Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions

In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and non- homogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.


Introduction
In this paper we consider the one dimensional semilinear hyperbolic-parabolic system on a finite planar network, where λ, β, D, b > 0 and a ≥ 0 .
The system has been proposed as a model for chemosensitive movements of bacteria or cells; the unknown u stands for the cells concentration, λv denotes their average flux and ψ is the chemo-attractant concentration produced by the cells themselves; the individuals move at a constant velocity, whose modulus is λ, towards the right or left along the axis; β is the friction coefficient while D, a, b are respectively the diffusion coefficient, the production rate and the degradation one for the chemoattractant .
Systems like (1.1) are adaptations of the so-called Cattaneo equation to the chemotactic case, introducing the nonlinear term uψ x in the equation for the flux [20,8], and their solutions have been studied in [14,15,10]; they are included among hyperbolic models which have been recently introduced in contrast to the parabolic ones considered before, since they give rise to a finite speed of propagation and allow better observation of the phenomena during the initial phase.
In recent years, one dimensional models on networks have been developed in order to describe particular chemotactic phenomena like the process of dermal wound healing and the behavior of the slime mold Physarum polycephalum as a model for amoeboid movements. Actually, during the healing process, the stem cells in charge of the possible stationary solution. Here a fondamental role is played by a suitable condition stated for the transmission coefficients, which allows to express the jumps of the densitiy u at each inner node as linear combinations of the values of the fluxes at the same node. This fact and assumptions on the data provide a control of the evolution in time of the L ∞ -norm of the density which permits to remove some conditions on the parameters a i and b i considered in [11,9]. When the boundary data for the fluxes λ i v i are constant functions, the hypothesys necessary to prove the a priori estimates imply that the sum of the fluxes incoming in the network have to equal the sum of the outgoing ones and that the initial mass of cells has to equal the mass of the stationary solution.
If a stationary solution (U (x), V (x), Ψ(x)) exists and the quantities U ∞ and Ψ x ∞ are small, the a priori estimates provide a bound, uniform in time, for a norm of the solutions having small perturbations of the stationary one as initial and boundary data; in this way, after the proof of real existence of stationary solutions, we would obtain the existence of global solutions for a class of initial and boundary data and would identify the stationary solutions as the asymptotic profiles for such class of solutions. For this reason we devote part of this paper to study the existence of stationary solutions. In the cases of acyclic networks we prove two results, under different smallness conditions on the boundary data and on the total mass; in particular we give conditions which ensure the existence of a stationary solution with nonnegative density U . For general networks we exhibit some stationary solutions in very particular cases for the parameters of the problem.
The paper is organized as follows. In Section 2 we give the statement of the problem and, in particular, we introduce the transmission conditions and the assumption on the data, while in Section 3 we prove the local existence result. Section 4 is devoted to the a priori estimates and to the consequent global existence and asymptotic behaviour results, under the assumption that a small stationary solution exists. In Section 5 we prove the results of existence of stationary solutions in the case of acyclic networks. Finally, in Section 6 we present the global existence and asymptotic behaviour results under assumptions which ensure the real existence of stationary solutions.

Statement of the problem
We consider a planar finite connected graph G = (Z, A) composed by a set Z of n nodes (or vertexes) and a set A of m oriented arcs, A = {I i : i ∈ M = {1, 2, ..., m}}. Each node is a point of the plane and each oriented arc I i is an oriented segment joining two nodes.
We use e j , j ∈ J , to indicate the external vertexes of the graph, i.e. the vertexes belonging to only one arc, and by I i(j) the external arc incident with e j . Moreover, we denote by N ν , ν ∈ N , the internal nodes; for each of them we consider the set of incoming arcs A ν in = {I i : i ∈ I ν } and the set of the outgoing In this paper, a path in the graph is a sequence of arcs, two by two adjacent, without taking into account orientations. Moreover, we call acyclic a graph which does not contains cycles, i.e. for each couple of nodes there exists a unique path connecting them, whose arcs are covered only one time.
Each arc I i is considered as a one dimensional interval (0, L i ). A function f defined on A is a m-tuple of functions f i , i ∈ M, each one defined on I i . The expression f i (N ν ) means f i (0) if N ν is the starting point of the arc I i and f i (L i ) if N ν is the endpoint, and similarly for f (e j ).
We consider the evolution of the following problem on the graph G In order to set boundary and transmission conditions, we introduce the following parameters: The boundary conditions for v, at each outer point e j , are while for ψ we set the Robin boundary conditions In addition, at each internal node N ν we impose the following transmission conditions for the unknown ψ ji for all i, j ∈ M ν , and the following ones for the unknowns v and u Motivations for the above constraints on the coefficients in the transmission conditions can be found in [11] . These kind of transmission conditions were introduced in [12] in a parabolic model for the description of passive transport through biological membranes and they are known as Kedem-Katchalsky permeability conditions. Finally, we impose the following compatibility conditions First we are going to prove that the problem (2.1)-(2.7) has a unique local solution On the other hand, the proofs of the existence of global solutions and of the existence of stationary solutions on acyclic graphs, carried out in the last sections, require the following further conditions on the transmission coefficients (2.8) for all ν ∈ N , for some k ∈ M ν , σ ν ik = 0 for all i ∈ M ν , i = k , in addition to suitable smallness and smoothness assumptions on the data. Finally, we remark that the transmission conditions (2.5) imply the continuity of the flux of ψ at each node, for all t > 0, and the conditions (2.6) ensure the conservation of the flux of the density of cells at each node N ν , for t > 0, which corresponds to the following condition for the evolution in time of the total mass i∈M Ii

Local solutions
In order to prove the existence and the uniqueness of a local solution to problem (2.1)-(2.7) we need to introduce some auxiliary functions.
We introduce the functions V(x, t) and Φ(x, t), defined on the network as follows where η j is defined in the previous section. Let the triple (u, v, ψ) be a solution to (2.1)-(2.7) and let the triple (u, w, φ) satisfies the following system and transmission conditions where α ν ij and σ ν ij are as in (2.5) and (2.6). We are going to prove an existence and uniqueness result for local solutions to the above problem; as a consequence we will obtain the result for problem (2.1)-(2.7).
Let X := (L 2 (A)) 2 and Y := (H 1 (A)) 2 ; we consider the unbounded operator A 1 : D(A 1 ) → X: Proof. The proof for the operator A 1 can be achieved as in [11] (see the proof of Proposition 4.2), taking into account that the transmission conditions imply For the operator A 2 we notice that the transmission conditions (3.6) imply that (3.10) while the homogeneous Robin boundary conditions provide the equality Then by standard methods, A 2 reveals to be a dissipative operator [4]. In order to prove that the operator in m-dissipative, we introduce the bilinear form a(φ, χ) : the form is continuous and coercive, hence, by the Lax-Milgram theorem, we know that, for each ϕ ∈ L 2 (A), there exists a unique φ ∈ H 1 (A) such that, for all taking χ i ∈ H 1 0 (I i ) for all i ∈ M, we obtain that φ ix ∈ H 1 (I i ), then taking χ i ∈ C ∞ 0 (I i ), as in [11], we prove the equality moreover, thanks to suitable choices of χ i (N ), χ i (a i ), we obtain that φ satisfies the right boundary and transmission conditions to belong to D(A 2 ).
Thanks to the above proposition we conclude that the operator A 1 is the generator of a contraction semigroup T 1 in (L 2 (A)) 2 while the operator A 2 is the generator of a contraction semigroup T 2 in L 2 (A).
Then there exist two positive constants L 1K , L 2K , depending on K , such that where β := max{β i } i∈M ; then the first inequality in the claim follows with L 1K = c S K + sup As regard to the second inequality we have where c S are Sobolev constants; then, setting L 2K = c S (K + c Φ ) + β, where c Φ is a suitable positive quantity depending on Φ, we obtain the second inequality.
Proof. We set a := max We consider the problem (3.2)-(3.7) and we set U 0 := (u 0 , w 0 ) and Let L 1K , L 2K be the constants in Lemma 3.1 and let T ≤ T .
Let consider the set (3.13) we equip B MK with the metric generated by the norms of the involved spaces, obtaining a complete metric space .
We define a map G on B MK in the following way: given where au I (t) := {a i u I i (t)} i∈M , and U is the solution to where we used the notation (3.11). First we prove that G is well defined and ) we can use the theory for nonhomogeneous problems in [4] and we infer the existence and uniqueness of a solution φ to problem (3.14) given by [4]) and Then we have Since the first and the second terms on the right hand side are non positive and u I , φ ∈ C 1 ([0, T ]; L 2 (A)) and Z 3 ∈ H 1 (0, T ); L 2 (A)), the above inequality implies that φ ∈ H 1 ((0, T ); H 1 (A)); moreover, for h → 0 and then t 1 → 0, t 2 → T , we have , so that φ satisfies the last condition in (3.13). Now we consider the problem (3.15) and we set F (t) := F φU I (t). We know that Z ∈ W 1,1 ((0, T ); X) and, from Lemma 3.1, F ∈ W 1,1 ((0, T ); X), then there exists a unique solution U ∈ given by [4]; moreover, using Lemma 3.1 and choosing T ≤ (2L 1K ) −1 , we obtain the following inequality then we can argue as we did before for φ, using [4] and Lemma 3.1, to obtain which implies, choosing T sufficiently small and setting λ := min Finally, using Lemma 3.1, Now we are going to prove that G is a contraction mapping on B MK , for small values of T . Let Then we have, arguing as for the previous estimates for φ, Then, using , Lemma 3.1, where c S is a Sobolev constant; moreover, arguing as in (3.17) and using (3.12) Hence, if T is sufficiently small, then G is a contraction mapping in B MK and the unique fixed point (U, φ) ∈ B MK is the solution to problem (3.2)-(3.7). The existence and uniqueness of local solutions for problem (2.1)-(2.7) follow from (3.1).

A priori estimates
In this section we assume the condition (2.8) . Moreover,we assume that there exists a stationary solution (U (x), V (x), Ψ(x)) to problem (2.1),( 2.5), (2.6), verifying the boundary conditions we notice that, integrating the first equation in (2.1) and using the conservation of the flux (2.10) at each inner node, it turns out to be necessary that j∈J W j = 0 .
We set Due to the assumption of existence of the stationary solution (U (x), V (x), Ψ(x)), the triple is the local solution of the following problem complemented with the transmission conditions (2.6) and (2.5). We set We are going to prove some a priori estimates for the solution to the above problem, assuming suitable conditions on the data. If the stationary solution and the data in (4.3) are small in some suitable norms, then these estimates provide a global existence result for problem (4.3). In this way, after the proof of the real existence of stationary solutions in the next section, the results in the following propositions will be the tools to prove the existence of global solution to problem (2.1)-(2.7). We assume the following relations among the initial and boundary data of (u, v, ψ) and the stationary solution: +∞)) , for all j ∈ J , (4.6) P j ∈ H 2 ((0, T )) for all T > 0 , P j − P j ∈ H 1 ((0, +∞)) , for all j ∈ J , First we remark that the assumption (2.8) implies that the condition (2.6) can be rewritten as follows for suitable γ ν ij (see the proof of Lemma 5.9 in [11]). This equality allow to prove the following estimate for u i (·, t) ∞ .
Set |A| := i∈M L i .
Proof. We consider two consecutive nodes, N ν and N h , and let I l be the arc linking them. For all x ∈ I l , t ∈ [0, T ] (by N ν we mean 0 if N ν is the starting node of I l and we mean L l otherwise); let k ν , k h the indexes relative to the nodes, N ν and N h in condition (2.8), then, using (4.8), we can write for all t ∈ [0, T ], Since each node of the network is connected with the node N 1 , the above relation implies that, for all p ∈ N , we can express the value of u kp (N p , t) in the following way where k p and k 1 are the indexes in condition (2.8) relative to N p and N 1 respectively, and For all i ∈ M 1 , thanks to condition (2.8)), we have, for all and, thanks to the previous computations, a similar expression can be derived for all i ∈ M p , for all p ∈ N : Obviolusly, each u i has not a unique expression; in all cases, for all i ∈ M we can write where N q is one of the extreme points of I i and Γ i (t) is a suitable quantity verifying Integrating on I i the equality (4.10), we obtain whence, summing for i ∈ M and using (4.11), we infer that, for all t, Now we use this inequality in (4.10) to obtain the claim. Now we are going to obtain a priori estimates necessary to prove the uniform (in time) boundedness of some norms of (u, v, ψ) when the data are small. Similar results are proved in [11] in the case of homogeneous boundary conditions, when ai bi does not change with i, and some of the proofs have minor differences from the ones in that paper .
where c S are Sobolev constants and c 1 is a suitable constant. depending on Sobolev's constants, on L i and on the quantity γ in Proposition 4.1 .
Proof. We multiply the first equation in (4.3) by u i , the second one by v i and we sum them; after summing up for i ∈ M, we obtain the claim, taking into account that from Proposition 4.1 we have where c is a suitable constant depending on L i , γ and Sobolev constants, and that the transmission conditions (2.6) imply condition (3.9) holding for u and v, hence the sum of the terms at nodes is non positive .
where c S are Sobolev constants.
Summing the above two equations and integrating over I i ×(δ, τ ), for 0 < δ < τ < T , |h| ≤ min{δ, T − τ } we obtain Using condition (3.9) and the boundary conditions we can compute so that, after dividing the equalities (4.12) by h 2 , summming them for i ∈ M and letting first h and then δ go to zero, we obtain the claim.
where c S depends on Sobolev constants .
Proof. We multiply the second equation in (4.3) by u ix , we integrate over I i and we sum for i ∈ M; using the Cauchy-Schwartz inequality we obtain the claim.
where c S depends on Sobolev constants and c 2 is a constant depending on Sobolev constants, on L i and on γ .
Proof. We multiply the second equation (4.3) by u ix , we integrate over I i × (0, T ) and we sum for i ∈ M; using the Cauchy-Schwartz inequality and Proposition 4.1, we obtain the claim.
where c S are Sobolev constants, and c 3 , c 4 , c 5 are positive constants depending on λ i , β i , σ ij , and on Sobolev constants.
Proof. Using the same notations as in the proof of Proposition 4.3, by the second equation in (4.3) we obtain, for 0 < δ < τ < T , |h| ≤ min{δ, T − τ }, τ δ Ii Using the boundary conditions in (4.3) and (2.6) we can write (4.13) In order to treat the terms at the inner nodes, as in [11], we set H(t) = u j (N, t) − u i (N, t), and we have As regard to the terms at the boundary nodes, we argue as in the proof of Proposition 4.3. Then we obtain the claim letting h and then δ go to zero and τ go to T in (4.13).
where c 6 depends on D i , b i , a i and c S depends on Sobolev constants.
Proof. From the third equation in (4.3), using (3.10), and the boundary conditions in (4.3), we obtain, for 0 < δ < τ < T i∈M Ii as in the previous proof we can conclude dividing by h 2 , letting h go to zero and then δ go to zero and τ go to T .
where c 7 , c 8 , c 9 are positive constants depending on γ, L i , a i , b i , D i and Sobolev constants.
Proof. The first inequality can be achievd multiplying the third equation in (4.3) by Di bi ψ ixx , integrating on I i , summing for i ∈ M and using the Cauchy-Schwartz inequality and (3.10). Integrating over I i × (0, T ) and using Proposition 4.1 we obtain the second inequality .

Now we introduce the functional
The a priori estimates in the previous propositions allows to prove the following theorem.

then, if the quantities
and, for all i ∈ M, lim t→+∞ i∈M Proof. Using the estimates proved in Propositions 4.1-4.8, it is easy to prove the following inequality (4.14) and δ is any positive quantity. If we choose U ∞ , Ψ x ∞ , δ in such a way that C 2 < 1, and we choose C 0 , then the inequality (4.14) implies that F T (u, v, ψ) remains uniformly bounded for all T > 0; then the solution is globally defined. Moreover the set {u(t), v(t), ψ(t)} t∈[0,+∞) is uniformly bounded in (H 1 (A)) 2 × H 2 (A); thus, if we call E s the set of accumulation points of {u(t), v(t), ψ(t)} t≥s in (C(A)) 2 × C 1 (A), then E s is not empty and E := ∩ s≥0 E s = ∅. Letv(x) be such that, for a sequence t n → +∞, If we set ω i (t) := v i (t, ·) L 2 (Ii) then the estimates obtained for the functions v i imply that ω i ∈ H 1 ((0, +∞)) and, as a consequence, lim t→+∞ ω i (t) = 0. As lim n→+∞ v i (·, t n ) 2 = v i (·) 2 , we obtain v 2 = 0. The same argument can be applied to the functions u i and ψ i .

Stationary solutions on acyclic networks
In this section we study the real existence of stationary solutions to problem (2.1)-(2.8). Concening the uniqueness, we can notice that the results of the previous section imply that two stationary solutions with the same mass and the same boundary data , which are small in H 1 × H 1 × H 2 norm, have to coincide.
In this section we restrict our attenction to acyclic graphs and we approach the study of existence of stationary solutions (U (x), V (x), Ψ(x)) with mass and boundary data assuming conditions (2.8) and some suitable smallness conditions on |µ s |, |W j | and |P j | . Of course, for all i ∈ M, V i (x) is a constant function, V i (x) = V i ; moreover, we recall that a set of boundary data {W j } j∈J is compatible with the transmission conditions only if j∈J W j = 0 (see previous section). These facts holds true for general networks.
In the case of acyclic network, a set of admissible boundary values {W j } j∈J determines univokely the costant value of each function V i on the internal arc I i .
Actually, let consider an internal arc I ι and its starting node N η and the sets (5.3) Q = {ν ∈ N : N ν is linked to N η by a path not covering I ι } , J ′ = {j ∈ J : e j is linked to N η by a path not covering I ι } ; at each inner node the conservation of the flux (2.10) holds, then Since V i (x) is constant on I i for all i ∈ M, using the first condition in (5.2), the above equality reduces to Hence, a stationary solution to problem (2.1)-(2.6) satisfying (5.1) is a triple (U (x), V (x), Ψ(x)) where V is determined by the boundary conditions and the functions U and Ψ solve the following problem.
Find C i , i = 1, ...m, and Ψ ∈ H 2 (A) such that We are going to prove existence of solutions to problem (5.5) using a fixed point technique; we need some preliminary results.
Given f i ∈ H 2 (I i ), for i ∈ M, we introduce the functions i∈M Ii Proof. The conditions (5.6) can be rewritten as (4.8). Using such relations at the node N 1 , for i ∈ M 1 we can express the coeffcients C f i as linear combination of the values V i and C f k1 , where k 1 is the index in (2.8), .
we have ; now, if N ν and N 1 are two consecutive nodes, linked by the arc I l , arguing as before we infer that the coefficients C f kν and C f k1 have to satisfy the following relation , which expresses C f kν in terms of C f k1 ; so, for all i ∈ M ν , we have the expression Since there are no cycles in the network, iterating this procedure we can write univokely all the coeffcients i andÕ f i are suitable quantities depending on the function f and on the values V i . In other words, system (5.6) has ∞ 1 solutions given by (5.8), for C k1 ∈ R. In order to determine C f k1 we use condition (5.7): Now, given f ∈ H 2 (A) we consider the problem which has a unique solution (see the proof of Proposition 3.1). We set Θ := j∈J |P j | + µ s max{a i } i∈M ; then the following estimates hold .
Lemma 5.2. Let G be an acyclic graph. Let U f i (x) ≥ 0 and let Ψ ∈ H 2 (A) be the solution to problem (5.9). Then there exist two positive constants K 1 , K 2 , depending on the parameters b i , D i , L i , d j ( i ∈ M, j ∈ J ), such that Proof. We multiply the first equation in (5.9) by Ψ i , we integrate on the interval I i and we sum for i ∈ M ; using (3.10) to treat the terms evaluated at the internal nodes, we obtain i∈M Ii where c S i are Sobolev constants. This yields the first inequality in the claim. In order to obtain the second inequality, first we notice that, if e j is an external node and I i(j) is the corresponding external arc, then the following inequality holds Then we consider an internal arc I ι and its starting node N η , the sets Q, J ′ as in (5.3) and S = {i ∈ M : I ι is incident with N ν for some ν ∈ Q} ; at each node the conservation of the flux (2.9) holds, then Then, for all x ∈ I ι , using the above equality and the boundary conditions (2.4), we have Then , for all l ∈ M, and using the first inequality in (5.10) we obtain The previous results give the tools to prove the following theorem of existence for stationary solutions under smallness conditions for some data; in particular we remark that the condition on i∈M |V i | is a condition on W j , j ∈ J , thanks to (5.4). Proof. If a stationary solution (U (x), V (x), Ψ(x)) exists then V i are univokely determined by the boundary data W j , j ∈ J ; moreover U, Ψ satisfy (5.5) and U i (x) are univokely determined by Ψ i (x) and the values V i , σ ij and µ s (Lemma 5.1). We remark that, if the solution verifies the properties in the claim, then the estimates in Lemma 5.2 hold for Ψ.
Let G be the operator defined in D(A 2 ) (see (3.8)) such that, if Ψ I ∈ D(A 2 ) then Ψ = G(Ψ I ) is the solution of problem (5.9) where f = Ψ I and U Ψ I is the function in Lemma 5.1. We consider G on the set where K 1 , K 2 are the constants in Lemma 5.2, equipped with the distance d generated by norm of H 2 (A); (B Θ , d) is a complete metric space.
Using the expression of C f i given in the proof of Lemma 5.1 and setting we can write It is readily seen that there exist some positive quantities q Θ i , increasing in Θ, and some positive quantities q −Θ i , decreasing in Θ, depending also on the parameters of the problem, such that, for all f ∈ B Θ , they can be used in (5.16) so that (5.15) implies where q Θ is a quantity increasing with Θ, depending also on the parameters of the problem; hence, for µ s + j∈M |V j | small enough, G is a contraction mapping on B Θ . Let Ψ be the unique fixed point of G in B Θ and let U = U Ψ ; then (Ψ, U ) is a solution to Problem (5.5) and it is the unique one verifying U ≥ 0. Regularity properties follow by the equations in (5.5).
i.e. U (x) ≥ 0. In this case the stationary solution of the previous theorem is the unique stationary solution with mass µ s .
When the quantity i∈M |V i | is not small enough respect to µ s we do not have informations about the sign of U i (x); however, if the boundary data, µ s and the parameters of the problem satisfy some relations, a stationary solution with mass µ s exists. First, if we set a := max{a i } i∈M , as in Lemma 5.2 we prove that if Ψ ∈ H 2 (A) is the solution to problem (5.9), then there exist two positive constants K 1 , K 2 , depending on the parameters b i , D i , L i , d j (i ∈ M, j ∈ J ), such that Moreover, let β, λ be as in Section 3 and γ as in Lemma 4.1, and let ) is a stationary solution, using Proposition 4.1 and (5.17), we obtain Then, if we can conclude that µ ± > 0 and , if a stationary solution (U, V, Ψ) exists, then U 1 ≤ µ − or U 1 ≥ µ + . So, under suitable smallness conditons for the data and |µ s |, we are able to prove the existence of a stationary solution verifying U 1 ≤ µ − . Fixed Ψ I ∈ B Θ1 , U Ψ I is still given by (5.11) and the relations(5.12)-(5.14) hold, where the quantities q i here depend on Θ 1 .
Thanks to (5.17), we can prove that Ψ = G(Ψ I ) ∈ B Θ1 if we show that U Ψ I 1 ≤ µ − ; this inequality can be achieved arguing as in the computations performed before the claim of this theorem, taking into account that The last part of the proof is equal to the one of Theorem 5.1, using the quantity Θ 1 in place of Θ, since, for Ψ I , Ψ I ∈ B Θ1 the following inequality holds where q Θ1 depends on Θ 1 , β i , λ i , L i , γ ν ij , increases with Θ 1 . Let Ψ be the unique fixed point of G in B Θ1 and let U = U Ψ ; then (Ψ, U ) is the unique solution to Problem (5.5) such that U 1 ≤ µ − and the claim is proved.

Global solutions
Here we use the results of Sections 4 and 5 to prove the existence of global solutions to problem (2.1)-(2.7). First we assume that G is an acyclic graph, so that the existence of some stationary solutions holds.
Let µ s ≥ 0, let the assumptions of Theorem 5.1 hold and let (U (x), V, Ψ(x)) be the stationary solution; due to (5.10) we can control the size of the quantity U ∞ + Ψ x ∞ by means of the size of aµ s + j∈J |P j |, in order to satisfy the hypothesis in Theorem 4.1. So, such theorem yields the following one.
We set µ(t) := On the other hand, if the assumptions of Theorem 5.2 hold, then the hypothesis in Theorem 4.1 can be satisfied by controlling the size of aµ − + j∈J |P j |; then, similarly to the above result, we obtain the existence of global solutions corresponding to data which are small perturbations of the stationary solution of Theorem 5.2, assuming that aµ − + j∈J |P j | is suitably small.