we shall not attempt to give a complete bibli- ography.

In this paper, we study the (parabolic-elliptic)-Keller-Segel system (1.1) on the clas- sical model of Riemannian manifold of constant negative curvature −1, namely the 2D- hyperbolic space. We present our study in the Poincar´ e disk model B

. Let

n : [0, +∞) × B

→ R

(t, x) → n

(x) = n(t, x)

be a non-negative function satisfying the following Keller-Segel system of equations

(1.1)



 

 

∂

∂t

n

(x) = ∆

n

(x) − χdiv

(n

(x)∇

c

(x)) , x ∈ B

, t > 0 ,

−∆

c

(x) = n

(x), x ∈ B

, t > 0, n(t = 0, x) = n

(x) ≥ 0, x ∈ B

.

Date: October 19, 2018.

2010Mathematics Subject Classification. 35R01, 47J35, 58J35, 43A85, 35J08, 35B45, 35D35, 35B44.

Key words and phrases. Keller-Segel equations, non compact Riemannian manifolds, negative cur- vature, hyperbolic space, Green function, logarithmic Hardy-Littlewood-Sobolev inequality, entropy method, dispersive estimates, smoothing estimates, global well-posedness.

1

We shall denote n

or n(t) indifferently. The subscript H on differential operators refers to the operators associated with the Riemannian metric of the Poincar´ e disk. More details will be given below in Section 2.

We shall understand the second condition in (1.1) on the function c

as c

(x) = (−∆

)

n

(x) =

Z

B²

G

(x, y) n

(y) dV

, x ∈ B

, where G

is the (Dirichlet)-Green function of −∆

given by

G

(x, y) = − 1

2π log(tanh ρ(x, y)/2), x, y ∈ B

.

We denote by ρ(x, y) the hyperbolic distance between x and y in B

(see Section 2).

The Cauchy problem for the analogous Keller-Segel system (1.1) in R

is now very well-understood (see [6],[3],[2]). The natural framework in dimension two is to work in L

which is the Lebesgue space invariant by the scaling of the equation and with non- negative solutions. The mass M = R

R²

n

dx is then a preserved quantity. Nevertheless because of the scaling critical aspect of L

, the conservation of the mass is not enough to ensure global well-posedness. A simple ”virial” type argument that we shall recall in Section 4 allows to prove that the solution blows up if the initial mass is such that χM > 8π. When χM < 8π, global existence results were proven in [3] by using the gradient flow structure of the equation. More precisely, the Keller-Segel system can be seen as the gradient flow of the free energy

F [n] = Z

n log n − χ 2

Z n c

and thus, if the initial data has finite entropy then we get a control of the free energy for all times. The use of the sharp logarithmic Hardy-Littlewood-Sobolev inequality in R

then allows to get an a priori estimate in LlogL of the solution of (1.1) which is enough to propagate any higher L

regularity. In the case χM = 8π, it was latter shown in [2]

that concentration occurs in infinite time.

As we see in this brief reminder, the results in R

are sharp but use very deeply the structure of the system and the dilation structure of the Euclidean space. If the system is perturbed a little bit, for example by replacing the Poisson equation by the equation

−∆c+αc = n with α > 0, or by replacing the Poisson equation by the parabolic equation

∂

c − ∆c = n, then the results are much less complete, see for instance [4]. In the same way, we can expect that any change in the geometry will also change some results. This type of problems was already studied in bounded domains of R

with various boundary conditions. The aim of our work is to investigate the influence of the geometry and in particular the curvature on these results. Note that, for larger mass, an additional condition for blow-up appears with respect to the Euclidean case. The main blow-up result of our Theorem 4.1 will be the following.

Theorem 1.1. There exists a weight p : B

7→ [0, +∞[ such that, if the following two conditions:

χM > 8π, with M := R

B²

n

dV , and Z

B²

p n

dV < M

r χM 8π − 1

!

2

are satisfied, then a smooth solution n : [0, T

) × B

→ R

of the Keller-Segel system (1.1) with initial condition n

can exist only on a finite interval [0, T

).

Note that in R

, the sufficient criterion for blow-up is obtained by studying the 2- moment

I(t) = Z

R²

|x|

n(t, x) dx.

The first difficulty we had to face was to find the appropriate substitute for the weight |x|

on the hyperbolic space. According to the exponential growth of the hyperbolic space, here the good quantity to use is the following weight of exponential type

p(ρ) := p(x) := 2|x|

1 − |x|

= 2 sinh

(ρ/2) = cosh ρ − 1 ≥ 0, x ∈ B

,

where ρ := ρ(x, 0) is the distance from x ∈ B

to 0. Because of the hyperbolic geometry, as we shall see in Section 4, the proof is more involved than in the Euclidean case. Our second main result is the following global well-posedness in the case χM < 8π.

Theorem 1.2. For every n

∈ L

( B

), with I

= R

B²

p n

dV < ∞ and χM < 8π, we have global well-posedness on X

∩ C( R

, L

( B

, (1 + p)dV )), for every T > 0 of the Keller-Segel system (1.1), where

X

= {n : [0, +∞) × B

7→ R | sup

[0,T]

t

kn

k

< +∞}, with

< q < 2.

A crucial ingredient in our proof is a logarithmic Hardy-Littlewood-Sobolev type in- equality on the hyperbolic space, that we deduce from a Hardy-Littlewood-Sobolev type inequality on B

, see [13]. To build the solution, by using dispersive and smoothing esti- mates, we shall propose a different approach than the one used in [3] which is based on the construction of weak solutions by compactness arguments. More precisely, we shall use the fixed point method popularized by Kato for parabolic equations (including the Navier-Stokes system) to prove local well-posedness in space X

∩ C

L

, see (3.3). We shall then use a priori estimates which can be deduced from the free energy dissipation to prove that the solution cannot blow-up in finite time.

Note that in the hyperbolic space, the case χM > 8π and

Z

B²

p n

dV ≥ M

r χM 8π − 1

!

is not covered by Theorem 1.1 and Theorem 1.2 above. It will be interesting to analyze this case further. A similar situation occurs in R

in the case studied by V. Calvez and L.

Corrias in [4], where the Poisson equation is replaced by −∆c + αc = n with α > 0. Note that this case shares some similarities with our case since the Laplacian has a spectral gap on the hyperbolic space, σ(∆

) = [1/4, +∞).

2. The hyperbolic space and some useful formulas

In this section, we shall recall the main geometric and analytic objects on the Poincar´ e disk and some useful formulae that we need. It is well-known that the Poincar´ e disk is one of the models of the hyperbolic space, which is a non-compact Riemannian manifold with constant negative curvature -1. Of course, our results can be translated in the other models as far as they are isometric to B

. For more details, we refer to Analysis and

3

Riemannian geometry textbooks [20],[7],[11] for example.

Let B

= {x ∈ R

, |x| < 1} the 2-dimensional hyperbolic disk endowed with its Rie- mannian metric

ds

= 4(dx

+ dx

) (1 − |x|

)

.

The hyperbolic distance between x ∈ B

and 0 = (0, 0) ∈ B

is given by ρ := ρ(x, 0) = log

1 + |x|

1 − |x|

(equivalent to |x| = tanh(ρ/2)). More generally, the hyperbolic distance between x and y in the disk is given by

ρ(x, y) = ρ(T

(y), 0) = ρ(T

(x), 0) = log

1 + |T

(y)|

1 − |T

(y)|

, where T

(y) is the M¨ oebius transformation

T

(y) = |y − x|

x − (1 − |x|

)(y − x) 1 − 2x · y + |x|

|y|

,

with x · y = x

y

+ x

y

denoting the scalar product on R

(see [20]). We have several useful relations

|T

(y)|

= |x − y|

1 − 2x · y + |x|

|y|

, T

(T

(x)) = x,

sinh ρ(T

(y))

2 = |T

(y)|

p 1 − |T

(y)|

= |x − y|

p (1 − |x|

)(1 − |y|

) , cosh ρ(T

(y))

2 = 1

p 1 − |T

(y)|

=

p 1 − 2x · y + |x|

|y|

p (1 − |x|

)(1 − |y|

) .

We can consider three different systems of coordinates: x = (x

, x

) (cartesian coor- dinates), (r, θ) with r = |x| (polar coordinates) and (ρ, θ) (spherical coordinates) with

|x| = tanh(ρ/2) where |x| is the Euclidean norm. So, we can write

x = |x|(cos θ, sin θ) = |x|e

= tanh(ρ/2)e

= tanh(ρ/2)(cos θ, sin θ).

We shall denote indifferently f (x) = f (r, θ) = f (re

) = f(ρ, θ) for simplicity. We shall also use indifferently in the same equation both variables x and ρ.

Let us denote by g

(X, Y ) the metric tensor on two vector fields X(x) = X

(x) ∂

∂x

+ X

(x) ∂

∂x

, Y (x) = Y

(x) ∂

∂x

+ Y

(x) ∂

∂x

, evaluated at x ∈ B

, which is given by

g

(X, Y ) =

2 1 − |x|

X

i=1

X

Y

.

The Riemannian element of volume (measure) of the exponential growth is given by dV (x) =

2 1 − |x|

dx = sinh ρ dρdθ,

4

where dx = dx

dx

is the Lebesgue measure on B

. In cartesian coordinates, we now define classical differential operators. First of all, the gradient ∇

with respect to the Riemannian structure is defined by

∇

f(x) :=

1 − |x|

2

∇

f(x), x ∈ B

, where ∇

f = (

1

,

2

) is the Euclidean gradient. For a radial function f, we note f (x) = f(ρ(x)) and have

∇

(f (ρ(x))) = f

(ρ) 2 cosh

(ρ/2)

x

|x| = f

(ρ) sinh(ρ) x, where f

(ρ) is the derivative of f with respect to ρ. In particular

g

(∇

f, ∇

f) =

1 − |x|

2

∇

f · ∇

f, where ∇

f · ∇

g =

1

∂g

∂x1

+

2

∂g

∂x2

. For a radial function f , we have

|∇

f (ρ(x)) |

= g

(∇

f, ∇

f) = |f

(ρ)|

. The divergence of the vector field Z(x) = Z

(x)

1

+ Z

(x)

2

is defined by div

Z(x) = ∇

· Z (x) := 4

(1 − |x|

)

2

X

i=1

x

Z

(x) +

2

X

i=1

∂

Z

(x), x ∈ B

.

We also define the Laplace-Beltrami operator on B

∆

f (x) :=

1 − |x|

2

∆

f (x), x ∈ B

, where ∆

is the Euclidean Laplacian

∆

= ∂

∂

x

+ ∂

∂

x

,

with x = (x

, x

) ∈ B

. For radial function f (ρ), the Laplace-Beltrami operator takes the form

∆

f(ρ) = f

(ρ) + coth(ρ)f

(ρ) = 1 sinh ρ

∂

∂ρ

sinh ρ ∂

∂ρ f

. The (Dirichlet)-Green function of −∆

is given by

(2.2) G

(x, y) = −k

log tanh(ρ(x, y)/2) = − k

2 log |T

(y)|

, with k

=

(see [20]).

For any function u and vector field Y defined on B

, we define in the usual way kuk

=

Z

B²

|u(x)|

dV

, kY k

= Z

B²

|g

(Y, Y )|

dV

.

5

3. Local well-posedness

3.1. Auxiliary estimates. Let us first state the dispersive and smoothing estimates that we shall use for the heat equation associated with the Laplace-Beltrami operator.

The proof of these estimates can be found for example by V. Pierfelice in [17].

Lemma 3.1. For every 1 ≤ p ≤ q, we have the estimate

ke

u

k

≤ c

(t)

e

ku

k

, t > 0,

where γ

= δ(

−

+

(1 −

)) ≥ 0 and c

(t) = CMax (1, t

) for some δ, C > 0.

We shall also use the next lemma.

Lemma 3.2. For every 1 ≤ p ≤ q, and every vector field Y on B

, we have the estimate ke

div

Y k

≤ c

(t)

e

kY k

, t > 0,

where γ

= δ(

−

+

(1 −

)) ≥ 0 and c

(t) = CMax (1, t

) for some δ, C > 0.

We shall also need to estimate c := −∆

n. To do so, in the next Lemma, we use the boundedness of the Riesz transorm, Sobolev embedding and Poincar´ e inequality on the hyperbolic space.

Lemma 3.3. For every s > 1, 1 < q < 2 such that

=

−

, we have the estimate k∇

ck

. knk

.

Proof. We want to estimate ∇

c = −∇

∆

H

n. Note that we can write

∇

c = −(∇

∆

1 2

H

)∆

1 2

H

n

therefore, by using the continuity of the Riesz transform on L

(see [21] for example), we obtain

k∇

ck

. k∆

1 2

H

nk

.

Next, by using the Sobolev embedding W

⊂ L

and the Poincar´ e inequality on the hyperbolic space, we obtain

k∇

ck

. k∆

H

nk

. k∇

∆

1 2

H

nk

.

By using again the continuity of the Riesz transform on L

, we finally obtain that k∇

ck

. knk

,

which is the desired estimate.

3.2. Local well-posedness results. Let 1 ≤ q ≤ +∞ and T > 0 be fixed. We recall the following Banach space

(3.3) X

= {n : [0, +∞) × B

7→ R | sup

[0,T]

t

kn

k

< +∞}, with norm

knk

= sup

[0,T]

t

kn

k

.

Theorem 3.4. For every n

∈ L

and

< q < 2, there exists T > 0 such that there is a unique solution n of the Keller-Segel system (1.1) in X

∩ C ([0, T ], L

( B

)).

6

Remark 3.5. Note that as usual, the arguments in the proof can also be used to get global well-posedness for sufficiently small data in L

.

Proof. Without lost of generality, we can assume that χ = 1 in (1.1). By using Duhamel formula, solving the system (1.1) is equivalent to look for

(3.4) n = e

n

+ B(n, n),

with

B(n, n) = − Z

0

e

div

(n(s)∇

c(s)) ds, c = −∆

n.

We shall use the following classical variant of the Banach fixed point Theorem:

Lemma 3.6. Consider X a Banach space and B a bilinear operator X × X −→ X such that

∀u, v ∈ X, kB(u, v)k

≤ γkuk

kvk

,

then, for every u

∈ X, such that 4γku

k

< 1, the sequence defined by u

= u

+ B(u

, u

), u

= 0

converges to the unique solution of

u = u

+ B(u, u) such that 2γkuk

< 1.

We can always assume that T ≤ 1 such that, in the following, we will use Lemma 3.1 and Lemma 3.2 with c(t) = C/t.

Step 1. Existence of solutions in X

.

1.1. Smallness of u

= e

n

in X

for some T > 0. On one hand, by applying Lemma 3.1, we have

(3.5) N

(n

) := ke

n

k

≤ ckn

k

, for any n

∈ L

( B

) and q ≥ 1. On the other hand, we have

lim

t

ke

n

k

= 0,

for any n

∈ L

∩ L

and q > 1. Hence lim

N

(n

) = 0. Then, by density of L

∩ L

in L

, we get

(3.6) lim

T→0

ke

n

k

= 0,

for any n

∈ L

( B

). So, for all γ > 0 and all n

∈ L

( B

), there exists T > 0 such that 4γke

n

k

< 1.

1.2. Boundedness of B(n, n). Next we shall study the continuity of B(n, n) on X

. By using successively Lemma 3.2 and the H¨ older inequality, we obtain

kB(n, n)k

. sup

[0,T]

t

Z

0

1 (t − s)

kn(s)∇

∆

n(s)k

ds

. sup

[0,T]

t

Z

0

1 (t − s)

kn(s)k

k∇

∆

H

n(s)k

ds,

7

with

=

−

and p ≤ q. By using Lemma 3.3 with c = −∆

H

n, we have . sup

[0,T]

t

Z

0

1 (t − s)

kn(s)k

ds,

with

=

−

. Thus for 1 ≤ p ≤ q such that

=

−

and hence q ≥

, we have

(3.7) kB(n, n)k

. I

knk

T

, where

(3.8) I

= sup

[0,T]

t

Z

0

1 (t − s)

1 s

ds.

By using the change of variables s = tw, we find I

= sup

[0,T]

t

Z

0

1

t

(1 − w)

t dw t

w

= Z

0

1 (1 − w)

1 w

dw

assuming

< 1 i.e. q > 1 and 2(1 −

) < 1 i.e. q < 2, we ensure I

< ∞ and hence (3.9) kB(n, n)k

. I

knk

T ,q

, for 4

3 ≤ q < 2.

1.3. Conclusion. By using Lemma 3.6 with X = X

with T > 0 given by Step 1.1, (3.6) and (3.9), we find a solution of (1.1) in X

i.e for Keller-Segel problem with initial data n

in L

for T sufficiently small.

Step 2. Proof of n ∈ L

([0, T ], L

( B

)) and Conclusion.

Since by Lemma 3.1 we have

(3.10) ke

n

k

≤ kn

k

,

it remains to estimate kB(n, n)k

. By using successively Lemma 3.2 and the H¨ older inequality, we obtain

kB(n, n)k

. sup

[0,T]

Z

1

(t − s)

kn(s)∇

∆

n(s)k

ds . sup

[0,T]

Z

1

(t − s)

kn(s)k

k∇

∆

H

n(s)k

(B²)

ds.

Thus we have (3.11)

kB(n, n)k

. sup

[0,T]

Z

1 (t − s)

s

k∇

∆

H

n(s)k

(B²)

ds

!

knk

. By Lemma 3.3 we have

k∇

∆

n(s)k

(B²)

. kn(s)k

, with 1 η = 3

2 − 1 q .

Since

< q < 2, we obtain that 1 < η < q and hence we can use the following interpola- tion inequality

kn(s)k

. kn(s)k

kn(s)k

, with θ = (

−

) (1 −

)

8

to obtain (3.12)

kB(n, n)k

. sup

[0,T]

Z

1 (t − s)

s

kn(s)k

kn(s)k

ds

!

knk

. Thus we have

(3.13) kB(n, n)k

. I

knk

knk

T ,q

, where

(3.14) I

= sup

[0,T]

Z

1 (t − s)

1 s

ds = sup

[0,T]

Z

ds (t − s)

s

. As before, by using the change of variables s = tw, we find

I

= sup

[0,T]

Z

tdw t (1 − w)

w

which is finite. This yields

knk

. kn

k

+ knk

knk

T ,q

, with 0 < θ < 1. By using the Young inequality we have

knk

. kn

k

+ knk

2−θ 1−θ

XT ,q

,

which proves that n ∈ L

([0, T ], L

( B

)) ∩ X

. By a classical argument we deduce that n ∈ C ([0, T ], L

( B

)) ∩ X

.

We can also deduce the uniqueness of n ∈ C([0, T ], L

( B

)) ∩ X

from similar argu- ments. This produces automatically local well-posedeness result.

Similar to Theorem 3.4, we shall obtain the next result of existence and uniqueness of the solution of the Keller-Segel system (1.1) with initial data n

in L

( B

) for all q such that

≤ q < 2 and a uniform positive lower bound on the existence time T independent of the initial data n

in any fixed ball of L

.

Theorem 3.7. For every n

∈ L

( B

) and

≤ q < 2, there exists T > 0 such that there exists a unique solution n ∈ C([0, T ], L

( B

)) of the Keller-Segel system (1.1). Moreover for every R > 0 there exists T (R) > 0 such that T ≥ T (R) for every kn

k

≤ R.

Proof. We shall again use Lemma 3.6 to solve (1.1). We can always assume that T ≤ 1 such that in the following we will use Lemma 3.1 and Lemma 3.2 with c(t) = C/t. We have

(3.15) ke

n

k

≤ kn

k

.

To study the continuity of B(n

, n

) on C([0, T ], L

( B

)) we use successively Lemma 3.2 and the H¨ older inequality

kB(n, n)k

. sup

[0,T]

Z

1 (t − s)

kn(s)∇

∆

n(s)k

ds

. sup

[0,T]

Z

1 (t − s)

kn(s)k

k∇

∆

H

n(s)k

ds,

9

with

=

−

, p ≤ q. By using Lemma 3.3 with c = −∆

H

n, we have . sup

[0,T]

Z

1 (t − s)

kn(s)k

ds,

with

=

−

, so

≤ q < 2. Thus for 1 ≤ p ≤ q such that

=

−

, we have (3.16) kB(n, n)k

. I

knk

,

where

(3.17) I

= sup

[0,T]

Z

1 (t − s)

ds.

As before, by using the change of variables s = tw, we find I

= sup

[0,T]

Z

tdw

t

(1 − w)

= sup

[0,T]

t

Z

0

1 (1 − w)

dw

which is finite for q > 1. Hence there exists a constant C

> 0 independent of T and n such that

(3.18) kB(n, n)k

≤ C

T

knk

for any q such that

≤ q < 2.

Let X = C([0, T ], L

( B

)) and γ := C

T

. Imposing the conditions 0 < T ≤ 1 and 4C

T

kn

k

< 1, this implies 4γke

n

k

< 1 by (3.15). By using Lemma 3.6 with u

:= e

n

and Duhamel formula (3.4), we obtain a unique solution n ∈ C([0, T ], L

( B

)) of the Keller-Segel system (1.1). This produces automatically local well-posedeness result.

We now prove the uniformity result. Choose T (R) := (8C

R)

such that T (R) ≤ 1 for R positive and large enough. The same arguments as above prove the existence of a unique solution n defined on the fixed interval [0, T (R)] for all initial conditions n

∈ L

( B

) such that kn

k

≤ R. This finishes the proof of our theorem.

4. Blow-up

In the case of R

, the blow-up for the Keller-Segel system is quite easy to prove. In fact, under the assumption R

R²

(1 + |x|

)n

(x) dx < ∞, we have the following ”virial” type identity

(4.19)

Z

R²

|x|

n(t, x) dx = Z

R²

|x|

n

(x) dx + 4M

8π (8π − χM )t, ∀t > 0, where M = R

R²

n

(x) dx. If χM > 8π and t large enough, the right-hand side of (4.19) is negative, contradicting the non-negative left-hand side of the equation. Thus the solution cannot exist for t > T

for some finite T

.

In this section, our goal is to study the blow-up phenomenon for the solution of (1.1) on B

. Because of the geometry of the hyperbolic space, our main difficulty here is to find an appropriate weight to obtain a ”virial” type argument for blow-up. Thanks to our choice of a weight of exponential type, we are able to replace the identity (4.19) by the inequality (4.22) below. This inequality will allow to prove blow up for M = R

B²

n

(x) dV > 8π/χ under an additional condition on the moment, with a suitable weight p of exponential type, R

B²

p n

dV . As noted before, an additional condition for blow-up on the 2-moment

10

R

R²

|x|

n

dx was also needed in [4][Thm.1.2 eq.(1.2)] where in particular the Keller-Segel system on R

with the Laplacian replaced by the operator −∆ + α, α > 0 was studied.

Let us recall the expression of the weight p that we shall use in our blow-up argument (4.20) p(ρ) := p(x) := 2|x|

1 − |x|

= 2 sinh

(ρ/2) = cosh ρ − 1 ≥ 0, x ∈ B

.

Note that the expression of the weight p = 2 sinh

(ρ/2) = cosh ρ − 1 is the same in any isometric representation of the Poincar´ e disk (for instance in the Poincar´ e upper-half space model). Next, we shall need the following relations

+ 1 =

and

(4.21) ∆

p = 2p + 2.

Our blow-up result is the following statement.

Theorem 4.1. Let n : [0, T

) × B

→ R

be a solution of the Keller-Segel system (1.1) with T

≤ +∞ such that n ∈ C([0, T

), L

( B

, (1 + p)dV ). Then we have

(1) For all t ∈ [0, T

), (4.22)

Z

B²

p n

dV + M

≤ Z

B²

p n

dV + M

− χ 8π M

!

e

+ χ 8π M

, with M = R

B²

n

(x) dV

.

(2) If the two conditions χM > 8π and (4.23)

Z

B²

p n

dV < λ

(M ), where

(4.24) λ

(M ) = M

r χM 8π − 1

! ,

are satisfied, then the solution n

can exist only on a finite interval [0, T

) with (4.25) T

≤ T

:= 1

4 log



 M

8π (χM − 8π)

"

χM

8π −

M +

Z

B²

p n

dV

#



 . In particular, a smooth solution n

does not exist for t > T

.

Proof. 1. Let I(t) = R

B²

p n

dV . Formally, the derivative of I(t) is given by I

(t) =

Z

B²

p ∂

∂t n

(x) dV = Z

B²

p ∆

n

(x) dV

− χ Z

B²

p∇

· (n

(x)∇

c

(x)) dV

= Z

B²

∆

p n

(x) dV

+ χ Z

B²

g

(∇

p(x), n

(x)∇

c

(x)) dV

. The second integral has been integrated by parts. Thus we obtain

I

(t) = Z

B²

∆

p n

(x) dV

+ χ Z

B²

2 1 − |x|

∇

p(x) · ∇

c

(x)n

(x) dV

. Here again X · Y denotes the Euclidean scalar product. We express the hyperbolic gradient with the Euclidean gradient and get

I

(t) = Z

B²

∆

p n

(x) dV

+ χ Z

B²

1 − |x|

2

∇

p(x) · ∇

c

(x)n

(x) dV

.

11