UNIFORM L ∞ BOUNDEDNESS FOR A DEGENERATE PARABOLIC-PARABOLIC KELLER-SEGEL MODEL

. This paper investigates the existence of a uniform in time L ∞ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diﬀusion exponent 0 < m < 2 − 2 d in the multi-dimensional space R d under the condition that the L d (2 − m ) 2 norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution u ( x,t ) satisﬁes mass conservation when m > 1 − 2 d . We also prove the local existence of weak entropy solutions and a blow-up criterion for general L 1 ∩ L ∞ initial data.

norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution u(x, t) satisfies mass conservation when m > 1 − 2 d . We also prove the local existence of weak entropy solutions and a blow-up criterion for general L 1 ∩ L ∞ initial data.
1. Introduction. We study the following quasilinear parabolic-parabolic Keller-Segel model in d ≥ 3: where the diffusion exponent m is taken to be supercritical in this paper, i.e. 0 < m < 2 − 2 d . The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [11] [14]. u(x, t) represents the cell density, and v(x, t) represents the concentration of the chemical substance. In this model, cells are attracted by the chemical substance and also able to emit it. Without loss of generality, we suppose v(x, 0) = 0 which is reasonable with the meaning that there is no chemical substance at the beginning, and then it is generated by cells.
For 1 < m < 2− 2 d , the associate free energy of problem (1) involves a conservative variational function u and a non-conservative variational function v, Model (1) can be recast into the following mixed conservative and non-conservative gradient flow This mixed variational structure is known as the Le Châterlier Principle and it formally possesses the following entropy-dissipation equality In the original parabolic-parabolic Keller-Segel model (m = 1, d = 2), there exists a critical mass 8π for the initial data u 0 (x). If the initial mass R 2 u 0 (x)dx = M < 8π, there exists a global weak non-negative solution [5].
By a natural extension to the quasilinear parabolic-parabolic Keller-Segel model, the diffusion exponent m plays an important role. 0 < m < 1 is called the fast diffusion and m > 1 is called the slow diffusion to describe the limiting behaviors of the diffusivity coefficient in the diffusion term ∆u m = ∇ · (mu m−1 ∇u).
When 0 < m < 2 − 2 d which is called the supercritical case, the aggregation dominates the diffusion for the high density (large λ) which leads to the finite-time blow-up [3,4,9,18], and the diffusion dominates the aggregation for the low density (small λ) which leads to the infinite-time spreading [1,18,20]. While m > 2 − 2 d which is called the subcritical case, the aggregation dominates the diffusion for the low density (small λ) which prevents spreading, while the diffusion dominates the aggregation for the high density (large λ) which prevents blow-up [12,19,20].
The model (1) has been widely studied in the slow diffusion case. Sugiyama [19,20] proved the global in time existence of weak solutions without any restriction on the size of the initial date for m ≥ 2. Then Ishida and Yokota [12] improved the global existence result from m ≥ 2 to m > 2 − 2 d . For the blow-up result in the slow diffusion case, Ishida and Yokota [13] proved that every radially symmetric energy solution with large negative initial energy blows up in either finite or infinite time when 1 ≤ m < 2 − 2 d . However, in the fast diffusion case, i.e. 0 < m < 1, few work has been done for the parabolic-parabolic Keller-Segel model.
In the supercritical case 0 < m < 2 − 2 d , there is an L p space, where p = d(2−m) 2 . The p is crucial when studying the existence and blow-up results of (1) and almost all the results are related to u 0 L p (R d ) . In fact, this critical L p space is widely used in studying the parabolic-elliptic Keller-Segel models [1,2,20], especially p = d 2 for the original parabolic-parabolic Keller-Segel model (m = 1) in R d [7].
For 0 < m < 2 − 2 d , if u 0 L p (R d ) < C d,m , where C d,m is a universal constant depending on d and m, then we prove that there exists a global weak solution (u, v) with the properties that u(x, t) preserves mass when 1 − 2 d < m < 2 − 2 d , and extincts at a finite time when 0 < m < 1 − 2 d . Furthermore, for m > 1, this weak solution is also a weak entropy solution satistying energy inequality if the initial second moment is bounded and u 0 ∈ L m (R d ). With the initial condition u 0 ∈ L 1 + ∩ L ∞ (R d ), we can prove that the weak solution is bounded uniformly in time by using bootstrap iterative method(See [2], [16]). With no restriction of the L p norm on initial data, we prove the local existence of a weak entropy solution for 1 < m < 2 − 2 d . This result also provides a natural blow-up criterion that all u L q (R d ) blow up at exactly the same time for q ∈ (p, +∞).
The results concerning the finite-time blow-up for the solutions of the Keller-Segel model in multi-dimension have only been proved for its parabolic-elliptic type until Winkler made a breakthrough in [21] to introduce a new method in fully parabolic problem when m = 1. There is few paper containing the finite time blow-up result for the solutions when m = 1. This is still an open problem.
The paper is organized as follows. In Section 2, we define a weak solution and introduce some crucial inequalities about semigroup theory and some lemmas. In Section 3, we propose a priori estimates of a weak solution. In Section 4, we prove our main theorem about uniformly in time L ∞ bound of weak solutions using a bootstrap iterative method. In Section 5, we construct a regularized problem to prove the existence of a weak solution. Finally, in Section 6, we prove the local existence of weak entropy solutions and a blow-up criterion.

2.
Preliminaries. The generic constant will be denoted by C, even if it is different from line to line. At the beginning, we define a weak solution of (1).
We use semigroup theory in this paper. The following definition and estimates are standard(See [12,17]). Consider the following Cauchy problem: (2) is the unique mild solution of problem (2) on [0, T ]. The heat semigroup operator e t∆ is defined by Using Young's inequality of the convolution and property of Gamma function, we immediately obtain that , where C is a positive constant depending on p, q and d, for and h 0 ∈ L p (R d ), using two inequalities above and Bochner Theorem in [8, pp.650 where C is a positive constant depending on p, q and d.

Remark 1.
It is well known that the mild solution defined above is also a weak solution. In fact, for any test function φ ∈ C ∞ c [0, T ) × R d , multiply φ t to both sides of (3) and integrate over [0, T ) × R d to obtain where in the last equality, we use the regularity in (5).
Then recall the following well-known maximal L p -regularity result for the heat kernel: for all f ∈ L p 0, T ; L p (R d ) .
The lemma above is a special case of the famous maximal L p -regularity Theorem which was proved by Hieber and Prüss in [10]. We can use the maximal L p result in our paper since the space R d and elliptical operator ∆ satisfy the conditions of the Theorem 3.1 in [10], and we consider v 0 (x) = 0. We also refer the readers to a thorough review on maximal L p -regularity for parabolic equation [15].
The following four lemmas which are proved in [1] are useful for later estimations.
where S d is the sharp constant in Sobolev inequality for d ≥ 3.

WENTING CONG AND JIAN-GUO LIU
With the additional condition that y(0) is bounded, we have Lemma 2.8 which can be proved by contradiction arguments.

3.
A priori estimates of weak solutions. In this section, we prove Theorem 3.1 which is concerning a priori estimates of weak solutions for (1).
. C p is the positive constant in (7). Under the assumption that u 0 ∈ L 1 + ∩ L p (R d ) and . Furthermore, the following a priori estimates hold true: (iii) For 1 − 2 d < m < 2 − 2 d , the solution u(x, t) satisfies mass conservation and u(·, t) L p (R d ) decays in time

And for any
For any p < q < ∞, u(x, t) has hyper-contractive property 2 , and C is a constant depending on m, d, q, η and u 0 L 1 (R d ) .
Proof. Step 1. (L p estimate for 0 < m < 2 − 2 d ). Multiplying the first equation in model (1) by pu p−1 and integrating it over R d , we obtain Now we estimate the second term on the right hand side. Using Hölder's inequality, we have Define Integrating (13) from 0 to t, it follows that Next, using Hölder's inequality and Lemma 2.3, we obtain where C p is the constant in Lemma 2.3. Substituting (15) into (14), we see that From Lemma 2.4 with q = p, then (16) turns to where By contradiction arguments, we can prove that for all t > 0, Therefore, combining (17) and (19), we obtain In the same time, from Lemma 2.4, we have Step 2. (L p decay estimates). From the fact u(·, t) . (20) Substituting (20) into (17), we see that Define For any small 0 > 0, we have Then from two equations above, we obtain that In the similar way of obtaining (21), integrating from t to t+ 0 instead of integrating from 0 to t, we see that It means that y(t) is a non-increasing function in time, i.e.
Then we have the conclusion that where Step 3. (Hyper-contractive estimate for any In the similar way of obtaining (23), we obtain .
Hyper-contractive estimates of L q norm for q ≥ r.
Combining (9) and (16) with q = p, we have Then in the similar way of obtaining (23), (28) turns to . Using Lemma 2.7 and choosing t 0 = t 2 , we obtain that for any t > 0 where C is a constant depending on m, d, q, η and u 0 L 1 (R d ) , satisfies (25).
Step 4. (L q decay estimate for any (24), we obtain that for any t > 0,

by using interpolation inequality and
(31) Step 5. (Mass conservation for u(x, t) For 1 − 2 d < m < 1, we can estimate the first term on RHS by using Hölder's inequality Using young's inequality, the second term on RHS of (32) goes to Recalling the second equation of (1) v t = ∆v − v + u, multiplying it by −∆v and integrating from 0 to t and over R d , we have where the last inequality can be obtained from (7). From (34) and (35), by using interpolation inequality, Hölder's inequality and Therefore, collecting (32), (33) and (36) together, it shows that by the dominated convergence theorem. For 1 ≤ m < 2 − 2 d , also using interpolation inequality and Hölder's inequality, we have the following estimate

WENTING CONG AND JIAN-GUO LIU
Then from (36) and (37), we have i.e. At the beginning of this section, we prove the following proposition concerning L q norm estimates of the weak solution for 1 < q < ∞.
where C depends on p, q and where C q u is a constant depending on d, m, q, u 0 L 1 (R d ) and u 0 L q (R d ) , satisfies where C ∞ v is a positive constant depending on C d+1 u .
Proof. Actually, the proof of Proposition 1 is almost the same as the proof of Theorem 3.1, except for the different initial condition u 0 ∈ L 1 Step 1 is L q estimate for u(x, t) and Step 2 is the uniform estimate for v(x, t). We omit some details which are similar to the proof of Proposition 1 in [2].
Step 1. (L q estimate for u(x, t))We have obtained the uniform L p estimate for Then for 1 < q ≤ p, using interpolation inequality, we have which is (39) by taking C(p, q, u 0 For p < r ≤ q, it is not hard to see that u(·, t) L r (R d ) ≤ u 0 L r (R d ) for any t > 0. By the similar way of obtaining (29), we have where δ = 1 + 1+q−r r−p andβ = 2mq(q−1) . Using Lemma 2.8 and interpolation inequality, we can obtain where satisfies Step 2. (Uniform W 1,∞ estimate for v(x, t)). From (4) and (5) with v 0 (x) = 0, choosing p = ∞ and q = d Next, we will prove the uniformly in time L ∞ boundness of u(x, t) by using a bootstrap iterative technique [2,16] with Proposition 1 and an additional initial condition u 0 ∈ L ∞ (R d ).
Cp is a universal constant, suppose (u, v) be a non-negative weak solution of (1). Then for any t > 0, Proof.
Step 1. (The L q k estimate). We denote Multiplying the first equation in (1) by q k u q k −1 and integrating, we have where the inequality holds from (41). By using Young's inequality and interpolation inequality, we obtain where inequalities hold since 1 < q k −m+1 < q k +1 and q k > m. Then substituting (45) into (44) yields to where In order to change the form of (46) into what we want, firstly we try to estimate u(·, t) q k +1 L q k +1 by using interpolation inequality and Sobolev inequality, where We can see that (q k +1)(1−θ) . Then using Young's inequality, we obtain By some simple computations, we know that a → 1 + d, b → 1+d d as k → ∞. Then C 2 (q k ) is uniformly bounded as k → ∞. Substituting (48) into (46), we obtain where Secondly, we will estimate ∇u .
From interpolation inequality, it shows that where .

It is shown that
. Using Young's inequality for (50), we have We can check that C 3 (q k ) is uniformly bounded as k → ∞. Substituting (51) into (49), we obtain where . Since C 2 (q k ) and C 3 (q k ) are all uniformly bounded as q k → ∞, we can choose a constant C 5 > 1 which is an upper bound of C 2 (q k ), C 3 (q k ) and C 4 u 0 L 1 (R d ) . Then by q k > 1 and a > 1, we have L q k estimate Step 2. (Uniform L ∞ estimate). Let y k (t) = u(·, t) q k L q k (R d ) and multiply e t to both sides of (53) Solving this ODE, we obtain for t ≥ 0 We have where C 0 is an appropriate positive constant. Combining (54) and (55) together, we can see where C 6 = 3C 0 C 5 . Then after some iterative steps, we have and max 1, Taking power 1 q k to both sides of (56) and letting k → ∞, we obtain Then (57) turns to u(·, t) L ∞ ≤ C(m, d, K 0 ).

5.
Global existence of weak entropy solutions. In this section, we prove a theorem of the existence of a weak entropy solution by constructing a corresponding regularized problem.
Then there exists a non-negative global weak solution (u, v) of (1), such that all the a priori estimates in Theorem 3.1 hold true. Furthermore, for which is non-increasing in time, (ii) with an extra assumption that u 0 ∈ L m (R d ) when 2d d+2 < m < 2 − 2 d , for all 1 < m < 2 − 2 d , the weak solution of (1) also satisfies energy inequality a.e. t > 0.
Proof. We separate the proof of Theorem 5.1 into nine steps. In Step 1, we construct the regularized problem of (1) and show that all the a priori estimates in Theorem 3.1 hold true. In Step 2-5, by applying Aubin-Lions-Dubinskiȋ Lemma, we prove that the non-negative weak solution of regularized problem (59) converges strongly to a non-negative weak solution of (1) in a bounded region which shows the existence of a non-negative weak solution of (1) in R d . Then in Step 6, with a little improvement of initial data, we extend the strong convergence to the whole space R d through the proof of the second moments are finite when 1 < m < 2− 2 d . In Step 7 and 8, we show the convergence of the free energy and the lower semi-continuity of the dissipation term. Furthermore, In Step 9, we prove that the global weak solution satisfies energy inequality.
Step 1. (Regularized problem and a priori estimates). We consider the regularized problem of (1) for > 0, is a sequence of approximation for u 0 (x), which satisfies that there exists δ > 0 such that for all 0 < < δ, For the existence of a strong solution of problem (59), we refer to [20,Section 3]. Our existence result of regularized problem can be obtained by almost the same way of proving Theorem 7 in [20], except for some small details. Then the regularized problem has a global strong solution (u , v ) with u ∈ W 2,1 Then we will prove that all the a priori estimates in Theorem 3.1 hold true for our regularized problem. Multiplying the first equation of (59) by pu p−1 ψ R (x) and integrating over R d × (0, t), where ψ R (x) is the cut-off function defined before, we obtain In order to estimate the right hand side of (60), we should have estimates of v at first. Multiplying ∂ t v = ∆v − v + u by −∆v and integrating over R d and from 0 to t, we have In the same way, multiplying ∂ t v = ∆v − v + u by v and integrating over R d and from 0 to t, we have Combining (61) with (62), we see that since u ∈ L 2 R + ; L 2 (R d ) . Then using Hölder's inequality, we obtain which means that we can use the dominated convergence theorem for this term as R → ∞ for any small . Next, we prove that last three terms on the right hand side of (60) go to 0 as R → ∞. Firstly, from u ∈ L ∞ R + ; L r (R d ) , for any t > 0 and small , we have Using the dominated convergence theorem, when R → ∞, (60) turns to which is same to (11) by the method of obtaining (23). From all above, we have the conclusion that all the a priori estimates in Theorem 3.1 hold true for the solution of the regularized problem. Then we have following estimates, ∇u m+r−1 2 Letting r = 3 − m − 2 d , we know that 1 < r ≤ p since 0 < m < 2 − 2 d . From (66), by using interpolation inequality and Sobolev inequality, we have Step 2. (Time regularity of u ). In this step, we estimate ∂ t u in any bounded domain in order to use Aubin-Lions-Dubinskiȋ Lemma. For any test function ϕ(x) where the last inequality holds since 2m(p+1) p+3 ≤ p + 1 and 2(p+1) p+3 ≤ p + 1 from 0 < m < 2 − 2 d . Choosingp = min p+1 m , p + 1 > 1, for any T > 0, we obtain Step 3. (Application of Aubin-Lions-Dubinskiȋ Lemma). Before using Aubin-Lions-Dubinskiȋ Lemma, we introduce the definition of Seminormed non-negative cone in a Banach space which can be found in [6].
Let {B k } ∞ k=1 ∈ R d be a sequence of balls centered at 0 with radius R k , and R k → ∞ as k → ∞. By a standard diagonal argument, there exists a subsequence {u } without relabeling, such that the following uniformly strong convergence holds true Step 4. (Strong convergence of v ). From the second equation of (1), using (67) and (69), for any test function ϕ(x) which satisfies ϕ ∈ W 2,2 (Ω) and ϕ W 2,2 (Ω) ≤ 1, we have Then for any T > 0, we obtain i.e. ∂ t ∇v Also let {B k } ∞ k=1 ∈ R d be a sequence of balls centered at 0 with radius R k , and Step 5. (Existence of a global weak solution). Next, we will prove that (u, v) is a weak solution of problem (1). The weak formulation for u is that for any test function ψ(x) ∈ C ∞ c (R d ) and any 0 < t < ∞, Firstly, we try to prove that u m → u m in L 1 0, T ; L 1 (Ω) , by using strong convergence (72). For 0 < m ≤ 1, using Hölder's inequality, we have For 1 < m < 2 − 2 d , also using Hölder's inequality, we obtain Then (82) turns that Owing to (81) and (83), passing limit → 0, one has that for any 0 < t < ∞, The weak formulation for v is that for any test function ψ(x) ∈ C ∞ c (R d ) and any 0 < t < ∞, From strong convergences we have obtained for u and v , it is easy to see that T 0 Ω |u − u| dxds → 0, as → 0.
Then passing limit → 0, one has that for any 0 < t < ∞, Now we have the conclusion that (u, v) is a global weak solution of (1).
Step 6. (Strong convergence in R d for the weak solution). For 1 < m < 2 − 2 d , we estimate the second moments of u and v at first. From (59), one has that Then using Gronwall's inequality, (89) turns to since e −t < 1 from t > 0. By using interpolation inequality for 1 < m < p + 1, we can obtain that for any t ∈ (0, T ]. Next we estimate (40) in Proposition 1, we have From Sobolev inequality and (69), one has that Combining two estimates above and using Hölder's inequality, we obtain Until now, we have m 2 (u (·, t)) ≤ C(T ) for any 0 < t ≤ T . From the second equation of (59), it shows that By using Gronwall's inequality, we have Then for m 2 v (·, t) , one has that i.e. m 2 v (·, t) ≤ C(T ) for any 0 < t ≤ T .
By using m 2 u (·, t) ≤ C(T ) and m 2 v (·, t) ≤ C(T ), we obtain that for any where 1 r2 = 1−θ2 1 + θ2 2 . By weak semi-continuity of L m+p−1 0, T ; L r1 (|x| > R) and From (73), (77) and Hölder's inequality, one has that Thus we have the following strong convergence in R d for the weak solution Step 7. (Convergence of the free energy for m > 1). The free energy of the regularized problem is In this step, we want to prove that as → 0, , v(·, t) , a.e. in (0, T ).
By denoting q := max{m, p} and using the similar method in Step 1 of Theorem 3.1, we have for any T > 0 ∇u m+r−1 2 The dissipation term satisfies From (107) by taking r = m and (94), we have for any T > 0 Then the first term in dissipation is uniformly bounded, i.e.
Furthermore, there exists a subsequence of 2m 2m−1 ∇u m− 1 2 − √ u ∇v without relabeling which weakly converges to f in L 2 0, T ; L 2 (R d ) . By the lower semi-continuity of L 2 norm, we obtain for any T > 0, Now we will prove that the weak limit f = 2m 2m−1 ∇u m− 1 2 − √ u∇v.

For any test function
From (81) by taking m − 1 2 instead of m which is reasonable since we consider Next from (75) and (81), we obtain Combining (109) and (110), we have proved (108), i.e. f = 2m 2m−1 ∇u m− 1 2 − √ u∇v. Then for any T > 0, we obtain lower semi-continuity of the first term in dissipation Next we will use the same method to prove the lower semi-continuity of the second term in dissipation. From the second equation of (1), using (67) and (69), we have Then there exists a subsequence of ∂ t v without relabeling which weakly converges to g in L 2 0, T ; L 2 (R d ) . Also by the lower semi-continuity of L 2 norm, we obtain that for any T > 0 g L 2 0,T ;L 2 (R d ) .
We will prove g = ∂ t v. Choosing any test function ψ ∈ C ∞ c [0, T ) × R d , we have From (111) and (112), the dissipation term satisfies for any T > 0 Step 9. (Weak entropy solution with the energy inequality for 1 < m < 2 − 2 d ). Multiplying the first equation in (59) by m m−1 u m−1 − v and integrating over R d shows that Multiplying the second equation in (59) by ∂ t v and integrating over R d turns that Then from two equations above, integrating from 0 to t, we have From (67) and (69), one has that for any t > 0 t 0 R d ∇u · ∇v dxds ≤ u L 2 0,t;L 2 (R d ) ∆v L 2 0,t;L 2 (R d ) ≤ C.
Then combining the convergence of the free energy and the lower semi-continuity of dissipation term, by letting → 0, there exists a global weak entropy solution which satisfies the energy inequality ≤ F(0), a.e. t > 0.
6. Local existence of a weak entropy solution and a blow-up criterion. In this section, we prove that for u 0 ∈ L 1 + ∩ L ∞ (R d ), a weak entropy solution of (1) exists locally without any restriction for the size of initial data. Furthermore, we also prove that if a weak solution blows up in finite time, then all L q -norms of the weak solution blow up at the same time for q ∈ (p, +∞). . Assume u 0 ∈ L 1 + ∩ L ∞ (R d ) and the initial second moment R d |x| 2 u 0 (x)dx < ∞. Then there are T > 0, such that (1) has a weak entropy solution in 0 < t < T with properties Proof. Take any fixed q > p. Using the same way of obtaining (16) and taking q = r > p in (9), we have Solving the inequality (116) shows that Denoting T q := q−p C(q,d) u 0 q L q (R d ) 1 p−q , then for any fixed q, we choose 0 < T < T q .
Next by the same way of proving Theorem 5.1, there exists a local in time weak entropy solution with properties where the second one is obtained by (96).
Proposition 2. (Blow-up criterion) Under the same assumptions as Theorem 6.1 and r = p + where is small enough, let T r max be the largest L r -norm existence time of a weak solution, i.e. u(·, t) L r (R d ) < ∞, for all 0 < t < T r max , lim sup t→T r max u(·, t) L r (R d ) = ∞, and T q max be the largest L q -norm existence time of a weak solution for q ≥ r > p. Then if T q max < ∞ for any q, T q max = T r max , for all q ≥ r.
Proof. Since u(·, t) L 1 (R d ) ≤ u 0 L 1 (R d ) , by using interpolation inequality, we know that for q ≥ r, T q max ≤ T r max . If T q max < T r max for any q ≥ r, then we will have contradiction arguments. T q max < T r max implies lim sup t→T q max u(·, t) L r (R d ) =: A < ∞.
Then using the similar way of obtaining (116) and taking q ≥ r > p, we have i.e. u(·, t) L q (R d ) ≤ C q, r, A, u 0 L q (R d ) , T q max , for t ∈ (0, T q max ) , which contradicts with lim sup t→T q max u(·, t) L q (R d ) = ∞.
Thus we have the conclusion that T q max = T r max for all q ≥ r > p, i.e. L q -norms blow up at the same time.