Decay and Asymptotic Behavior of a Solution of the Keller-segel System of Degenerated and Non-degenerated Type

We classify the global behavior of the weak solution of the Keller-Segel system of degenerated type. For the stronger degeneracy the weak solution exists globally in time and it shows the time uniform decay under some extra conditions. If the degeneracy is weaker the solution exhibit a finite time blow-up if the data is non-negative. The situation is very similar to the semi-linear case. Some additional discussion is also presented. 1. Keller-Segel system 1.1. Survey for Non-degenerated Case. This note is concerning the temporal behavior of a global solution of the degenerated parabolic elliptic system. Before introducing the problem we consider, let us start from the original model of the chemotaxis called as the Keller-Segel system introduced in [16]. The semilinear type of the original Keller-Segel system is the following form: For λ ≥ 0,   ∂tu− ∆u + ∇(u∇ψ) = 0, x ∈ R, t > 0, ∂tψ − ∆ψ + λψ = u, x ∈ R, t > 0, u(0, x) = u0(x), x ∈ R, ψ(0, x) = ψ0(x), x ∈ R. (1.1) Here the unknown function u(t, x);R+ ×Rn → R+ denotes the density of a mucus amoeba and ψ(t, x);R+ × Rn → R stands for the potential of chemical substances. In order to exploit the contrast between the existence and non-existence of the solution, Jäger-Luckhaus [14], Wolansky [38] and Nagai [22] considered the parabolic-elliptic version of the above system:   ∂tu− ∆u + ∇(u∇ψ) = 0, x ∈ R, t > 0, − ∆ψ + λψ = u, x ∈ R, t > 0, u(0, x) = u0(x), x ∈ R. (1.2) It has been studied in detail for the asymptotic behavior of the solutions for the above systems ([14], [40], [1], [23], [9]). In fact this system (1.2) has a strong connection with the self-interacting particles that studies largely by Biller [1], [2] and reference therein. 1 The above systems are also connected to a simplest model equation of the semiconductor devise simulation of bipolar type (cf. [21], [15]):   ∂tn− ∆n−∇(n∇ψ) = 0, x ∈ R, t > 0, ∂tp− ∆p + ∇(p∇ψ) = 0, x ∈ R, t > 0, − ∆ψ = ε(p− n) + g, x ∈ R, t > 0, n(0, x) = n0(x), p(0, x) = p0(x), x ∈ R, (1.3) where n(t, x) and p(t, x) denote the density of the negative and positive charge, respectively and g(x) denotes the background charge density which is a given function. When the background charge can be neglected, the equation is considered as the two species version of the Keller-Segel model except the sign of the nonlinear interaction. The semi-conductor devise model chooses a stabler sign of the nonlinearity that makes the system admits large data global solutions. Note that the unstable case, there is an analogous blow up result holds for the above two species system (see Kurokiba-Ogawa [20] and Kurokiba-Nagai-Ogawa [19]). In the both cases (1.2) and (1.3), the critical case for the equation is n = 2 in the scaling point of view. This is corresponding to the well known Fujita exponent 1 + 2/n for the semilinear heat equation ([12]) and the two dimensional case the quadratic nonlinearity is exactly corresponding to the critical situation. The existence, the uniqueness and the regularity theory for the corresponding problem in a bounded domain has already been done by many authors. Here we concentrate the Cauchy problem in R2 to examine the scaling invariance point of view. The result for the global existence for the Keller-Segel system (1.1) can be summarized as follows: Theorem 1.1. ([25]) Let λ > 0 be constants and n = 2. Suppose (u0, v0) ∈ (L1(R2)∩L2(R2))× H1(R2) are positive. Then under the condition either for (1.1), ∫ R2 u0(x)dx < 4π (1.4) or for (1.2) ∫ R2 u0(x)dx < 8π, (1.5) then the positive solution to (1.1) (or (1.2)) exists globally in time. Namely (u, v) ∈ C([0,∞); (L2∩ L1) × (H1 ∩ L1)) ∩ C1((0,∞);H2 ×H2) and it satisfies that for all T > 0, there exists a finite constant C = C(T ) such that ∫ R2 {(1 + u(t)) log(1 + u(t))} + 1 2 ‖∇v(t)‖2 + 1 2 λ‖v(t)‖2 ≤ C(T ), t ∈ [0, T ]. (1.6) In the both cases, the role of the generalized free energy (1.6) is important to obtain the time apriori estimate for the solutions. Note that it has already proved that if the initial data satisfies ∫ R2 u0(x)dx > 8π then the positive solution blows up in a finite time (cf. Biler [1], Nagai [22] and Nagai-SenbaYoshida [27]). 2 On the other hand, to discuss the analogous result for the simpler system λ = 0 of (1.2) we encounter a different kind of technical difficulty. For this case, it is also known that the solution with u0 ≥ 0 blows up in a finite time if ∫ R2 u0(x)dx > 8π (Biler [1], Nagai [22], [24] and NagaiSenba-Yoshida [26]). For the whole space case, the restriction that the solution having the finite second moment ∫ R2 |x|2u(t)dx < ∞ is removed by the scaling method in Kurokiba-Ogawa [20]. Besides when the domain is bounded in R2 with the Neumann boundary condition, SenbaSuzuki [31] showed that the L1 density shows a concentration with the measure 8πδ0 if the data is the radially symmetric. This can be generalized for the non-radial case by Senba-Suzuki [32]. The second system (1.2) with λ = 0 also has analogous property of its structure. However the proof of the global existence is rather complicated since the behavior of the solution of the second equation is different from the first one. Namely we can not use the free energy functional directly to derive any a priori bound for the solution which is not considered in the literatures before. We discuss on this direction in [25] in details. One may summarize those existence and non-existence result for the whole space case as follows: Theorem 1.2. ([25], [20]) Let λ = 0 in (1.2). Suppose u0 ∈ L1(R2) ∩ L2(R2) is non-negative everywhere. (1) Then under the condition ∫ R2 u0(x)dx < 8π, (1.7) the positive solution to (1.2) exists globally in time. Namely (u, ψ) ∈ C([0,∞); (L2 ∩ L1) × Ẇ 1,∞) ∩ C1((0,∞);H2 × Ẇ 2,1) and it satisfies that for all T > 0 there exists a finite constant C = C(T ) such that ∫ R2 { (1 + u(t)) log(1 + u(t)) − u(t) } dx ≤ C(T ), t ∈ [0, T ]. (2) On the other hand, if the positive initial data satisfies ∫ R2 u0(x)dx > 8π, (1.8) then the solution does not exists globally. Namely it blows up in a finite time. The threshold case ‖u0‖1 = 8π is considered recently by Biler-Karch-Laurençot-Nadzieja [5] for the radially symmetric case. 1.2. Degenerated Case. The second problem we would like consider here is the degenerated version of the modified model of the Keller-Segel system.   ∂tu− ∆u + ∇(u∇ψ) = 0, x ∈ R, t > 0, − ∆ψ + λψ = u, x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, (1.9) where α ≥ 1 and λ > 0. An analogous variant of the semiconductor system like (1.3) is also our motivation. In that case, the stabler sign of the nonlinear interaction is chosen. 3 The striking difference between the semilinear system (1.2) and the degenerated case (1.9) is that the equation essentially includes the hyperbolic structure in it and the finite propagation of the support of the solution may occur. If the solution is strictly positive, the solution is considered similarly as the semi-linear case. As is mentioned for the semilinear case, there exists a finite time blow up solution for a certain initial data and analogously the finite time blow up possibly occurs for the degenerated case. More precisely, when the data is positive and have the large initial value in the sense of L1, then the solution for the modified version of the Keller-Segel system blows up in a finite time ([22], [1]) when n = 2 and for higher dimensional cases, the condition is getting weaker since the system is less stable. For the degenerated case, we expect an analogous situation. If there is a point where the solution varnish, the equation is essentially degenerated and therefore the notion of weak solution is required. Definition. Let α ≥ 1. Given u0 ∈ L1∩Lα(Rn) with u0(x) ≥ 0 for x ∈ Rn, we call (u(t, x), ψ(t, x)) as a weak solution of the system (1.9) if there exists T > 0 such that i) u(t, x) ≥ 0 for any (t, x) ∈ [0, T ) × Rn, ii) u ∈ C(Rn × [0, T )) with ∇uα ∈ L2(Rn × [0, T )), iii) For arbitrary test function φ ∈ C1,1(Rn × [0, T )), ∫