Kinetic and Related Models doi:10.3934/krm.2016012 c American Institute of Mathematical Sciences Volume 9, Number 4, December 2016 pp. 687{714 A DEGENERATE p-LAPLACIAN KELLER-SEGEL MODEL Wenting Cong∗ School of Mathematics Jilin University Changchun 130012, China and Department of Physics and Department of Mathematics Duke University Durham, NC 27708, USA Jian-Guo Liu Department of Physics and Department of Mathematics Duke University Durham, NC 27708, USA (Communicated by Tao Luo) Abstract. This paper investigates the existence of a uniform in time L1 bounded weak solution for the p-Laplacian Keller-Segel system with the su- 3d d percritical diffusion exponent 1 < p < d+1 in the multi-dimensional space R d(3−p) under the condition that the L 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 \ L1 initial data. 1. Introduction. In this paper, we study the following p-Laplacian Keller-Segel model in d ≥ 3: 8 p−2 d <> @tu = r · jruj ru − r · (urv) ; x 2 R ; t > 0; d (1) −∆v = u; x 2 R ; t > 0; > d : u(x; 0) = u0(x); x 2 R ; 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, 8 2 < @tu = ∆u − r · (urv); x 2 R ; t > 0; −∆v = u; x 2 R2; t > 0; (2) 2 : u(x; 0) = u0(x); x 2 R : 2010 Mathematics Subject Classification. Primary: 35K65, 35K92, 92C17. Key words and phrases. Chemotaxis, fast diffusion, critical space, global existence, monotone operator, non-Newtonian filtration. The first author is supported by NSFC grant 11271154. ∗ Corresponding author: Wenting Cong. 687 688 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); where 1 − 2π log jxj; d = 2; Φ(x) = 1 1 ; d ≥ 3; d(d−2)α(d) jxjd−2 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. One natural extension of the original Keller-Segel model is the degenerate Keller- Segel model in the multi-dimension with m > 1, 8 m d < @tu = ∆u − r · (urv); x 2 R ; t > 0; −∆v = u; x 2 Rd; t > 0; (3) d : u(x; 0) = u0(x); x 2 R ; which has been widely studied [2,4,7,8, 15, 22, 23, 24, 25]. Another natural exten- sion 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 phe- nomena 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 diffusiv- ity depending on the gradient of the unknown. The comprehensive and systematic study for these two equations can be found in V´azquez[27], DiBenedetto [10] and Wu, Zhao, Yin and Li [28]. In the p-Laplacian Keller-Segel model, the exponent p plays an important role. 3d When p = d+1 , if (u; v) is a solution of (1), constructing the following mass invariant scaling for u and a corresponding scaling for v 1 uλ(x; t) = λu λ d x; λt ; (4) 1− 2 1 vλ(x; t) = λ d v λ d x; λt ; 3d then (uλ; vλ) is also a solution for (1) and hence p = d+1 is referred to the critical exponent. For the general exponent p,(uλ; vλ) satisfies the following equation ( 1+ 1 p−3 p−2 u = λ( d ) r · jruj ru − r · (urv) ; t (5) −∆v = u: 1 If 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 1 infinite-time spreading. If 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. q d(3−p) In the supercritical case, there is a L space, where q = p . The q is crucial when studying the existence and blow-up results of the p-Laplacian Keller-Segel A DEGENERATE p-LAPLACIAN KELLER-SEGEL MODEL 689 model and almost all the results are related to the initial data ku0(·)k q d . Also L (R ) considering model (1), if (u; v) is a solution, then 3−p uλ(x; t) = λu λ p x; λt ; 3− 6 3−p vλ(x; t) = λ p v λ p x; λt ; is also a solution of (1). Furthermore, the scaling of u(x; t) preserves the Lq norm 3d ku k q = kuk q . For 1 < p < , if ku k q d < C , where C is a universal λ L L d+1 0 L (R ) d;p d;p constant depending on d and p, then we will show that there exists a global weak 1 1 d solution. Since the initial condition u0 2 L+ \ L (R ), 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 Lq norm on initial data, we prove the local existence of a weak solution. This result also provides a natural blow-up criterion for 3d 1 < p < that all kuk h d blow up at exactly the same time for h 2 (q; +1). d+1 L (R ) 3d In the subcritical case p > 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, intro- duce 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 L1 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. 1 1 d Definition 2.1. (Weak solution) Let u0 2 L+ \ L (R ) be initial data and T 2 (0; 1). v(x; t) is given by the fundamental solution 1 Z u(y; t) v(x; t) = d−2 dy: d(d − 2)α(d) d R jx − yj Then (u; v) is a weak solution to (1) if u satisfies (i) Regularity: 2d 1 1 d p 1;p d 2 d+2 d u 2 L 0;T ; L+(R ) \ L 0;T ; W (R ) \ L 0;T ; L (R ) ; p −2; p d @tu 2 L p−1 0;T ; W p−1 (R ) : 1 d (ii) 8 (x; t) 2 Cc [0;T ) × R , Z T Z Z T Z p−2 u(x; t) t(x; t) dxdt = jru(x; t)j ru(x; t) · r (x; t) dxdt d d 0 R 0 R 690 WENTING CONG AND JIAN-GUO LIU 1 Z T Z Z r (x; t) − r (y; t) · (x − y) u(x; t)u(y; t) − 2 d−2 dxdydt 2dα(d) d d 0 R R jx − yj jx − yj Z − u0(x) (x; 0)dx: (6) d R The following lemma is a Sobolev inequality with the best constant which was identified by Talenti [26] and Aubin [1]. Lemma 2.2. (Sobolev inequality) Let 1 < p < d. If the function u 2 W 1;p(Rd), then kuk p∗ d ≤ K(d; p)kruk p d ; (7) L (R ) L (R ) dp where p∗ = d−p and 1 1 " # d 1− p d − 1 − 1 p − 1 Γ(1 + 2 )Γ(d) K(d; p) = π 2 d p : (8) d − p d d Γ( p )Γ(1 + d − p ) Next two lemmas are proposed by Bian and Liu [2].