Global stabilization of the full attraction-repulsion Keller-Segel system

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system \begin{equation}\label{ARKS}\tag{$\ast$} \begin{cases} u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),&x\in \Omega, ~~t>0, v_t=D_1\Delta v+\alpha u-\beta v,&x\in \Omega, ~~t>0, w_t=D_2\Delta w+\gamma u-\delta w,&x\in \Omega, ~~t>0,\\ u(x,0)=u_0(x),~v(x,0)= v_0(x), w(x,0)= w_0(x)&x\in \Omega, \end{cases} \end{equation} in a bounded domain $\Omega\subset \R^2$ with smooth boundary subject to homogeneous Neumann boundary conditions. %The parameters $D_1,D_2,\chi,\xi,\alpha,\beta,\gamma$ and $\delta$ are positive. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system \eqref{ARKS} with large initial data. Precisely, we show that if the parameters satisfy $\frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\}$ for all positive parameters $D_1,D_2,\chi,\xi,\alpha,\beta,\gamma$ and $\delta$, the system \eqref{ARKS} has a unique global classical solution $(u,v,w)$, which converges to the constant steady state $(\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0)$ as $t\to+\infty$, where $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0dx$. Furthermore, the decay rate is exponential if $\frac{\xi\gamma}{\chi\alpha}>\max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\}$. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. $D_1\ne D_2$) in multi-dimensions.

When ξ = 0, the variable w can be decoupled from the system (1), where the variables u and v satisfy the classical attractive Keller-Segel (KS) system u t = ∆u − χ∇ · (u∇v), x ∈ Ω, t > 0, v t = D 1 ∆v + αu − βv, x ∈ Ω, t > 0. (2) The KS model (2) has been extensively studied in the past four decades in various perspectives and massive results are available (cf. survey articles [6,1] and references therein). One of the mostly studied topics for the KS model (2) is the boundedness and blowup of solutions in two or higher dimensions [22,30,7] based on the following Lyapunov function: If χ = 0, the variable v can be decoupled and (u, w) satisfies the following repulsive Keller-Segel model u t = ∆u + ξ∇ · (u∇w), x ∈ Ω, t > 0, w t = D 2 ∆w + γu − δw, x ∈ Ω, t > 0.
Compared to the attractive KS model (2), the results on the repulsive KS model (3) are much less. The global existence of classical solutions in two dimensions and weak solutions in three or four dimensions were established in [4] based on the following Lyapunov function which is difference from the one for the attractive KS model. A further investigation on the repulsive KS model was made in [26]. Roughly speaking, the attraction-repulsion Keller-Segel model (1) can be regarded as a superposition of the attractive and repulsive KS models. Hence one may expect the ARKS model should behave more or less the same as the attractive or repulsive models. However it is not straightforward to justify this suspicion due to the interaction between attraction and repulsion. In particular, as we recalled above, the understanding of the attractive and repulsive KS models heavily rely on the finding of Lyapunov functions. Therefore to have a comprehensive understanding for the ARKS model, finding appropriate Lyapunov function is indispensable. This is by no means an easy work for a strongly coupled cross-diffusion system of PDEs like ARKS model. A sequence of works thus have been stimulated to reveal the mystery underlying the model gradually. The first such progress was made by Tao and Wang [27] who found that the solution behavior of the ARKS model was essentially determined by the sign of which is an index measuring the competition between attraction and repulsion. More precisely, they showed that if D 1 = D 2 = 1 and τ 1 = τ 2 = 0, the ARKS system (1) has a unique classical solution with uniform-in-time bound if θ 1 ≥ 0 (i.e. repulsion dominates or cancels attraction) in higher dimensions (n ≥ 2). The main idea of [27] was a transformation s = ξw − χv which may significantly simplify the system and has become a major source for many of subsequent researches on the ARKS model. For the opposite case θ 1 < 0 (i.e. attraction dominates repulsion), it was shown that the solution of system (5) may blow up in finite time if initial mass is large [5,14] and exist globally for small initial mass [5] in two dimensions. If τ 1 = 1 and τ 2 = 0, Jin and Wang [11] constructed a Lyapunov function to establish the global existence of uniformly-in-time bounded classical solutions in two dimensions for large initial data if θ 1 ≥ 0. Conversely if θ 1 < 0, they showed there exists a critical mass m * such that the solution blows up if Ω u 0 > m * and globally exists if Ω u 0 < m * . If the three equations of the ARKS model (1) are all parabolic (i.e. τ 1 = τ 2 = 1), it is much harder to study and much less results are available. We recall the known results below. In one dimension, the global existence of classical solutions, nontrivial stationary state, asymptotic behavior and pattern formation of the system (1) have been studied in [10,19,20]. In two dimensions, when D 1 = D 2 , it was shown in [27] that global classical solutions exist for large data if β = δ and for small data if β = δ when θ 1 ≥ 0 (i.e. repulsion dominates or cancels attraction). Subsequently the global existence of large-data solutions was extended to the case β = δ in [8,18]. Moreover, for β = δ, when cell mass is small, it was shown that the global classical solution will exponentially converge to the unique constant steady state (ū 0 , α βū 0 , γ δū 0 ) withū 0 = 1 |Ω| Ω u 0 in [15,16], which was further elaborated by assumingū in [17] wherein the convergence rate was, however, not given. Whether or not the same results holds for large initial data in multi-dimensions still remains unknown. Part of above-mentioned results have been extended to the multi-dimensional whole space in [9,25]. We should underline that all existing results in two or higher dimensions recalled above for the case τ 1 = τ 2 = 1 are essentially based on the assumption D 1 = D 2 so that the idea of making a change of variable s = ξw − χv introduced in [27] can be employed. To the best of our knowledge, no result for the case τ 1 = τ 2 = 1 and D 1 = D 2 has been available to (1) in multi-dimensions to date. It is the purpose of this paper to exploit this challenging case and contribute some results, where the corresponding ARKS model (1) reads as The main challenge is that when D 1 = D 2 the conventional approach of using the transformation in [27] is no longer effective and new ideas are desirable. Here we shall construct a Lyapunov functional for (5) which allows us to establish the global boundedness and asymptotic behavior of solutions to (5) in some parameter regimes. Specifically, the following results are obtained in the paper.

Remark 2.
The results of Theorem 1.1 hold for all D 1 , D 2 , α, β, ξ, γ > 0 without any smallness conditions on initial data under the parameter regime given by (6). In the case D 1 = D 2 = 1 and β = δ, the same result was recently obtained in [17] under the essential assumption (4) where the initial cell mass can not be arbitrarily large and parameter regime depends upon the initial data. Hence our results not only improve those of [17], but also cover the case D 1 = D 2 for which no results have been known so far.
Outline of proof: We first establish the boundedness criterion of solution for system (5) such that the boundedness of u L ∞ can be reduced to prove the boundedness of u L p with p > max{1, n 2 }. Motivated by the results in [8,18], we know that the boundedness of u L 2 holds in two dimensions if there exists a constant c 1 > 0 such that Hence to show the global existence of classical solutions in two dimensions, we only need to prove (8). When D 1 = D 2 and θ 1 > 0, using the transformation s = ξw−χv as in [27], one can derive the following entropy inequality (cf. [18,8]) which can be used to derive (8) and hence the boundedness of solutions. However, when D 1 = D 2 , the transformation idea fails to work. Luckily, we are able to find a different Lyapunov function E(u, v, w) (see the definition in (31)) for the system (5) under the condition (6), which satisfies where F (u, v, w) is defined by (32). We remark that the form of E(u, v, w) is quite different from the one (9) for D 1 = D 2 . To prove E(u, v, w) is a Laypunov function, we organize the estimates into a quadratic form which is the new idea developed in the paper. Then using (10), we show that under the condition (6), there exists a constant c 3 > 0 such that The former estimate leads to the boundedness of solutions in two dimensions and the later estimate gives the convergence properties of u. The convergence of v and w can be derived by the parabolic comparison principle. To study the decay rate, we first show that there exists a constant µ > 0 such that which together with (10) gives Using the definition of E(u, v, w) and noting the fact u −ū L 1 ≤ 2ū Ω u ln ū u in Lemma 2.1, the exponential decay of u −ū L 1 under the condition (6) is obtained. Then using the ideas in [27] or [15], we derive the decay rate of u −ū L ∞ and hence the exponential decay rate of v − α βū 0 L ∞ and w − γ δū 0 L ∞ . In the end of this section, we remark that Theorem 1.1 only present some firsthand results on the full ARKS model for D 1 = D 2 under the parameter regime given in (6) and leave out many interesting questions due to technical difficulty. For example, whether the condition (6) is necessary for global existence of solutions and how solutions behave (in particular whether solutions blow up) if the condition (6) fails remain unsolved in our paper. We hope our studies in this paper will provide useful clues to further explore the ARKS model in future.

Some basic inequalities.
In what follows, without confusion, we shall abbreviate Ω f dx as Ω f for simplicity. Moreover, we shall use c i (i = 1, 2, 3, · · · ) to denote a generic constant which may vary in the context. For reader's convenience, we present some known inequalities for later use.
Proof. Using the Csiszár-Kullback-Pinsker inequality (see [3] ), one has On the other hand, choosing ψ = f f and using the fact that ψ ln Then the combination of (12) and (13) gives (11).

Boundedness criterion and Lyapunov function.
3.1. Local existence. The local existence theorem of system (5) can be proved by the fixed point theorem and maximum principle along the same line as in [27]. Hence we only present the results without proof for brevity. 3 . Then there exist a T max ∈ (0, ∞] such that the system (5) has a unique solution (u, v, w) of nonnegative functions from Furthermore, the cell mass is conservative: 3.2. Boundedness criterion. To extend the local solutions to global ones, we derive a boundedness criterion for the solution of system (5). The idea of our proof is essentially inspired by [1, lemma 3.2] and we present necessary details below for clarity.
then one can find a constant C > 0 independent of t such that Furthermore, there exists σ ∈ (0, 1) such that for all t > 1 Proof. Since u(·, t) L p ≤ M 0 , then applying the parabolic regularity estimates in [12, Lemma 1] to the second and third equations of system (5) we have ∇v(·, t) L r + ∇w(·, t) L r ≤ c 1 , for all t ∈ (0, T max ) where Without loss of generality, we assume that n 2 < p ≤ n which yields np n−p > n. Then we can find a constant r > 0 with n < r < np n−p such that (23) holds. Now, for each T ∈ (0, T max ), we define which is finite due to the local existence results in Lemma 3.1. Next, we will estimate M (T ). Fix t ∈ (0, T ) and let t 0 = (t−1) + . Then applying the variation-of-constants formula to the first equation of system (5), we get We first estimate the term I 1 . If t ≤ 1, then t 0 = 0 and we can use the maximum principle for the heat equation to obtain whereas in the case t > 1 and t 0 = t − 1, we use the standard L p -L q estimates for (e τ ∆ ) τ ≥0 to derive Moreover, since r > n, we can fix a number q > n satisfying q ∈ ( r r+1 , r). Then by the Hölder inequality, interpolation inequality and (25), we can find ζ = r(q−1)+q rq ∈ (0, 1) such that Similarly, we have q−n thanks to q > n. Then by the smoothing properties of (e τ ∆ ) τ ≥0 (see [29,Lemma 1.3]), we derive Substituting (27), (28) and (29) into (26), we can find a constant c 10 > 0 such that Since 0 < ζ < 1, from (30) one has M (T ) ≤ max c 10 c 9 1 ζ , (2c 9 ) 1 1−ζ , for all T ∈ (0, T max ), which implies u(·, t) L ∞ ≤ c 11 for all t ∈ (0, T max ). Furthermore the combination of (23) and (24) gives (21). At last, from (21) we know that χu∇v and ξu∇w are bounded in L ∞ (Ω×(0, ∞)). Then applying the standard parabolic regularity theory (e.g. see [24,Theorem 1.3] and [28, Lemma 3.2]) and parabolic Schauder theory [13], we immediately obtain the estimate (22). Then the proof of Lemma 3.2 is completed.
3.3. Lyapunov function. As mentioned in Remark 1, we consider the case D 1 = D 2 or β = δ which implies that θ 1 > 0 from (6). When D 1 = D 2 , the boundedness of solutions shown in Theorem 1.1 has been proved in [27] with β = δ and in [18,8] with β = δ by constructing entropy inequality based on an idea of using the transformation s = ξw − χv. However this transformation is no longer helpful for the case D 1 = D 2 . Hence, we need to find a new way. Here we achieve our results by constructing a Lyapunov function for the system (5). First, we define and where θ 1 := ξγ −χα and θ 2 := ξγ +χα. Then, we will show that E(u, v, w) is indeed a Lyapunov function under (6). More precisely, we have the following results. where E(u, v, w) and F (u, v, w) are defined by (31) and (32), respectively. Moreover, if (6) holds, then E(u, v, w) ≥ 0 and F (u, v, w) ≥ 0 for all t > 0.
Proof. Multiplying the first equation of system (5) by ln ū u , we have Similarly, we multiply the second and third equations of system (5) On the other hand, the second and third equations of system (5) give us that The combination of (38) and (39) gives (33). Next, we will show the nonnegative of E(u, v, w) and F (u, v, w) under (6). First, we rewrite E(u, v, w) in (31) as where Θ T 1 denotes the transpose of Θ 1 and Since θ 1 > 0, one has θ 2 2 > θ 2 2 − θ 2 1 = 4ξγχα. This implies the matrix A 1 is positive definite and hence there exists a constant c 1 > 0 such that where we have used the fact Ω u ln ū u ≥ 0 from Lemma 2.1. Similarly, we rewrite F (u, v, w) as where Clearly, the matrix A 2 is nonnegative definite if Similarly, the matrix A 3 is nonnegative definite under the condition Hence, the nonnegativity of the matrices A 2 and A 3 are satisfied simultaneously if One can check that (43) However, the latter is impossible due to θ 1 > 0. Hence, if (6) holds, one has E(u, v, w) ≥ 0 and F (u, v, w) ≥ 0. The proof of (34) is completed.
4. Proof of Theorem 1.1. In this section, we are devoted to proving Theorem 1.1 based on the Lyapunov function constructed in Lemma 3.3.

Boundedness of solutions.
In this subsection, we show the boundedness of solutions for system (5) under the condition (6). First, we give a core lemma concerning the boundedness and asymptotical behavior of solution for system (5) in two dimensions.
and t 0 Ω where C > 0 is a constant independent of t.
Proof. The nonnegativity of E(u, v, w) and F (u, v, w) has been proved in Lemma 3.3 under the condition (6). Then integrating (33) and using (40) and (41), along with the nonnegativity of A 2 and A 3 , we have two positive constants c 1 , c 2 such that which, together with the fact Ω u ln ū u ≥ 0 from Lemma 2.1, gives On the other hand, from (46), we directly obtain which, along with the fact −u ln u ≤ 1 e for all u ≥ 0, gives Then the combination of (47) and (48) gives (44). Hence the proof of this lemma is completed. where the constant C > 0 is independent of t.
Proof. Multiplying the first equation of system (5) by u and integrating it by parts, we have 1 2 Noting the fact u ln u L 1 ≤ c 2 and u L 1 ≤ c 3 , one can find a small ε > 0 such that where we have used the following fact (see [22]): when n = 2, for any ε > 0, there exists a constant C ε such that On the other hand, noting the facts ∂v ∂ν ∂Ω = ∂w ∂ν ∂Ω = 0 on ∂Ω and using the boundedness of ∇v L 2 and ∇w L 2 (see (44)), from Lemma 2.2, one has Then combining (51) and (52), and using Young's inequality and noting the fact ε > 0 is small, we find a small η > 0 such that Substituting (53) into (50) gives Differentiating the second equation of system (5) once, and multiplying the result by −∇∆v, and then we integrate the product in Ω to obtain 1 2 which yields Similarly, we have the following estimates for w: Letting ρ = α 2 D2+γ 2 D1
Next, we will show the existence of global classical solutions. Proof. From Lemma 4.2, we know that there exists a constant c 1 > 0 such that u(·, t) L 2 ≤ c 1 . Noting n = 2 and using Lemma 3.2, one has which together with the local existence results in Lemma 3.1 completes the proof of this lemma.

4.2.
Convergence. In this subsection, we will show the convergence of solutions.
Proof. The combination of (7) and (45) implies that there exist a constant c 1 > 0 such that Noting the conservation of cell mass and using the Poincaré inequality, we will derive u(·, t) −ū 0 Combining (61) and (62), one can find a constant c 3 > 0 such that then (60) follows directly. We shall show (64) by the argument of contradiction. Suppose that (64) is wrong, then for some constant c 4 > 0, there exist some se- From Lemma 3.2, we know u −ū 0 is uniformly continuous in Ω × (1, ∞). Then there exist r > 0 and T 1 > 0 such than for any j ∈ N, for all x ∈ B r (x j ) ∩ Ω and t ∈ (t j , t j + T 1 ).
Because of the smoothness of ∂Ω, we can get a constant c 5 > 0 such that Using (65) and (66), for all j ∈ N, we have tj +T1 tj Ω However, by the fact t j → ∞ as j → ∞, we have from (63) that tj +T1 tj Ω which contradicts (67). Hence (64) holds by the argument of contradiction. Thus the proof of Lemma 4.4 is completed.
Next, we will show the convergence of v and w by the comparison principle.
Then from the second equation of (5), one has Let φ * (t) be the solution of ODE problem The application of the comparison principle show that φ * (t) is a super-solution of problem (68) and satisfies Similarly, we can prove that φ(x, t) ≥ −φ * (t) for all x ∈ Ω, t > 0. Hence, one has On the other hand, using the fact u(·, t) −ū 0 L ∞ → 0 as t → ∞ and from (69) we have φ * (t) → 0 as t → ∞, Similar arguments applied to the third equation of system (5) yield which completes the proof of Lemma 4.5.
Lemma 4.6. Suppose that the conditions in Lemma 4.1 hold. If ξγ χα > max β δ , δ β , then there exist two constants C > 0 and λ > 0 such that u(·, t) −ū 0 L 1 ≤ Ce −λt for all t > 0. (73) Proof. The nonnegativity of E(u, v, w) and F (u, v, w) has been proved in Lemma 3.3 under the condition (6). Next, we show that if ξγ χα > max β δ , δ β , there exists a constant µ > 0 which will be chosen later such that In fact, using the definition of E(u, v, w) and F (u, v, w) in (31) and (32), respectively, we derive that where To show the nonnegativity of D(u, v, w), we first show the nonnegativity of first term on the right hand of (75). From Lemma 2.1, we have On the other hand, using (62) and the fact u L ∞ ≤ c 1 , one derives which combined with (76) gives where µ 1 = c 1 c 2 . Hence, we can choose µ ≥ µ 1 such that the first term on the right hand of (75) is nonnegative. Next, we will show the nonnegativity of D 1 (u, v, w). In fact, we can rewrite D 1 (u, v, w) as where A 2 and Θ 2 are defined in (42). The condition (6) gives ξγ χα ≥ max D1 D2 , D2 D1 . Then hence the matrix A 2 is nonnegative definite and hence D 1 (u, v, w) ≥ 0 for any µ > 0.
Similarly, to show the nonnegativity of D 2 (u, v, w), we rewrite it as .
Next, we will derive the decay rate of solutions in L ∞ -norm based on the decay rate of u(·, t) −ū 0 L 1 . Lemma 4.7. Let (u, v, w) be the global classical solution of system (5). Suppose that there exist two positive constant C, λ such that then the solution (u, v, w) will exponentially decay to (ū 0 , α βū 0 , γ δū 0 ) with L ∞ -norm as t → ∞.
Proof. With (79) in hand, we can use the Moser-Alikakos iteration procedure as in [27] or the semigroup estimate method in [15] to obtain u −ū 0 L ∞ ≤ c 1 e −c1t .
Then applying the comparison principle as in [27], one can show that there exists a constant c 2 > 0 such that Then the proof of this lemma is completed. converges to ( α βū 0 , γ βū 0 ) as t → ∞ in Lemma 4.5. Moreover, if ξγ > χα max β δ , δ β , then using Lemma 4.6, we can obtain u(·, t) −ū 0 L 1 ≤ c 3 e −λt , which, along with Lemma 4.7, gives the exponential decay rate as shown in Theorem 1.1. Then Theorem 1.1 is proved.