LOWER BOUNDS FOR BLOW-UP IN A PARABOLIC-PARABOLIC KELLER-SEGEL SYSTEM

. This paper deals with a parabolic-parabolic Keller-Segel system, modeling chemotaxis, with time dependent coeﬃcients. We consider non-negative solutions of the system which blow up in ﬁnite time t ∗ and an explicit lower bound for t ∗ is derived under suﬃcient conditions on the coeﬃcients and the spatial domain.


1.
Introduction. In 1970 Keller and Segel (see [6]) derived a mathematical model of a chemotaxis process by using a system of two parabolic equations, whose solutions describe the movement of cells in response to a presence of a chemical signal substance, with a non-homogeneous space distribution. Let u(x, t) represent the cell density and v(x, t) the concentration of the chemoattractant, then we have the following system (u∇v), (x, t) ∈ Ω × (0, ∞), where Ω is a bounded domain with smooth boundary, u ν (i.e. v ν ) is the derivative of u (i.e. v) with respect the unit normal ν to ∂Ω directed outward. The coefficients χ, τ, D, a, b, are positive constants. One interesting question is the "chemotactic collapse": from mathematical point of view it corresponds to the blow-up in finite time to solutions of system (1). There has been a large interest Keller-Segel type system with bounded or unbounded solutions: we refer to [2], [3], [4], [5], [16], [17], [18] and [20] and the references therein.
We point out that Payne and Song in [15] derive a lower bound for the blow up time to solutions of system (1) when Ω is a bounded domain in R 2 or R 3 . They achieve such estimate by introducing a suitable auxiliary function satisfying a first order differential inequality. Additionally, in [1] the authors provide a numerical method by means of which, approximations for the blow-up time of solutions to (1) can be obtained.
On the other and, natural observations and practical experiences show how in specific circumstances the parameters modeling the chemotaxis phenomena can also change in time. Specifically we consider the following problem where t * is the blow-up time (0 < t * < ∞), Ω is a bounded domain in R 3 with smooth boundary ∂Ω, u ν is the normal derivative on ∂Ω. The coefficients k i (t) (i = 1, 2, 3, 4) are positive and regular functions of t, u 0 (x) and v 0 (x) are assumed non-negative on Ω satisfying the compatibility conditions on ∂Ω. System (2) represents the following situation: the chemoattractant spreads diffusively and decays with rate k 2 (t) and k 3 (t), respectively; it is also produced by the bacteria with rate k 4 (t). The bacteria diffuse with mobility 1 and also drift in the direction of the gradient of concentration of the chemoattractant with velocity k 1 (t)|∇v|; k 1 (t) is called chemosensitivity. Moreover, the Neumann boundary conditions mean that no flux with the external boundary is permitted.
The object of this paper is to study solutions of (2) which blow up in finite time t * . It is well known that when blow-up occurs at t * , explicit estimates are of a great practical interest, since, mostly, it is not possible an exact (accurate) computation of t * . More precisely, we derive sufficient conditions on the data in order to obtain an explicit lower bound for t * .
2. Preliminaries and main results. As remarked in the introduction our aim is to derive a lower bound (possible explicit) for the blow-up time t * to solutions of (2).
We will make use of the following Sobolev type inequality, here presented in this general form.
Lemma 2.1. (Sobolev type inequality) Let v be a non-negative C 1 function, defined in a bounded domain Ω ∈ R 3 with the origin inside, assumed to be star-shaped and convex in two orthogonal directions. Then valid for n ≥ 1, with ρ 0 := min For the proof see [12] and [14]. From Lemma 2.1 we derive a bound for Ω v 3 dx, to be used in the proof of the main theorem.
with m 1 := 3 2ρ0 , m 2 := 1 + d ρ0 and an arbitrary positive function. Proof. We point out that (4) can be derived by (3), but we sketch the proof for reader convenience.
We use (3) with n = 2 and by using Schwarz inequality in the second integral, we get

LOWER BOUNDS FOR BLOW-UP IN A KELLER-SEGEL SYSTEM 811
By applying first the arithmetic inequality (a + b) 2 ), valid with a, b > 0, and the Hölder inequality we obtain where in the last inequality we use and is any positive function; the lemma is so proved.
Let us present our main result. First we introduce the following auxiliary function which value at t = 0 is W (0) = α(0) Ω u 2 0 dx + β(0) Ω (∆v 0 ) 2 dx; α(t) and β(t) in (6) are positive and derivable functions in [0, t * ), to be determined. Now we give the following definition Now we state our main result, i.e. we derive an explicit lower bound for t * . For brevity we write k i := k i (t), i = 1, 2, 3, 4.
Theorem 2.4. Let (u, v) be a solution of (2). Assume Ω a bounded domain in R 3 , with the origin inside, star-shaped and convex in two orthogonal directions. Let W defined in (6) and (u, v) becomes unbounded at some time t * in W -measure (7). Moreover assume that the coefficients k i (for i = 1, 2, 3, 4) satisfy the following relation and let be Then with H −1 the inverse function of H(t) := t 0 ω(τ )dτ, ω(τ ) being a positive function depending only on the data.

Proof of Theorem 2.4.
Proof. We show that W (t) defined on solution of the system (2) satisfies an appropriate differential inequality of the first order. By integrating such inequality we get the lower bound of t * .
By differentiating W (t) we have Now we focus our attention to the last two integrals in (10). By using the first equation in (2) and the divergence theorem, the first term can be written as Moreover, it can be checked that where in the last equality we have used the second equation in (2). First of all, we observe that Now by using Schwarz inequality and (5), we have where 1 is an arbitrary positive and time depending function to be determined. We now combine (13) and (14) in (12) to get Plugging (11) and (15) in (10) we lead to Regarding term Ω u 2 ∆vdx, by means of Hölder inequality and (5) we obtain with 2 another positive and time depending function to be chosen. We observe that we are now under the hypotheses of Lemma 2.2, which can be applied to both terms in (17). In fact we can write 3 Ω where 3 and 4 are any positive and time depending functions to be determined. Hence, by employing (18) and (19) in (17) By using (20) in (16) we obtain Now we choose α(t) and β(t) in (6) as and the arbitrary functions i (for i = 1, ..., 4) as By using the values in (22) and (23), the coefficients of Ω |∇u| 2 dx and Ω |∇∆v| 2 dx in (21) vanish. Moreover from hypothesis (8). In these circumstances we neglect in (21) the non-positive terms and drop the terms whose coefficients are zero due to the choice of i (t) and α(t) and β(t). By using the inequality a γ + b γ ≤ (a + b) γ , valid for γ > 1 and a and b non-negative, at the end we obtain  , α, β and i (i = 1, 2, 3, 4) given by (22) and (23).
With the aim to simplify (24), we compare the values of W (t) in the time interval [0, t * ) with the initial value W 0 = W (0).
We recall that W (t) is assumed blowing up at t * . If W (t) is non decreasing in [0, t * ), then W (t) ≥ W 0 , ∀t ∈ [0, t * ); on the contrary, if W is non increasing (possibly with some kind of oscillations), there exists a time t 1 ∈ (0, t * ) where W (t 1 ) = W 0 and as a consequence, W (t) ≥ W 0 , ∀t ∈ [t 1 , t * ). This fact implies that Inserting (25) in (24), we obtain the desired differential inequality, being