Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v),&(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u,&(x,t)\in\Omega\times (0,T),\\ \end{array} \right. \qquad \qquad (\star) \end{equation} is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and approaches the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_{\Omega} u_0$. Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.


Introduction
In this paper, we consider the initial-boundary value problem for two coupled parabolic equations, ∇v · ν = (∇u − uS(x, u, v) · ∇v) · ν = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω, (1.1) where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary and ν is the normal outer vector on ∂Ω. S is either a scalar number or S(u, v, x) = (s i,j (u, v, x)) n×n is supposed to be a matrix with s i,j ∈ C 2 ([0, ∞) × [0, ∞) ×Ω) (i, j = 1, 2) and satisfies |S(u, v, x)| ≤ C S with C S > 0. (1.2) Systems of this type describe the evolution of cell populations and their movement affected by the gradients of a chemical signal produced by the cells themselves, a mechanism commonly called chemotaxis. A classical chemotaxis system was proposed by Keller and Segel [10]. In the simplest form of their model, the first equation in (1.1) reads u t = ∆u − ∇ · (χu∇v), (1.3) where u, v denote the density of the cell population and chemical substance concentration, respectively.
The number χ ∈ R measures the sensitivity of the chemotactic response to the chemical gradients, and the second term on the right of (1.3) reflects the hypothesis that cells move towards higher densities of the signal. The second equation in (1.1) models the assumptions that the chemical substance is produced by cells and degrades. This kind of model has been widely studied during the last 40 years [15,6]. We also refer to the survey [5,4] for a broad overview.
Among the large quantity of the related researches, deciding whether solutions exist globally or blow up in finite time seems to be one of the most challenging mathematical topics [18]. In the two-dimensional setting, a critical mass phenomenon has been identified and studied in many works. In the case Ω u 0 < 4π, the solution is global and bounded [11], whereas if Ω u 0 > 4π, the occurrence of blow-up for some initial data is only detected when the second equation is replaced by an elliptic equation of the form −∆v + v − u = 0 or −∆v − (u −ū 0 ) = 0, which reflects a certain limit procedure [7]. In higher dimensions, there are many results for such simplified parabolic-elliptic versions. For instance, in [2], it is proved that the corresponding Cauchy problem possesses a global weak solution whenever u 0 L n 2 (R n ) < C with some C > 0. Considering the fully parabolic case, it is also shown that for small initial data, with some suitably small constant ε > 0, q > n 2 and p ≥ n, the solution exists globally and is bounded [3]. However, for the fully parabolic version in bounded domains, the same conclusion is up to now known to hold only for q > n 2 and p > n [17]. It is our goal to extend this result in the corresponding critical case, that is, for q = n 2 and p = n. In contrast to (1.3), a recent study suggests a more general model which allows a wider direction of the cells' movement, such as they move not to the higher density of chemical any more but with a rotation. Then in this system a sensitivity tensor S(u, v, x), instead of a scalar constant χ, is introduced to describe chemotactic motion [14]. The introduction of this tensor valued sensitivity is caused by a kind of complicated interactions between the cell motion speed and directional effects stemming from the action of gravity, for example.
In our study of this new model, we concentrate on the two-dimensional case, and we anticipate that small mass of u guarantees global existence, which indeed parallels the case of a scalar sensitivity. However, the classical way of proof in the scalar case strongly depends on the use of an energy inequality [11], which is apparently lacking in the general system. To the best of our knowledge, the only results on global existence and boundedness in a related case can be found in [9], but with the second equation being replaced by v t = ∆v − f (u)v, whereby v enjoys an a priori upper bound according to the obvious estimate v ≤ v 0 L ∞ (Ω) . Under mild assumptions on S and f , the authors in that work proved that the solution exists globally and is bounded if either v 0 L ∞ (Ω) or u 0 L 1 (Ω) is small enough.
Since the second equation of (1.1) has a production term, the method in [9] does not apply to the present situation. However, we may benefit from our approach developed above for the case of scalar sensitivities in order to prove global boundedness under the assumptions that (1.2) holds, and that both u 0 L 1 (Ω) and ∇v 0 L 2 (Ω) are small enough. We underline that this assumption on the initial data is still stronger than that in the case of scalar sensitivity, but our results include exponential convergence, which has not been found before without assuming ∇v 0 L 2 (Ω) small enough.
Our main result says that • if n ≥ 2, one can find an upper bound for u 0 in L n 2 (Ω) and ∇v 0 in L n (Ω), such that the solution (u, v) of (1.1) exists globally and is bounded.

Preliminaries
In this section, we recall some classical L p − L q estimates for the Neumann heat semigroup on bounded domains. Almost all of the results and their proofs can be found in [17,Lemma 1.3]. However, some of the estimates we use below go slightly beyond, and since we could not find a precise reference, we will give a short proof here.
holds for each w ∈ L q (Ω).
is true for all w ∈ W 1,p (Ω).
Proof (i) and (ii) are precisely proved in [17,Lemma 1.3]. Focusing on (iii), we note that it is obviously true for all t < 2 [17]. If t ≥ 2, letw = 1 |Ω| Ω w, we use (2.1), (2.2) and the Poincáre inequality to obtain Thus (iii) is valid for all t > 0. Note that k 3 is independent of p or q since the constant in Poincáre inequality could be independent of q.
for all t ∈ (0, T ) with k 3 = 3k 1 k 2 . Thus (2.4) is obtained by means of a unique extension to all of (W 1,p (Ω)) n .
Before going into the main part, we also recall some local existence and extensibility results for (1.1).
We refer to [ S is a scalar constant. Beyond these, we generalize as follows.
Now we give a sufficient condition for boundedness and global existence.
If the solution of (1.1) satisfies The following lemma presents an estimate for certain integrals, which we will frequently use in the next section. The proof can be found in [17, Lemma 1.2].

Smallness conditions in optimal spaces
Now having at hand the tools collected in the last section, we can prove global existence in the classical where Ω ⊂ R n with smooth boundary and n ≥ 2.
Our proof relies on a fixed-point type argument developed in [17]. Unlike in the original proof there, our assumption is that the initial data be suitably small in optimal spaces in the sense that we require that u 0 L n 2 (Ω) and ∇v 0 L n (Ω) are small.
This seems too weak to give an L ∞ (Ω) bound in a one-step procedure. However, we can first use the "weak" assumption to obtain the smallness condition in supercritical spaces, which meet the assumptions in [17, Theorem 2.1]. Finally, we can derive convergence in L ∞ (Ω), and obtain the convergence rate e −λ ′ t with any 0 < λ ′ < λ 1 , where λ 1 > 0 denote the first nonzero eigenvalue of −∆ in Ω under Neumann boundary conditions.
In Theorem 1, we show how we improve the smallness condition into a supercritical space.
With ε 0 > 0 to be determined below, we assume (3.2) holds for ε ∈ (0, ε 0 ), and define We observe that T is well defined and positive due to Lemma 2.2. It is sufficient to prove that T = ∞.
Since n ≥ 2, it is easy to see that Then by the definition of T and (2.1), for each θ ∈ [q 0 , θ 0 ], holds for all t ∈ [0, T ), where c 4 = 1 + 2k 1 only depends on Ω.
Remark 3.1 A careful re-inspection of the above argument shows that for the constant c 10 = c 10 (λ ′ ) satisfies c 10 (λ ′ ) → ∞ as λ ′ → λ 1 , where the constant ε 0 = ε 0 (λ ′ ) by Theorem 1 has the property that With the decay property in higher Lebesgue spaces obtained above, we can obtain a smallness condition which ensures stabilization of (u, v) in L ∞ (Ω).

System with rotational sensitivity
In this section, we consider the modified Keller-Segel system with general tensor-valued sensitivity as given by where Ω ⊂ R 2 with smooth boundary. The sensitivity S is now supposed to be a tensor-valued function satisfying (1.2). The non-flux and coupled boundary condition complicate the solvability. Following [9], we first regularize the system as below , which vanishes on the boundary ∂Ω if ρ η is a suitable cut-off function on Ω. Where ρ η ∈ [0, 1] and satisfies ρ η → 1 a.e. as η → 0. (4. 3) The first boundary condition of (4.1) is then reduced to ∇u η · ν = 0, so that local classical solvability of (4.1) can be obtained by the standard approach (Lemma2.2). Upon combining the idea in the previous section with a limiting procedure η → 0, we will derive the following.

then (4.1) possesses a global classical solution (u, v) which is bounded and satisfies
with someC > 0.
Before we proceed to prove Theorem 3, we start with studying the regularized problem (4.2). Since  If u 0 ∈ C 0 (Ω) and v 0 ∈ W 1,σ (Ω) with σ > 2 are nonnegative and satisfy for some ε ≤ ε 0 , where ε 0 depends on Ω and λ ′ . Then the classical solution (u η , v η ) of (4.2) exists globally and stays bounded. Moreover, there exists M > 0 depending on Ω and λ ′ such that and as well as ∇v η L 2 (Ω) ≤ M εe −λ ′ t for all t > 0. (4.10) Note that the above estimates are independent of η. Having obtained global existence and long time convergence for (4.2), we proceed to construct a solution of (4.1) upon letting η → 0. This limit procedure needs some compactness properties of (u η , v η ), which are proven in the following lemmata.
Proof The boundedness of ∞ 0 Ω |∇v η | 2 is an immediate consequence of (4.10). Next, we multiply the first equation of (4.2) by u η to see that for all t > 0. Rearranging and integrating over (0, ∞) imply Finally, we can choose C 1 > 0 in an obvious way to establish (4.11) and (4.12).
Now we can obtain the desired compactness properties of (u η , v η ) to prove Theorem 3.

Acknowledgement
The author would like to thank Professor Michael Winkler for guidance and useful comments on this paper, which largely improve the recent work.