Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors

We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hoelder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.


Introduction
The study of tumor growth models is a very current topic in mathematics. During the last four decades an increasing number of mathematical models have been proposed to describe the growth of solid tumors (see [3,7,9,14] and the literature therein). There is a three level approach in modeling the complex phenomena influencing and describing the processes inside a tumor. Models at sub-cellular level take into consideration that the evolution of a cell is determined by the genes in its nucleus, at cellular level they model cell-cell interaction and at macroscopic level, when the tumor is considered to consist of three zones: an external proliferating zone near high concentration of nutrient, an intermediate layer and an internal zone consisting of necrotic cells only. Very often models combine aspects from different scales. There are also, a large variety of different types of models: biological models, consisting of coupled ODE systems where the variables correspond to some biological properties of an entire population; mechanical models yield to determine the cell movement based on physical forces; the discrete models handle singlecell scale phenomena and the effects are then examined at macroscopic scale and moving boundary models when the macroscopic description of biological tissues is obtained from continuum mechanics or microscopic description at cellular level.
In this paper we deal with a moving boundary problem, which is obtained by combining aspects from the cellular and macroscopic scale, and possesses also characteristics of the mechanical model (Darcy's law). Cristini et al. obtained in [6], using algebraic manipulations, a new mathematical formulation of an existing model (see [5,14,18]), which describes the evolution of nonnecrotic tumors in all regimes of vascularisation. This new formulation has the advantage of considering different intrinsic-time and length-scales related to the evolution of the tumor and, by incorporating them in the modeling, provides a model describing both vascular and avascular tumor: in Ω(t), t ≥ 0, on ∂Ω(t), t > 0, (1.1) Hereby Ω 0 is the initial state of the tumor, V is the normal velocity of the tumor boundary, κ ∂Ω(t) the curvature of ∂Ω(t), and the constants A and G have biological meaning, namely G is the rate of mitosis (cell proliferation) and A describes the balance between the rate of mitosis and apoptosis (naturally cell death). The function f ∈ C ∞ ([0, ∞)) has the following properties f (0) = 0 and f ′ (ψ) > 0 for ψ ≥ 0. (1. 2) The tumor domain Ω(t) is an unknown of the problem and, together with the rate ψ at which nutrient is added to the tumor domain Ω(t) and the pressure p inside the tumor, is to be determined. Three different regimes of vascularisation are introduced by the constants A and G : if G ≥ 0 and A > 0 the tumor is low vascularised, G ≥ 0 and A ≤ 0 correspond to the moderate vascularised case, and if G < 0 the tumor is highly vascularised.
In [6] the special case f = id [0,∞) is analysed numerically. Moreover, in this situation, the first equation of the system is linear, and if the tumor domain is a sphere or an infinite cylinder, then the solution is known through an explicit formula. The radially symmetric case when tumors are circles is considered in [12], where we show that if A ∈ (0, f (1)), then there exists a unique radially symmetric stationary solution D(0, R A ) of (1.1). The radius of the stationary solution depends only on A and this circular steady-state is exponentially stable under radially symmetric perturbations in the avascular case when G > 0, and unstable in the high vascularisation regime G < 0, result established for f = id [0,∞) also in [6]. The analysis in [12] will serve us in the present paper as an ancillary tool when proving the local wellposedness of problem (1.1) and when studying the stability properties of D(0, R A ).
The model, presented in [5,14,18], has been studied extensively by different authors, see e.g. [4,[7][8][9]14,15] and the references therein. In particular, it is shown in these papers that if certain parameters belong to an appropriate range, then the mathematical formulation possesses a unique radially symmetric solution, result matching perfectly with [12]. Moreover, the stability properties of this solution under general perturbations, as well as bifurcation phenomena are studied. In contrast, for the model presented in [6,12], and which we consider herein, not many analytic results are available. We prove that also this model is locally well-posed in time, meaning that for appropriate smooth initial data Ω 0 , there exists a unique solution of (1.1), cf. Theorem 2.1. Though in the radially symmetric case the steady-state solution D(0, R A ) is exponentially stable if G > 0, we show in Theorem 2.2, by considering arbitrary initial data, that this solution is unstable also in the low vascularised case, provided G lies above a well-defined constant G * . This result matches the case γ < γ * in [8, Theorem 1.2], since G is inversely proportional to γ. The situation when G ∈ (0, G * ) is still an open problem. If G = 0 the problem is equivalent to the Hele-Shaw problem studied in [13] and the exponential stability result stated by [13,Theorem 4.3] holds true. As a new property we establish in Theorem 2.3 exponential convergence of D(0, R A ) for every G > 0 and initial data in a certain class which depends on G.
The outline of the paper is as follows: we introduce in the second section a parametrisation for the unknown tumor domain which permites us to present the main results Theorems 2.1-2.3. Section 3 is dedicated to the proof of Theorem 2.1, and the stability results stated in Theorems 2.2 and 2.3 are proved in Section 4.

The main results
Let R > 0 be fixed for the remainder of this section. Our goal is to show that if the tumor is initially close to D(0, R), then problem (1.1) possesses a unique classical Hölder solution. To this scope let h r (S), r ≥ 0, denote the closure of the smooth functions C ∞ (S) in the Hölder space C r (S). Hereby, S stands for the unit circle and we identify functions on S with 2π-periodic functions on R. The small Hölder spaces h r (S) have the nice property that the embedding h r (S) is densely and compactly in h s (S) for all 0 ≤ s < r. We fix α ∈ (0, 1) and we shall use functions ρ ∈ V, whereby to parametrise the boundary of the tumor domain. Obviously, V is an open neighbourhood of the zero function in h 4+α (S). Given ρ ∈ V, we define the Figure 1. Parametrisation of the tumor domain C 4+α -perturbation of the circle centred in 0 with radius R The simply connected component of R 2 which is bounded by the curve Γ ρ is the set with boundary ∂Ω ρ = Γ ρ . Given x ∈ Γ ρ , the real number ρ(x/|x|) is the ratio of the signed distance from x to the circle R · S and R (see Figure 1). It is suitable to represent Γ ρ as the 0−level set of an appropriate function. For this, let N ρ : A(3R/4, 5R/4) → R be the function defined by where A(3R/4, 5R/4) is the annulus centred in 0 with radii 3R/4 and 5R/4 A(3R/4, 5R/4) := {x ∈ R 2 : 3R/4 < |x| < 5R/4}. Obviously A(3R/4, 5R/4) is an open neighbourhood of Γ ρ and Γ ρ = N −1 ρ (0). Let ν ρ denote the outward normal at Γ ρ . Since Γ ρ is the 0−level set of N ρ , the gradient ∇N ρ and ν ρ must be collinear vectors. Moreover, N ρ is positive on the complement of Ω ρ , hence N ρ (x + λν ρ (x)) > 0 for all x ∈ Γ ρ and λ > 0. Differentiating this relation with respect to λ at λ = 0 yields that ∇N ρ · ν ρ > 0, hence To incorporate time let T > 0. Presuppose that the function ρ ∈ C([0, T ], V)∩ C 1 ([0, T ], h 1+α (S)) describes the evolution of the tumor, which at time t = 0 is located at Ω(0) = Ω ρ(0) . The normal velocity V (t) of the moving boundary Γ ρ(t) is then given by the expression This relation follows from the standard assumption that the interface moves along with the tumor and from relation Γ ρ = N −1 ρ (0). With this notation (1.1) is equivalent to the following system of equations and (ρ, ψ, p) solves the system (2.1) pointwise. Given ρ ∈ V, buc k+α (U ) stands for the closure of BUC ∞ (U ) in BUC k+α (U ). Of major interest is to determine the mapping ρ which describes the evolution of the tumor. The functions ψ and p can be then determined as solutions of Dirichlet problems, cf. Lemmas 3.1, 3.3, and 3.4. This is the reason why we shall also refer only to ρ as solution to (2.1). The first main result of this paper is the following theorem: is smooth.
When R = R A , we re-discover ρ ≡ 0, situation when the tumor is located at D(0, R A ) as the unique radially symmetric stationary solution of (2.1). Concerning the stability properties of this solution we already know from the radially symmetric case [12,Theorem 1.2] that this solution is unstable for G < 0. Moreover, we have: Theorem 2.2. Let R = R A and G * > 0 be the constant defined by (4.21). Then the radially symmetric equilibrium ρ ≡ 0 is unstable for all G > G * .
Additionally to Theorem 2.2 we have: whereby u 0 is the solution of (4.13) for n = 0. Given G > 0, there exists a positive integer l G ∈ N such that for all ω ∈ (0, µ 0 ) and l ≥ l G we find positive constants K l > 0 and δ l > 0 with the property that if ρ 0 C 4+α (S) ≤ δ l and ρ 0 is 2π/l−periodic, then the solution ρ to (2.1) exists in the large, and Moreover, the solution ρ is 2π/l−periodic for all t ≥ 0.
We will show in the Appendix that the condition (2.2) is satisfied particularly when f = id [0,∞) and R A = 1.

The well-posedness result
This section is dedicated to the proof of Theorem 2.1 and preparing Theorems 2.2 and 2.3. A fundamental difficulty in treating problem (2.1) is the fact that one has to work with unknown, variable domains Ω ρ . We overcome this difficulty by transforming problem (2.1) on the unitary disc Ω := D(0, 1). Therefore, we define for all ρ ∈ V the mapping Θ ρ : where the cut-off function ϕ ∈ C ∞ (R, [0, 1]) satisfies is strictly increasing and therefore bijective. The composition ρ(·/| · |) has the same regularity properties as ρ on any subset of R 2 which is bounded away from 0, and using the chain rule we have, cf. [13], that . Such a diffeomorphism was first introduced by Hanzawa in [19] to study the Stefan problem, and it is therefore called Hanzawa diffeomorphism. Additionally, we have that Θ ρ (S) = Γ ρ (see Figure 2). The push-forward operator induced by Θ ρ is defined by These operators allow us to transform the problem into an abstract Cauchy problem over S. General results of the theory of maximal regularity, due to Sinestrari [23], can be used to prove existence of a unique classical solution, . .
The transformed operators A(ρ) and B, are defined as follows. Given ρ ∈ V, A(ρ) : buc 2+α (Ω) → buc α (Ω) is the differential operator given by The operator A(ρ) is linear and uniformly elliptic, with Using (3.1) and the chain rule, we can determine the coefficients b ij (ρ) and b i (ρ), 1 ≤ i, j ≤ 2, explicitly in terms of ρ and ϕ, the cut-off function used when defining Θ ρ . Moreover, A depends analytically (3.4) with tr the trace operator on S, i.e. tr v = v| S for v ∈ BUC (Ω), and the curvature κ Γρ can be expressed in terms of ρ by the relation .
in Ω, The notion of solution for this problem is defined analogously to that of solution to (2.1). In fact the problems (2.1) and (3.5) are equivalent in the following sense: Proof. The analyticity is obvious. In order to compute the derivative one has only to calculate the gradient of a real valued function of three variables.
We introduce now solution operators to some semilinear, respectively linear Dirichlet problems related to our transformed problem (3.5). From the Leray-Schauder fixed point theorem (cf. [17,Theorem 11.3]) we obtain for each ρ ∈ V a solution u ∈ BUC 2+α (Ω ρ ) of problem with ρ ∈ V. Using the maximum principle as we did in the proof of [12, Theorem 2.6] we may prove the uniqueness of this solution. Consequently, we have: The mapping V ∋ ρ → T (ρ) ∈ buc 2+α (Ω) is smooth.
Proof. For details we refer to the proof of [22,Theorem 4.3.5].
We consider now the solution operator corresponding to the second, linear Dirichlet problem in system (3.5). We state: Proof. Given ρ ∈ V, the mapping It is well-known that the function mapping a bijective bounded linear operator onto its inverse is analytical; it can be expressed by a Neumann expansion in the neighbourhood of some other linear isomorphism. Hence, in view of Lemma 3.2 and equation (3.3) it follows that is analytic. Since S maps smooth functions on S into BUC ∞ (Ω) we also have S(ρ) ∈ buc 2+α (Ω) for all ρ ∈ V.
3.1. The nonlinear Cauchy problem. We use now the solution operators defined in Lemmas 3.3 and 3.4 to transform the system (3.5) into an abstract Cauchy problem on the unit circle S. We put in the third equation of (3.5) T (ρ), the solution to (3.7), for v, respectively S(ρ), the solution to (3.8), for q, to obtain the following abstract Cauchy problem where Φ( · ) := B( · , T ( · ), S( · )) (3.10) is a nonlinear and nonlocal operator of third order which depends smoothly on ρ. In order to prove Theorem 2.1 is suffices to show that ∂Φ(0) generates a strongly continuous analytic semigroup in . The operator ∂Φ(0) can be decomposed as the sum of its principal part, which has order three in ρ, with an operator of first order. More exactly:
In order to prove this theorem, we have to study first the regularity properties of the operator B, defined by (3.4). It is convenient to we write this operator as a sum where B 1 ∈ C ω (V, L(buc 2+α (Ω), h 1+α (S)) and B 2 ∈ C ω (V, h 1+α (S)) are the operators defined by Since by the chain rule for ρ ∈ h 4+α (S), we must not only show that B i , i = 1, 2, have the regularity mentioned above but also determine their derivatives in 0. We shall see that the first term of this sum is the important one (corresponding to the operator A 1 in Theorem 3.5) since it is a third order operator, and the last two terms are of lower order and play, as we shall see, no role when studying the well-posedness of the abstract evolution equation (3.9). Using relation (3.1) we get that for all ρ ∈ V. Particularly, we obtain find the following expression for B 2 wherefrom we can easily see that B 2 is analytic and that (3.14) Consider now the operator B 1 . From the weak maximum principle we find that the function T (0) is radially symmetric, and one can easily see that S(0) is constant. Hence it suffices to determine ∂B 1 (0)[ρ]v 0 for a radially symmetric function v 0 ∈ buc 2+α (Ω) and ρ ∈ h 4+α (S). We state: Lemma 3.6. The nonlinear operator B 1 is analytic, i.e.
Given v 0 ∈ buc 2+α (Ω) a radially symmetric function, we have that Proof. Let v 0 ∈ buc 2+α (Ω) be a radially symmetric function and ρ ∈ V. In view of (3.12) we have that wherefrom we obtain the regularity assumption stated in the lemma. Concering (3.15) a detailed proof can be found in [22,Lemma 4.4.2].
To finish the preparations for the proof of Theorem 3.5 one more step must be done. We have to determine the Fréchet derivative in 0 of the analytic solution operator defined in Lemma 3.4. in Ω, Proof. The proof is standard and we omit it.
We come now to the proof of the main result of this subsection: Proof of Theorem 3.5. The regularity assumption follows directly from Lemmas 3.3, 3.4, 3.6, and relation (3.13). Moreover since B 1 (0) = R −1 ∂ ν , we have that The operator T defined in Lemma 3.3 can be extended to T : {ρ ∈ h 2+α (S) : ρ C(S) < 1/4} → buc 2+α (Ω), because we only need there that ρ is of class C 2+α to guarantee existence of a solution to (3.6). Whence, the operator defined by . This completes the proof.
We conclude this section with the proof of our first main result Theorem 2.1.
Proof of the Theorem 2.1. The key point is showing that the operator A 1 , which can be seen as the principal part of the Fréchet derivative ∂Φ(0) generates a strongly continuous and analytic semigroup in L(h 1+α (S)) for all 0 < α < 1, i.e. −A 1 ∈ H(h 4+α (S), h 1+α (S)). To this scope we show that −A 1 is a Fourier multiplier. Given ρ ∈ h 4+α (S), we consider its the Fourier expansion of ρ = k∈Z ρ(k)x k , where ρ(k) := S ρ(x)x k dx is the k-th Fourier coefficient of ρ. The well-known Poisson integral formula yields then (∆, tr) −1 (0, ρ ′′ ) = k∈Z −k 2 r |k| ρ(k)x k for all r ≤ 1 and x ∈ S. Taking the derivative with respect to r, in r = 1, we finally obtain for all k∈Z ρ(k)x k ∈ h 4+α (S), where or simplicity Following the same steps as in the proof of [13, Theorem 3.5] we obtain that −A 1 ∈ H(h 4+α (S), h 1+α (S)) for all α ∈ (0, 1).

Stability properties
We study in this section the stability properties of the unique radially symmetric solution D(0, R A ) determined in [12] when A ∈ (0, f (1)) and G = 0. Therefore, we choose the constant R fixed at the beginning of Section 2 to be R = R A . Particularly, functions in V parametrise domains near the stationary tumor D(0, R A ). We rediscover ρ ≡ 0, situation when the tumor domain is the discus D(0, R A ), as the unique radially symmetric stationary solution of (2.1).
In order to study the stability properties of this equilibrium we have to determine the spectrum of the complexification of the Fréchet derivative ∂Φ(0), which we denote again by ∂Φ(0). The stability results established in Theorem 2.2 and Theorem 2.3 are then obtained by applying the principle of linearised stability to problem (3.9). Repeating the arguments presented in the proofs of Theorem 2.1 we see that the complexification of ∂Φ(0) generates a strongly continuous and analytic semigroup. Taking into consideration that the embedding h 4+α (S, C) ֒→ h 1+α (S, C) is compact, we deduce that the complexification of ∂Φ(0) has a compact resolvent. From [20, Theorem III. 8.29], we conclude that its spectrum consists only of eigenvalues of finite multiplicity, σ(∂Φ(0)) = σ p (∂Φ(0)).
Given ρ ∈ h 4+α (S), we look in the following for the Fourier expansion of ∂Φ(0) [ρ]. Having shown that ∂Φ(0) is a Fourier multiplier, the point spectrum of ∂Φ(0) is given by the symbol of this multiplier. The cornerstone of the analysis leading to Theorems 2.2 and 2.3 is the Theorem 4.1, which states that ∂Φ(0) is a Fourier multiplier operator with symbol explicitly determined.
Theorem 4.1. Given ρ ∈ h 4+α (S), we let ρ = k∈Z ρ(k)x k denote its associated Fourier series. We have that where the symbol (µ k ) k∈Z is given by the relation
where v 0 = T (0) and ϕ the cut-off function used to define the Hanzawa diffeomorphism.
Proof. The proof is standard though lengthy. For detailed calculations we refer to [22,Lemma 5.1.1].
The result of this lemma is not very useful yet. This is due to the fact that the first equation of (4.4) contains besides ρ and w also derivatives of ϕ and v 0 . Therefore, it is difficult to determine the Fourier expansion of w, the solution of (4.4), when knowing that of ρ. That is why we formally linearise the free boundary problem describing the stationary states of the full system (1.1) at the unique radially symmetric solution (0, ψ A , p A ), found in [12, Theorem 1.1], where we simply write ψ A := T (0) • Θ 0 for the solution of (3.6) and p A : By doing this we shall find a nice decomposition of the derivative ∂T (0) as a sum of two operators (see Lemma 4.3 below). Hence, we consider now perturbations of the radially symmetric solution of the form where we simply write ρ(s) = ρ(e is ) for all s ∈ R. Here, ε is a small parameter, and ψ, p, ρ are unknown functions. The linearisation of problem (1.1) is then the following free boundary problem (4.5) We look now for a connection between the linearisation (4.5) and the Fréchet derivative of Φ in 0. To this scope, we transform first the system (4.5) to the unitary disc, i.e. we set for x ∈ Ω, and substitute these expressions in (4.5). This leads us to the following system of equations Given ρ ∈ h 4+α (S), we let W(ρ) ∈ buc 2+α (Ω) denote the solution to the linear Dirichlet problem Further on, we want to determine a relation between ∂T (0)[ρ] and W(ρ). Therefore, we define the extension operator E : h 4+α (S) → buc 2+α (Ω) by for x ∈ Ω. Using these operators we can now write ∂T (0) as the sum of W and E. This decomposition is very useful because we got rid, in this way, of all the terms from the right hand side of first equation of (4.4).
Recall that our goal is to determine the Fourier expansion of ∂ ν (∂T (0)[ρ]) when ρ ∈ h 4+α (S). It turns out, cf. Lemma 4.3, that ∂ ν (E(ρ)) is collinear with ρ for all ρ ∈ h 4+α (S). Moreover, using ODE-techniques we are able to determine an expansion for W(ρ) for all ρ ∈ h 4+α (S), cf. (4.15). Indeed, we have: where α A := ∂ 2 U/∂r 2 (1, R 2 A ) > 0 and U is the solution of the parameterdependent problem: Proof. The proof follows by direct computation by taking into consideration . In virtue of Lemma 4.3, if we determine a Fourier expansion for W(ρ), then we completed the task of determining the expansion of ∂T (0)[ρ] for all ρ ∈ h 4+α (S). For this reasoning, we consider expansions of the form with r ∈ [0, 1] and x ∈ S. Substituting these expressions into (4.7), and comparing the coefficients of x k , we come to the following problem for the unknown function A k : (4.12) We have used here the relation ∂ ν v 0 = v ′ 0 (1) on S, where we identified v 0 with its restriction to the interval [0, 1]. In order to prove the existence and uniqueness of the solution to (4.12) we consider first the associated problem (4.13). The solution of (4.12) will be then expressed in terms of the solution of this new system. Thus, given n ∈ N, the problem has a unique solution u n ∈ C ∞ ([0, 1]).
With this notation we have: Lemma 4.4. Given k ∈ Z, problem (4.12) possesses a unique solution A k ∈ C ∞ ([0, 1]) explicitly given by (4.14) We denoted by u n , n ∈ N, the solution of (4.13).
Proof. That A k is a solution of (4.12) follows by direct computation. The uniqueness may be obtain by a contradiction argument.
We give now a short proof of the theorem stated at the beginning of this section.
Proof of Theorem 4.1. From Lemma 4.4 we obtain the following expansion for W(ρ) for all ρ ∈ h 4+α (S). Let us determine now the constant α A form Lemma 4.3. From the first equation of (4.10) we find that the constant α A satisfies the relation Moreover, from the same equation and relation (3.27) in [12] we have that hence In view of Theorem 3.5, 3.17, and (4.15) we conclude (4.1) and the proof is complete.

4.1.
Estimates for the symbol of the derivative of Φ. In order to study the stability properties of the unique radially symmetric equilibrium determined in [12] we need to study the sign of symbol (µ k ) k∈Z given by (4.2) in dependence of the parameters (A, G) ∈ (0, f (1)) × (0, ∞). We consider in here just the case when G > 0, since for G < 0 we already established in [12, Theorem 1.2. (d)] that this circular equilibrium is unstable. It is worth noticing that µ k = µ −k for all k ∈ Z, so that we consider just the terms µ k with k ∈ N. At first we have to as certain that 0 is in the spectrum of ∂Φ(0) for all (A, G) ∈ (0, f (1)) × [0, ∞).
Proposition 4.5. We have that Differentiating this equation with respect to r and setting w 0 := v ′ 0 we get Given r ∈ [0, 1], define w(r) = v ′ 0 (1)ru 1 (r)/u 1 (1), where u 1 denotes the solution of the problem (4.13) when n = 1. With c := v ′ 0 (1)/u 1 (1), we obtain by differentiation that w ′ (r) = c(u 1 (r) + ru ′ 1 (r)), w ′′ (r) = c(ru ′′ 1 (r) + 2u ′ 1 (r)), which in turn implies that w is a solution of the equation Moreover, for r = 1, we get w(1) = v ′ 0 (1). Since the solution of (4.18) is unique, we get If we choose r = 1 this last relation leads to In view of relations (4.16) and (4.17), and the definition of α A , we have Inserting this result in the expression of µ 1 yields Proposition 4.5 reveals that µ 1 = 0 belongs to the spectrum of ∂Φ(0). However, we show now that the sequence µ k → k→∞ −∞. This is obviously true if the sequence (u ′ n (1)/u n (1)) n∈N is bounded. Even more, we have: Lemma 4.6. It holds that Proof. Let n ∈ N and set v := u n+1 − u n ∈ C ∞ ([0, 1]). Recall that u n is an increasing function for all n ∈ N. This can be seen since relation which is obtained by multiplying the first equation of problem (4.13) with r 2n+1 and then integrating twice. From (4.13) we obtain which implies by (1.2) that v ′ (t) < 0 for t ∈ (0, δ) and some δ < 1. Thus, v is decreasing on (0, δ). Let now t ∈ [0, 1] and m t := min [0,t] v ≤ 0. A maximum principle argument shows that the non-positive minimum must be achieved at t, m t = v(t) which implies u n+1 (t) ≤ u n (t) for all t ∈ [0, 1]. Particularly, u n+1 (1) ≤ u n (1). It also holds that it follows that The dominated convergence theorem implies u ′ n (1) ց 0. A similar argument provides u n (1) ց 1. Therefore, u ′ n (1)/u n (1) → 0, and we are done. gives a false impression about the stability properties of the radially symmetric equilibrium. If k ∈ N is large enough, we find in view of the Lemma 4.6 a unique value G k of the parameter G such that µ k = 0 iff G = G k . More precisely, given k ∈ N such that we set . (4.20) To prove the statement of Theorem 2.2, we set Moreover, since G k → k→∞ ∞, the minimum must be achieved, i.e. we find at least an integer k 0 ∈ N such that and G * = G k 0 . With these preparations we are able to prove the instability result.
Proof of Theorem 2.2. Let G > G * be given and k 0 be an integer such that G * = G k 0 and (4.22) holds true. It follows that which implies µ k 0 > 0. We are left to check the following instability assumptions σ + (∂Φ(0)) = σ(∂Φ(0)) ∩ {λ ∈ C : Reλ > 0} = ∅, where ∂Φ(0) stands here for the complexification of ∂Φ(0). The first one is clear since µ k 0 > 0. Moreover, we infer from Lemma 4.6 that µ k → k→∞ −∞, and therefore the unstable spectrum σ + (∂Φ(0)) contains finitely many positive eigenvalues of the Fréchet derivative ∂Φ(0). Thus, we found out that the assumptions of [ 1) which correspond to initial data ρ 0 closed to the unique radially symmetric equilibrium and 2π/lperiodic. The positive integer l depends on the constant G, which is chosen to be positive, so that D(0, R A ) is the unique equilibrium of the problem (1.1). The main result of this section, Theorem 2.3 requires the following assumption A 2 which means that the eigenvalue µ 0 = µ 0 (G) is negative for all G > 0. This assumption is satisfied for example if R A = 1 and f = id [0,∞) (see Appendix).
In order to prove Theorem 2.3 we introduce first appropriate subspaces of the small Hölder spaces. Given k ∈ N and l ∈ N, n ≥ 2, we define the subspace of h k+α (S) consisting of 2π/l-periodic functions by where ρ(kl) is the kl−th Fourier coefficient of ρ. Our first objective is to prove that if ρ ∈ V l , then Φ(ρ) ∈ h 1+α l (S), that is (4.23) Having shown (4.23), by choosing l large enough we can exclude the eigenvalues µ k with k small from the spectrum of ∂Φ(0). These are the eigenvalues which we could not estimate whether they are negative or not. In this way we also eliminate µ 1 = 0 from the spectrum. Let ρ ∈ V l be given and let ψ : where v is the solution of (3.7). We prove that ψ satisfies ψ(x) = ψ(e 2πi/l x) for all x ∈ Ω ρ . Therefore, we must prove first that, if x ∈ Ω ρ , then xe 2πi/l belongs to Ω ρ . Indeed, given x ∈ Ω ρ , we have that |xe 2πi/l | = |x| < R 1 + ρ x |x| = R 1 + ρ xe 2πi/l |xe 2πi/l | , which implies that xe 2πi/l ∈ Ω ρ . Recall that the function ψ ∈ buc 2+α (Ω ρ ) is the unique solution of the Dirichlet problem ∆ψ = f (ψ) in Ω ρ , ψ = 1 on Γ ρ . for all x ∈ S. Summarising, B 1 (ρ, T (ρ)) xe 2πi/l = B 1 (ρ, T (ρ)) (x) for all x ∈ S, and the proof is completed.
The Fréchet derivative ∂Φ(0) of the mapping Φ ∈ C ∞ (V l , h 1+α l (S)) is, in view of (4.1), given by the relation where (µ kl ) k are defined by (4.2). We come now to the proof of the exponential stability result for 2π/l− periodic data: Proof of Theorem 2.3. Let G > 0 be given. Since µ |k| → |k|→∞ −∞, we find a positive integer l G such that µ |k| ≤ µ 0 for all |k| ≥ l G . Let l ≥ l G be fixed. In view of relation Lemma 4.7, we find that the restriction Φ ∈ C ∞ (V l , h 1+α l (S)) satisfies the assumptions of [21, Theorem 9.1.2]. Indeed, since l ≥ l G , it holds that µ kl ≤ µ 0 for all k ∈ N. Consequently, the spectrum of the complexification of the ∂Φ(0) consists only of the negative eigenvalues {µ kl : k ∈ N}, and is bounded away from the positive half plane by µ 0 . The assertion follows now immediately from [21, Theorem 9.1.1].

Appendix
We show now that the condition (2.2), meaning that µ 0 (G) < 0 for all G > 0, is not to restrictive.
Proof. In view of Proposition 4.5, our assertion is equivalent with Consequently, we have to show only that We assume now that u 0 , the solution of (4.13) when n = 0, is analytic and the Taylor series associated to u 0 in 0 a k x k , converges on [0, 1]. Problem (4.13) writes now as follows      xu ′′ 0 + u ′ 0 − xu 0 = 0, 0 ≤ x ≤ 1, u 0 (0) = 1, u ′ 0 (0) = 0. From the initial conditions of (4.13) it follows immediately that a 0 = u 0 (0) = 1 and a 1 = u ′ 0 (0) = 0. Plugging u 0 and its derivatives in the first equation of the system, one finds out that We make now the same assumption on u 1 , the solution of (4.13) when n = 1. We then get, that u 1 is the solution of the following system      xu ′′ 1 + 3u ′ 1 − xu 1 = 0, 0 ≤ x ≤ 1, u 1 (0) = 1, u ′ 1 (0) = 0. As above, we obtain ∞ k=1 k(k + 1)a k+1 x k + 3 , ∀k ∈ N. It is worth noticing that the Taylor series associated to u 0 and u 1 , respectively, in 0 define analytic functions on the whole real line, so that the representations (5.2) and (5.3) are valid.
With three exact decimals we have that which leads to the desired conclusion.