A DEGENERATE p-LAPLACIAN KELLER-SEGEL MODEL

This paper investigates the existence of a uniform in time L∞ bounded weak solution for the p-Laplacian Keller-Segel system with the supercritical diffusion exponent 1 < p < 3d d+1 in the multi-dimensional space Rd under the condition that the L d(3−p) p norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general L1 ∩ L∞ initial data.


Introduction.
In this paper, we study the following p-Laplacian Keller-Segel model in d ≥ 3: where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion.Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13,14].The original model was considered in 2D, x ∈ R 2 , t > 0, u(x, 0) = u 0 (x), x ∈ R 2 . ( 688 WENTING CONG AND JIAN-GUO LIU u(x, t) represents the cell density, and v(x, t) represents the concentration of the chemical substance which is given by the fundamental solution v(x, t) = Φ(x) * u(x, t), One natural extension of the original Keller-Segel model is the degenerate Keller-Segel model in the multi-dimension with m > 1, which has been widely studied [2,4,7,8,15,22,23,24,25].Another natural extension is the degenerate p-Laplacian Keller-Segel model in the multi-dimension since the porous medium equation and the p-Laplacian equation are all called nonlinear diffusion equations.Work in these two models has frequent overlaps both in phenomena to be described, results to be proved and techniques to be used.The porous medium equation and the p-Laplacian equation are different territories with some important traits in common.The evolution p-Laplacian equation is also called the non-Newtonian filtration equation which describes the diffusion with the diffusivity depending on the gradient of the unknown.The comprehensive and systematic study for these two equations can be found in Vázquez [27], DiBenedetto [10] and Wu, Zhao, Yin and Li [28].
In the p-Laplacian Keller-Segel model, the exponent p plays an important role.When p = 3d d+1 , if (u, v) is a solution of (1), constructing the following mass invariant scaling for u and a corresponding scaling for v u λ (x, t) = λu λ then (u λ , v λ ) is also a solution for (1) and hence p = 3d d+1 is referred to the critical exponent.For the general exponent p, (u λ , v λ ) satisfies the following equation If 1 + 1 d p − 3 < 0 which is called the supercritical case, the aggregation dominates the diffusion for high density(large λ) which leads to the finite-time blow-up, and the diffusion dominates the aggregation for low density(small λ) which leads to the infinite-time spreading.If 1 + 1 d p − 3 > 0 which is called the subcritical case, the aggregation dominates the diffusion for low density(small λ) which prevents spreading, while the diffusion dominates the aggregation for high density(large λ) which prevents blow-up.At the end of Section 5, we have the theorem of the existence of a global weak solution for (1) in the subcritical case.
In the supercritical case, there is a L q space, where q = d(3−p) p .The q is crucial when studying the existence and blow-up results of the p-Laplacian Keller-Segel model and almost all the results are related to the initial data u 0 (•) L q (R d ) .Also considering model (1), if (u, v) is a solution, then is also a solution of (1).Furthermore, the scaling of u(x, t) preserves the L q norm u λ L q = u L q .For 1 < p < 3d d+1 , if u 0 L q (R d ) < C d,p , where C d,p is a universal constant depending on d and p, then we will show that there exists a global weak solution.Since the initial condition u 0 ∈ L 1 + ∩ L ∞ (R d ), we can prove that weak solutions are bounded uniformly in time by using the bootstrap iterative method(See [3], [19]).With no restriction of the L q norm on initial data, we prove the local existence of a weak solution.This result also provides a natural blow-up criterion for 1 < p < 3d d+1 that all u L h (R d ) blow up at exactly the same time for h ∈ (q, +∞).In the subcritical case p > 3d d+1 , there exists a global weak solution of (1) without any restriction of the size of initial data.
In the process of proving the existence of a global weak solution of (1), we combine the Aubin-Lions Lemma with the monotone operator theory.The theory of monotone operators was proposed by Minty [20,21].Then the theory was used to obtain the existence results for quasi-linear elliptic and parabolic partial differential equations by Browder [5,6], Leray and Lions [17], Hartman and Stampacchia [12], DiBenedetto and Herrero [11].
The paper is organized as follows.In Section 2, we define a weak solution, introduce a Sobolev inequality with the best constant and some lemmas.In Section 3, we give the a priori estimates of our weak solution.In Section 4, we prove the 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 global weak solution.Finally, in Section 6, we discuss the local existence of weak 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) in this paper.
The following lemma is a Sobolev inequality with the best constant which was identified by Talenti [26] and Aubin [1].
where p * = dp d−p and Next two lemmas are proposed by Bian and Liu [2].
3. A priori estimates of weak solutions.In this section, we prove Theorem 3.1 which is concerning a priori estimates of weak solutions of (1).
. Under the assumption that ) .Furthermore, following a priori estimates hold true: has the finite time extinction.The extinct time T ext satisfies where , where C depends on d, p, A(d, p), u 0 L 1 (R d ) and u 0 L q (R d ) .
And for any .
For any q < h < ∞, u(x, t) has hyper-contractive property , where C is a constant depending on h, d, p, A(d, p) and u 0 L 1 , > 0 satisfies Proof.
Step 1. (The L q estimate for 1 < p < 3d d+1 ).Multiplying the first equation in problem (1) by qu q−1 and integrating it over R d , we obtain Now we estimate the second term on the right hand side.Firstly, by using the interpolation inequality, we obtain that where the last equality holds since . Then using the Sobolev inequality (7), (10) turns to where K(d, p) is given by (8).Substituting ( 11) into (9), we have Combining (11) with two estimates above, we obtain Step 2. (The L q decay estimate).By using the interpolation inequality and (11), we have i.e.
In the same way of obtaining ( 9)-( 11), we obtain and where the third equality holds since rd+pd+pq−qd−2d = 1 and q(pd+pr+p−3d) rd+pd+pq−qd−2d = 3 − p, and the last inequality holds from the Sobolev inequality.Then combining (22), ( 23) and ( 24) together, we have By using the interpolation inequality and ( 24), we have L r u 0 . Substituting ( 26) into ( 25) yields that Solving this inequality by using Lemma 2.3, we have Hyper-contractive estimates of L h norm for h ≥ r.For h ≥ r > q, using the interpolation inequality, Sobolev inequality and Young's inequality together, we obtain where dp(h Considering (9) with h = q, we have Substituting ( 28) into (30) yields that By the same way of obtaining (26), we obtain Then (31) turns to where , for any t > t 0 > 0, we have By choosing t 0 = t 2 , we obtain that for any t > 0 where C is a constant depending on h, d, p, A(d, p) and u 0 L 1 , satisfies (22).

4.
The uniformly in time L ∞ estimate of weak solutions.In this section, we prove our theorem about uniformly in time L ∞ boundness of weak solutions by using a bootstrap iterative method.At the beginning of this section, we prove the following proposition concerning L h norm estimates of weak solutions for 1 < h < ∞.
is a universal constant, let (u, v) be a non-negative weak solution of (1).Then u(x, t) satisfies for any t > 0 where C depends on h, q, and u 0 L 1 , and where C h u is a constant depending on d, p, h, u 0 L 1 and u 0 L h , > 0 satisfies Actually, the proof of Proposition 1 is almost the same as the proof of Theorem 3.1, except for the different initial condition Using the same method in Step 1 of Theorem 3.1, we have for all t > 0 Then we discuss in two different situations with respect to h.For 1 < h ≤ q, using the interpolation inequality, we have For q < h < ∞, letting r := q + ≤ h < ∞, there exists > 0 small enough such that (q + )p p K p (d, p)(q Then (25) also holds true, i.e.
for all t > 0. Combining (30), ( 32) and (39) together, we obtain where where satisfies Next, we prove the uniformly in time L ∞ boundness of u(x, t) by using a bootstrap iterative technique [3,19] with Proposition 1 and an additional initial condi- ) be a non-negative weak solution of (1).Then for any t > 0, Proof.We denote Multiplying the first equation in (1) by h k u h k −1 and integrating, we have Using the interpolation inequality and Sobolev inequality together, we obtain where , it is easy to see that p(h k +1)θ h k −2+p < p. Then using Young's inequality and (44), we have where We can see that where Next, we estimate ∇u . By using the interpolation inequality and Sobolev inequality, we have where Since it is easy to see that ph k β h k −2+p < p, then using Young's inequality, we have where We can also check that C 3 (h k ) is uniformly bounded as k → ∞.Combining ( 46) and ( 48) together, we have Since C 2 (h k ) and C 3 (h k ) are both uniformly bounded as k → ∞, we can choose a constant C 4 > 1 which is an upper bound of C 2 (h k ) and C 3 (h k ).Then by h k > 1 and b > 1, we have for any t > 0, Step 2. (The L h k estimate for 2 < p < 3d d+1 ) By changing form of (42), we have d dt where 0 k since h k > 1 and p < 3. Using Young's inequality and (44), we have where We can see that where Substituting (53) into (51), we obtain Next, by using Young's inequality and (47), we have where and where Combining (54) and (56) together, we have Since C 2 (h k ) and C 3 (h k ) are both uniformly bounded as k → ∞, we can choose a constant C 7 > 1 which is an upper bound of C 2 (h k ) and C 3 (h k ).Then by h k > 1 and 2b > b > 1, we have for any t > 0, Step 3. (The uniform L ∞ estimate for 1 < p < 3d d+1 ) Let and C 8 > 1 is an upper bound of C 4 and C 7 .Then (50) and (58) turn to Multiplying e t to both sides of (59), we have Solving this ODE, we obtain for t ≥ 0, It is easy to see that where C 0 is an appropriate positive constant.Combining (61) and (62) together, we can see for all 1 < p < 3d d+1 , where C 9 = 2C 0 C 8 .Then after some iterative steps, we have and for any 1 since lim k→∞ Taking the power 1 h k to both sides of (63) and letting k → ∞, we obtain where C = 3 Then (66

5.
Global existence of weak solutions.The following Lemma proved in [9, Lemma 2.1] is necessary for the existence of weak solutions of problem (1) in the supercritical case.
Lemma 5.1.For any η, η ∈ R d , there exists where C 1 and C 2 are two positive constants only depending on p.
Then there exists a non-negative global weak solution (u, v) of (1), such that all a priori estimates in Theorem 3.1 and the uniform L ∞ estimate in Theorem 4.1 hold true.
Proof.We separate the proof of Theorem 5.2 into four steps.In Step 1, we construct the regularized problem of (1) and show that all a priori estimates in Theorem 3.1 and the uniform L ∞ estimate in Theorem 4.1 hold true.Furthermore, we obtain the uniform estimate of ∇u .In Step 2 and 3, by applying the Aubin-Lions Lemma, we prove that a non-negative weak solution of the regularized problem (68) converges strongly to a non-negative weak solution of (1) in a bounded domain.Finally, in Step 4, using the weak convergence and strong convergence estimates obtained in Step 1-3, we prove the existence of a global weak solution of (1) with monotone operators.
Step 1. (The regularized problem and a priori estimates) We consider the regularized problem of (1) for > 0, where d ≥ 3, 1 < p < 3d d+1 and J A simple computation shows that v can be expressed by where α(d) is the volume of the d-dimensional unit ball.The initial condition ) is a sequence of approximation for u 0 (x), which satisfies that there exists δ > 0 such that for all 0 < < δ, According to the classical theory for parabolic equations [16], the regularized problem has a global smooth non-negative solution (u , v ) with the regularity for all Then we want to show that all a priori estimates in Theorem 3.1 hold true for our regularized problem.we take a cut-off function 0 ≤ ψ 1 (x) ≤ 1, satisfying where ) and choose a constant C 3 , such that ∇ψ Multiplying the first equation of (68) by qu q−1 ψ R (x) and integrating over R d , we obtain Integrating (70) from 0 to t yields that For the second term on the right hand side of (71), by using Hölder's inequality, we have since u ∈ L r+1 R + ; L r+1 (R d ) for any r ≥ 1.Then we can use the dominated convergence theorem as R → ∞ for any small > 0 later.Next, we want to prove that last three terms on the RHS of (71) go to 0 as R → ∞.First, by using Hölder's inequality and Young's inequality of convolution [18, pp.107], we obtain Then using the interpolation inequality, (73) yields to Third, by using Hölder's inequality, we have Then we should prove dxds is bounded in order to show (76) goes to 0 as R → ∞.Using Young's inequality for (76), we obtain since q − 2 + p ≥ 1. Combining (71), (72), (74), ( 75) and (77) together, we have Taking R large enough, we can see that q(q−1)p p since u 0 ∈ L q (R d ) when R is large enough.Substituting (79) into (76), we have Until now, we have proved that last three terms on the RHS of (71) go to 0 as R → ∞.Using the dominated convergence theorem, when R → ∞, (71) turns to i.e., for any t > 0, where last two inequalities can be obtained by the same method of (72) and (11).
Then we have which is same to (12), and all a priori estimates in Theorem 3.1 hold true for our solution of the regularized problem.We also have following uniformly bounded estimates, Additionally, since u 0 ∈ L ∞ (R d ), we let u 0 (x) also satisfy u 0 L ∞ (R d ) ≤ C, where C is a positive constant independent of .Then from the Theorem 4.1, we have the uniformly bounded estimate For q ≥ 2, i.e. 1 < p ≤ 3d d+2 , by taking r = 2 in (86), we have For 1 < r ≤ q < 2, i.e. 3d d+2 < p ≤ 3d d+1 , by using (87), we obtain where C is a positive constant.From two estimates above, we have ∇u for all 1 < p < 3d d+1 .
Step 2. (The time regularity of u ) In this step, we want to estimate ∂ t u in any bounded domain in order to use the Aubin-Lions Lemma.For any test function ϕ(x) which satisfies ϕ ∈ W 2,p (Ω) and ϕ W 2,p (Ω) ≤ 1, we have Then for any ≤ C.
Step 3. (The application of the Aubin-Lions Lemma) It is easy to see that where Ω is any bounded domain.Then we obtain that u ≤ C. By the Sobolev Embedding Theorem, we have ≤ C, and W 1,p (Ω) → → L p (Ω) → W −2, p p−1 (Ω).By the Aubin-Lions Lemma, there existes a subsequence of {u } without relabeling such that u → u, in L p 0, T ; L p (Ω) . (91) Step 4. (The existence of a global weak solution) Next, we will show that (u, v) is a weak solution of the problem (1).The crucial idea in this step follows the proof of Theorem 2.2.1 in [28, p171].The weak formulation for u is that for any test function dxdydt. (92) Next, we separate the proof of this step into three parts.(i) Since u → u in L p (0, T ; L p (Ω)), using Hölder's inequality, we have • u (x, t)u (y, t) − u(x, t)u(y, t) dxdydt In order to estimate I 1 , we have since is small enough.Then using Hardy-Littlewood-Sobolev inequality, I 1 satisfies For I 2 , also using Hardy-Littlewood-Sobolev inequality, we have For 2d d+2 ≤ p < 3d d+1 , by using the interpolation inequality and Hölder's inequality, we obtain → 0, as → 0. (98) Combining ( 98) and (99) shows that, for all 1 < p < 3d d+1 , Then we have I 2 → 0, as → 0. Until now, we obtain as → 0.
(iii) Finally, we will prove as → 0. Since ∇u There exists a χ such that Letting → 0 in (92), we have Then we will prove to finish the proof of the existence of a weak solution for (1).Choosing φ(x, t) ∈ C ∞ c [0, T ) × R d with 0 ≤ φ ≤ 1, multiplying the first equation in (68) by u φ and integrating, we obtain For any ω ∈ L p 0, T ; W 1,p (R d ) to be determined later, we can obtain the following inequality by using Lemma 5.1 i.e.
Combining ( 107) and ( 109) together, we have Next we estimate terms in (110) one by one.Since and which means that (u, v) is a global weak solution of (1).
For the subcritical case, we have the following theorem of the existence of a global weak solution.Since the proof is almost identical as that for the supercritical case, we omit details.
then there exists a non-negative global weak solution (u, v) of (1).

6.
Local existence of a weak solution and a blow-up criterion.In this section, we prove that for u 0 ∈ L 1 + ∩ L ∞ (R d ), a weak solution of (1) exists locally without any restriction for the size of initial data.Furthermore, we also prove that if a weak solution blow up in finite time, then all L h -norms of the weak solution blow up at the same time for h > q.Then there are T > 0, such that (1) has a weak solution in 0 < t < T .
Proof.Take any fixed r > q.Using the same way of obtaining (30) and taking h = r > q, we have Denoting T r := r−q C(r) u 0 r L r (R d ) 1 q−r , then for any fixed r, we choose 0 < T < T r .
Next by the same way of proving Theorem 5.2, (1) has a local in time weak solution in 0 < t < T .

1 |x| d− 2
, d ≥ 3, α(d) is the volume of the d-dimensional unit ball.In this model, cells are attracted by the chemical substance and also able to emit it.