Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic--elliptic Keller--Segel system in the whole $n$-dimensional space is studied. For this model, every constant $A \in \mathbb{R}$ is a stationary solution. The main goal of this work is to show that $A<1$ is a stable steady state while $A>1$ is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.


Introduction
There are several mathematical works on the chemotaxis model introduced by Keller and Segel [15]. Here, we refer the reader only to the monograph [24] and the reviews [3,12] for a discussion of those mathematical results as well as for additional references. In this paper, we consider the following minimal parabolic-elliptic Keller-Segel system where u = u(t, x) denotes the density of cells and ψ = ψ(t, x) is a concentration of chemoattractant. In these equations, all constant parameters are equal to one for simplicity of the exposition. System (1.1) was already studied in the whole space e.g. in the papers [4,5,6,13,14,16,17,19], where several results either on a blow up or on a large time behavior of solutions have been obtained. For each constant A ∈ R, the couple (u, ψ) = (A, A) is a stationary solution of system (1.1) and, since the domain is unbounded, it does not belong to any Lebesgue L pspace with p ∈ [1, ∞). Thus, in Theorem 2.1 and in Section 3, we develop a mathematical theory concerning local-in-time solutions to the initial value problem for system (1.1) in the uniformly local Lebesgue spaces L p uloc (R n ). Then, we consider a constant stationary solution (u, ψ) = (A, A) with A ∈ [0, 1) and we show in Theorem 2.3 that a small L p -perturbation of such an initial datum gives a global-in-time solution which converges toward (A, A) as t → ∞. On the other hand, we prove in the Theorem 2.4 that the constant solution is unstable in the Lyapunov sense if A > 1.
A stability of constant solutions of chemotaxis models has been already studied in bounded domains. For example, the paper [10] describes dynamics near an unstable constant solution to the classical parabolic-parabolic Keller-Segel model in a bounded domain and obtained results are interpreted as an early pattern formation. Another work [22] is devoted to the system u t − ∆u + ∇ · (u∇ψ) = 0, −∆ψ + µ = u, µ ≡ 1 in the ball of radius R > 0 with Neumann boundary condition. Here, constants are also stationary solutions and it is shown in the work [22] that there exists a critical number m c such that at mass levels above m c the constant steady states are extremely unstable and blow-up can occur. On the other hand, for m < m c there exist infinitely many radial solutions with a mass equal to m.
Notation. The usual norm of the Lebesgue space L p (R n ) with respect to the spatial variable is denoted by · p for all p ∈ [1, ∞]. In the following, we use also the uniformly local Lebesgue space L p uloc (R n ) with the norm · p,uloc defined below by formula (2.3). Any other norm in a Banach space Y is denoted by · Y . The letter C corresponds to a generic constant (always independent of t and x) which may vary from line to line. We write C = C(α, β, γ, ...) when we want to emphasize the dependence of C on parameters α, β, γ, ... . We use standard definition of Fourier transform f (ξ) = (2π) −n/2 R n e −iξ·x f (x) dx.

Results and comments
Our goal is to study properties of solutions to the Cauchy problem for the simplified parabolic-elliptic Keller-Segel model of chemotaxis with n ≥ 1. We solve the second equation with respect to ψ to obtain ψ = K * u, where K is the Bessel function (see Lemma 3.5 below) which reduces problem (2.1) to the following one We begin by a result on an existence of local-in-time solutions to problem (2.2) in the uniformly local Lebesgue spaces

4)
and every u 0 ∈ L p uloc (R n ), there exists T > 0 and a unique mild solution The more-or-less standard proof of Theorem 2.1 is based on the Banach contraction principle applied to an integral representation of solutions to problem (2.2) (see Section 3 for more details). Now, we formulate a simple consequence of Theorem 2.1 in the case when an initial condition is an L p -perturbation of a constant A ∈ R.
Corollary 2.2. Let p satisfy conditions (2.4). For every A ∈ R and every v 0 ∈ L p (R n ) there exists a unique local-in-time mild solution u = u(t, x) of problem (2.2) (as stated in Theorem 2.1) corresponding to the initial datum This corollary is an immediate consequence of the uniqueness of solutions established in Theorem 2.1 combined with the uniqueness result of solutions to the perturbed problem considered in Proposition 5.1, below.
Next, we show that one can construct global-in-time solutions around each constant solution A ∈ [0, 1). 1). Assume that the exponent p satisfies conditions (2.4) and moreover p ≤ n. Fix q ∈ (n, 2p]. There exists ε > 0 such that for The smallness assumption imposed on initial conditions in Theorem 2.3 seems to be necessary. This is clear in the case A = 0, where sufficiently large initial data lead to solutions which blow-up in finite time, see e.g. [4,6,14,16] for blow-up results for solutions of system (1.1) considered in the whole space.
Next, we deal with A > 1 which appears to be the unstable constant stationary solution. In this theorem, we do not claim that solutions corresponding to L p -perturbations of A > 1 are global-in-time. We show only that if they are global-in-time then they cannot be stable.
To conclude this section, we notice that a constant A < 0 is linearly stable which we comment in Remark 4.8, below. The proof of nonlinear stability of this constant can be obtained by the method used in the proof of Theorem 2.3. We add some comments on the linear stability of the constant solution A = 1 in Remark 4.9, below.

Local-in-time solutions in uniformly local Lebesgue spaces
We find solutions to problem (2.2) via its formulation in the integral form This construction requires auxiliary results which we are going to gather and prove below. Then, Theorem 2.1 is proved at the end of this section. We begin by recalling properties of the heat semigroup {e t∆ } t≥0 acting on the L p uloc -spaces. Proposition 3.1 ( [2,18]). For all 1 ≤ q ≤ p ≤ ∞, k ∈ Z + and α ∈ Z n + there exist numbers C = C(n, p, q, k, α) > 0 such that for every f ∈ L p uloc (R n ) and all t > 0. In particular, when p = q and 1 ≤ p ≤ ∞, it holds that Remark 3.2. Proposition 3.1 generalizes the following well-known estimates of the heat semigroup acting on the Lebesgue space L p (R n ) Remark 3.3. It follows from estimates (3.2) applied with k = 0, α = 0 and p = q that e t∆ f ∈ L ∞ (0, ∞); L p uloc (R n ) for each f ∈ L p uloc (R n ) and, in general, this mapping in not continuous in time. This continuity holds true in the smaller space where BUC(R n ) is a space of all bounded uniformly continuous functions on R n . In fact, the following statements are equivalent: (1) f ∈ L p uloc (R n ).
(3) e t∆ f − f p,uloc → 0 as t → 0. We refer the reader to [18,Proposition 2.2] for the proof of these properties. |f (y)| p dy 1/p for each ρ > 0. However, by a simple scaling property, we can show that all these norms are in fact equivalent.
Let us also recall properties of the Bessel kernel which are systematically used in this work.
Lemma 3.5. Denote by ψ ∈ S ′ (R n ) a solution of the equation −∆ψ + ψ = u for some u ∈ S ′ (R n ). The following statements hold true.
Sketch of the proof. Property 1 is well known. Item 2 can be obtained by a direct calculation. For the proofs of properties 3 and 4 we refer the reader to [1, Sec. 1.2.5.]. In particular, in order to show property 4, we recall that ) stands for the modified Bessel function of the third kind, that is In particular, we observe that is radially symmetric and decreasing in |x|.
Our goal is to estimate the nonlinear term on the right hand side of equation (3.1). We begin with the following elementary lemma.

Thus we have
We define cubes corresponding to y k ∈ Q 1/2 (k) for k ∈ G. Hence, there exists ℓ ∈ N and We call such a cube the corresponding cube and we denote By [18, line 5 from above, p.384], for each fixed k ∈ G, the set P k consists of at most 5 n points. Therefore, and thus where 1 + 1 p = 1 q + 1 r for 1 ≤ r < n n−1 . Thanks to Lemma 3.6, we obtain ∇K * f p,uloc ≤ (90 n ∇K 1 + 54 n ∇K r ) f q,uloc , which completes the proof.
for all u, v ∈ L p uloc (R n ) and τ > 0. Proof. Using the heat semigroup estimates from Proposition 3.1 and the Hölder inequality (which also holds in L p uloc -norm) we have Moreover, applying Proposition 3.7, we obtain Let us show that for every p satisfying condition (2.4) we can always choose k ∈ (n, ∞] and r ≥ 1 satisfying the conditions above. Indeed, if n > 1 then for every k ∈ [p, 2p] we have 1 Analogously if n = 1 then the inequalities holds true for every k ∈ [p, ∞]. Next, the condition r ≥ 1 is equivalent to k ∈ [p/(p−1), ∞]. We obtain a solution to integral equation (3.1) from the Banach fixed point theorem formulated in the following way. Proposition 3.9. Let X be a Banach space and let Q = Q[·, ·] : X × X → X be a bounded bilinear form with Assume that δ ∈ (0, 1/(4C 1 )). If y 0 X ≤ δ, then the equation has a solution with u X ≤ 2δ. This solution is unique in the set u ∈ X : u X ≤ 2δ and stable in the following sense: if y 0 , y 0 ∈ X satisfy y 0 X ≤ δ, y 0 X ≤ δ then for the corresponding solutions u, u ∈ X we have where C 2 > 0 is independent of u and u.
Proof of Theorem 2.1. For T > 0, we introduce X T ≡ L ∞ (0, T ); L p uloc (R n ) which is a Banach space with the norm u X T ≡ sup t∈(0, T ) u(t) p,uloc . In order to apply Proposition 3.9, it suffices to estimate the bilinear form By Lemma 3.8, for some k > n, we obtain Therefore, we have Since inequality (3.3) in Proposition 3.1 provides e t∆ u 0 X T ≤ u 0 p,uloc , we obtain a solution to the integral equation via Proposition 3.9 for sufficiently small T > 0. In order to show that u ∈ C (0, T ); L p uloc (R n ) it suffices to follow the arguments from [18, p. 388]. This solution is unique by a usual reasoning.
To show that a solution is non-negative in the case of non-negative initial datum, we pass through an approximation process with a sequence of smooth solutions {u ε } ε>0 corresponding to the smooth, uniformly bounded non-negative initial conditions u ε 0 (x) = (e ε∆ u 0 )(x). Here, u ε ≥ 0 by the classical maximum principle, see e.g. [9, Theorem 9, p.43]. To complete this reasoning, we show that u ε (t) − u(t) p,uloc → 0 as ε → 0 for each t ∈ (0, T ). Indeed, from inequality (3.3) in Proposition 3.1, it holds that u ε 0 p,uloc ≤ u 0 p,uloc for all ε > 0. Hence, by the above construction of a local-intime solutions via Proposition 3.9, there exists a constant M > 0 such that u X T ≤ M and u ε X T ≤ M. Now, computing the norm u ε (t) − u(t) p,uloc and using the integral representation (3.1) of these functions as well as a second inequality in estimate (3.6) we obtain u ε (t) − u(t) p,uloc ≤ e t∆ u ε 0 − e t∆ u 0 p,uloc Applying the Volterra type inequality from [24, Ch. 9] we conclude that u ε (t) − u(t) p,uloc ≤ e t∆ u ε 0 − e t∆ u 0 p,uloc + C

Linearized problem
4.1. Preliminary properties. The linearization procedure described below in Section 5 leads to the following linear problem where A ∈ R is an arbitrary constant and the operator ∆ − A∆K * can be expressed by the Fourier transform as follows We begin by presenting preliminary properties of this operator.
This lemma is an immediate consequence of the fact that the operator −∆K * can be represented by a convolution with a finite measure on R n . We skip the proof of this classical result from the harmonic analysis, see e.g.

, ∞). This semigroup is defined by the Fourier transform by the formula
Proof. It is well-known that Laplacian generates an analytic semigroup of linear operators on L p (R n ) for every p ∈ [1, ∞). A bounded perturbation of such an operator maintains the same property, see e.g. [7, Chapter III, Theorem 2.10]. The Fourier representation of this semigroup can be obtained by routine calculations.

Lemma 4.3.
Assume that A ∈ R and choose the constant L from inequality (4.2). Then for each 1 ≤ q ≤ p ≤ ∞, there exists a constant C = C(p, q, n) > 0 such that Proof. Here, we use the notation S A (t)v 0 = e t∆ (e −A∆K * v 0 ). Using the L p -L q estimates of the heat semigroup (see Remark 3.2) and Lemma 4.1, we obtain The proof for the second inequality is analogous. Theorem 4.4. Assume that A ∈ [0, 1). For all exponents satisfying 1 ≤ q ≤ p ≤ ∞ there exist constants C = C(p, q, n, A) > 0 such that

Decay estimates when
The proof of this theorem is based on the following lemmas.

by the properties of the function h(ξ)
and k + 1 − (ℓ + 1)/2 ≤ (N + 1)/2. Now we group coefficients t k and |ξ| ℓ in the following way, t k |ξ| ℓ = t k−ℓ/2 | √ tξ| ℓ thus, by the assumption t ≥ 1 and induction, we have t k−ℓ/2 ≤ t N/2 . We obtain an estimate where P (s) is a polynomial of degree N. By the same inequality as in the case N = 0 and properties of the exponential function, for some δ ∈ (0, 1 − A). Calculating the L 2 -norm of both sides of inequality (4.6) we obtain the result.
Lemma 4.7. Assume that A ∈ [0, 1). For every p ∈ [1, ∞] there exists a constant C > 0 such that for all t > 0 and all v 0 ∈ L p (R n ).
Proof. For t ∈ [0, 1], this is an immediate consequence of Lemma 4.3. For t ≥ 1, the function µ A is from the Schwartz class in variable ξ. Thus, by the Young inequality, In order to estimate µ A (t) 1 , we recall a well known fact that both quantities D N v 2 and |α|=N ∂ α x v 2 are comparable for each N ∈ N. Combining Lemma 4.5 and Lemma 4.6, for N > n/2 and N ∈ N, we obtain Proof of Theorem 4.4. We begin with inequality (4.4). Let us choose ε ∈ (A, 1). By the standard heat semigroup estimates (see Remark 3.2), Now we substitutet = εt to obtain tA =t(A/ε) =tÃ and 0 ≤Ã < 1. Thus, by Lemma 4.7, We prove inequality (4.5) analogously using the formula Remark 4.8. The L q -L p estimates (4.4)-(4.5) of the semigroup S A (t) t≥0 hold true for A < 0 as well. They can be proved by the same reasoning as above using the obvious inequality Remark 4.9. For the completeness of this work, we notice that constant solution A = 1 is linearly stable in L 2 (R n ). Indeed, since e −t(|ξ| 2 −|ξ| 2 /(1+|ξ| 2 )) ≤ 1 for all ξ ∈ R n and t ≥ 0, by the Plancherel formula, we obtain S 1 (t)v 0 2 ≤ v 0 2 for all v 0 ∈ L 2 (R n ). We skip a discussion of a stability of this constant solution for p = 2.

4.3.
Exponential growth when A > 1. Next, we study an instability of solutions to linear problem (4.1).
Proof. It follows from Lemma 4.10 that a = ( √ A − 1) 2 lies on the boundary of the spectrum of the operator ∆ − A∆K * and such elements belong to the approximate point spectrum (see e.g. [20,Lemma 1]). Thus, for each ε > 0 there exists v ε ∈ L p (R n ) (in fact, v ε belongs to the domain of the closure in L p (R n ) of the operator ∆ − A∆K * ) such that for some constant C > 0 from inequality (4.3) with p = q. Therefore, inequality (4.10) with ε = γ CT e (|A|L+a)T provides the estimate Since a > 0, the second inequality in (4.9) can be obtained immediately from the first one by choosing γ = 1.

Perturbations of constant solutions
We study a solution u = u(t, x) of problem (2.2) which is a perturbation of the constant stationary solution A ∈ R. Thus, the function v(t, x) = u(t, x) − A satisfies Here, in fact, we consider a mild solution to this problem satisfying the integral equation where the semigroup {S A (t)} t≥0 has been studied in Section 4. For the proof of this proposition, one should follow the reasoning in the proof of Theorem 2.1. In particular, a solution is obtained via Proposition 3.9 where the required estimate of the bilinear form is a direct consequence of the following lemma.
Lemma 5.2. For each p satisfying condition (2.4) there exist k ∈ (n, ∞] and positive numbers C 1 = C 1 (p, k, n, ∇K) and C 2 = C 2 (p, k, n, ∇K) such that . If A < 1 then C 1 = 0 and if A ≥ 1 then C 1 can be an arbitrary constant satisfying We skip the proof of this lemma, because it is the same as the proof of Lemma 3.8. In particular, it is based on the semigroup estimates from Lemma 4.11.

5.2.
Global-in-time solutions for A ∈ [0, 1). The proof of Theorem 2.3 requires the following extension of Lemma 5.2.
• each q > n if n ≥ 2 and q ≥ 1 if n = 1 satisfying q ∈ [p, 2p], • each r ≥ 1 such that r ∈ q 2 , p , there exists a constant C > 0 such that for all v, w ∈ L q (R n ) and τ > 0.
Proof. First, we use Theorem 4.4 and the Hölder inequality to estimate with 1 ≤ r ≤ p satisfying 1 r = 1 q + 1 k . Next, we apply the Young inequality ∇K * w k ≤ C ∇K q 1 w q with 1 q 1 + 1 q = 1 + 1 k . Let us show that q 1 ∈ 1, n n−1 in order to have ∇K ∈ L q 1 (R n ). Indeed, by the assumption on p, q, r we have where, for n ≥ 2, we use also the inequality 1 − 1 q > 1 − 1 n . Proof of Theorem 2.3. It is sufficient to construct a global-in-time solution to problem (5.1) formulated in the mild form (5.2), because u = A + v by the uniqueness of solutions from Proposition 5.1. The solution is obtained via Proposition 3.9 applied to equation (5.2) in the Banach space with n 2p − n 2 1 q + 1 k + 1 2 ≥ 0 and, by Lemma 5.3, Therefore, there exists a constant C > 0 independent of t > 0 such that Finally, it follows from inequalities (5.7) and (5.8) that for a positive number η independent of t, v and w. Hence, if v 0 p is sufficiently small, by inequality (5.4) and Proposition 3.9, there exists a mild solution of problem (5.1) in the space X . This solution is unique by Proposition 5.1.

5.3.
Instability for A > 1. In this section, we prove that the constant solution u = A of problem (2.2) is unstable in L p (R n ) if A > 1. Here we apply the classical so-called linearization principle which was used e.g. in [8,20].
Proof of Theorem 2.4. We begin with arbitrary δ ∈ (0, 1) and arbitrary v 0 ∈ L p (R n ) with v 0 p = 1 to be chosen later on. Under the assumpion on p, by Corollary 2.2, there exists a unique local-in-time mild solution v ∈ C [0, T max ); L p (R n ) to problem (5.1), with the initial datum δv 0 . Suppose that this solution is global-in-time and for a = ( √ A − 1) 2 define two numbers T = sup t : v(τ ) − S A (τ )δv 0 p ≤ δ 2 e aτ for all τ ∈ [0, t] (5.9) and T ′ = 1 a log 2 δ , hence δe aT ′ = 2. If either T > T ′ or T = ∞, then the zero solution is unstable. Indeed, by Lemma 4.12, for each γ > 0 we may choose v 0 ∈ L p (R n ) with v 0 p = 1 such that S A (T ′ )δv 0 − e aT ′ δv 0 p ≤ γδ v 0 p = γδ. In particular, v(T ′ ) p ≥ 1 2 for γ = 1 2 . Next, suppose that T ≤ T ′ and consider the mild representation of the solution of problem (5.1) with the initial condition δv 0 v(t) − S A (t)δv 0 =  Finally, we apply Lemma 4.12 with γ = 1 4k 0 C ≤ 1 for some fixed k 0 ≫ 1 and T ′ (recall that T * ≤ T ≤ T ′ ) in order to obtain v 0 ∈ L p (R n ) with v 0 p = 1 such that The proof of instability is completed because the right hand side is independent of δ.
Remark 5.4. The last inequality in (5.11) follows from a direct calculation which we present for the reader convenience. For a fixed η ∈ (0, T ), we obtain