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Research article

Existence of a positive radial solution for semilinear elliptic problem involving variable exponent

  • This paper consider that the following semilinear elliptic equation

    {Δu=uq(x)1,  in  B1,u>0,  in  B1,u=0,  in  B1,

    where B1 is the unit ball in RN(N3), q(x)=q(|x|) is a continuous radial function satifying 2q(x)<2=2NN2 and q(0)>2. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.

    Citation: Changmu Chu, Shan Li, Hongmin Suo. Existence of a positive radial solution for semilinear elliptic problem involving variable exponent[J]. Electronic Research Archive, 2023, 31(5): 2472-2482. doi: 10.3934/era.2023125

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  • This paper consider that the following semilinear elliptic equation

    {Δu=uq(x)1,  in  B1,u>0,  in  B1,u=0,  in  B1,

    where B1 is the unit ball in RN(N3), q(x)=q(|x|) is a continuous radial function satifying 2q(x)<2=2NN2 and q(0)>2. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.



    In this paper we consider a semilinear hyperbolic-parabolic system evolving on a finite planar network composed from nodes connected by m oriented arcs Ii,

    {tui+λixvi=0,tvi+λixui=uixϕiβivi,t0,xIi,i=1,...,m;tϕi=Dixxϕi+aiuibiϕi, (1.1)

    the system is complemented by initial, boundary and transmission conditions at the nodes (see Section 2).

    We are interested in the study of stationary solutions and asymptotic behaviour of global solutions of the problem.

    The above system has been proposed as a model for chemosensitive movements of bacteria or cells on an artificial scaffold [12]. The unknown u stands for the cells concentration, λv is the average flux and ϕ is the chemo-attractant concentration. In particular, the model turns out to be useful to describe the process of dermal wound healing, when the stem cells in charge of the reparation of dermal tissue (fibroblasts) create an extracellular matrix and move along it to fill the wound, driven by chemotaxis; tissue engineers use artificial scaffolds, constituted by a network of crossed polymeric threads, inserting them within the wound to accelerate the process (see [13,20,25]). In the above mathematical model, the arcs of the graph mimic the fibers of the scaffold; each of them is characterized by a tipical velocity λi, a friction coefficient βi, a diffusion coefficient Di, and a production rate ai and a degradation one bi; the functions ui,ϕi are the densities of fibroblasts and chemoattractant on each arc.

    Starting from the Keller-Segel paper [18] in 1970 until now, a lot of articles have been devoted to PDE models in domains of Rn for chemotaxis phenomena. The parabolic (or parabolic-elliptic) Patlak-Keller-Segel system is the most studied model [17,23,22]; in recent years, hyperbolic models have been introduced too, in order to avoid the unrealistic infinite speed of propagation of cells, occurring in parabolic models [8,9,14,23,24,1,15,16,11].

    In [11] the Cauchy and the Neumann problems for the system in (1.1), respectively in R and in bounded intervals of R, are studied, providing existence of global solutions and stability of constant states results.

    Recently an interest in these mathematical models evolving on networks is arising, due to their applications in the study of biological phenomena and traffic flows, both in parabolic cases [2,6,21] and in hyperbolic ones [10,7,26,12,3].

    We notice that the transmission conditions for the unknowns, at each inner node, which complement the equations on networks, are crucial characteristics of the model, since they are the coupling among the solution's components on each arc.

    Most of the studies carried out until now, consider continuity conditions at each inner node for the density functions [7,6,21]; nevertheless, the eventuality of discontinuities at the nodes seems a more appropriate framework to decribe movements of individuals or traffic flows phenomena [5].

    For these reason in [12], transmission conditions which link the values of the density functions at the nodes with the fluxes, without imposing any continuity, are introduced; these conditions guarantee the fluxes conservation at each inner node, and, at the same time, the m-dissipativity of the linear spatial differential operators, a crucial property in the proofs of existence of local and global solutions contained in that paper.

    In this paper we focus our attention on stationary solutions to problem (1.1) complemented by null fluxes boundary conditions and by the same transmission conditions of [12] (see next section and Section 3 in [12] for details). We consider acyclic networks and we prove the existence and uniqueness of the stationary solution with fixed mass of cells mi=1Iiui(x)dx, under the assumption that the mass is suitably small. If the quantity aibi does not vary with the index i, we easily show that such solution is a constant state on the whole network. We notice that, in the case of acyclic networks, although the transmission conditions do not set the continuity of the density u at the inner nodes, the fluxes conservation at those nodes and the boundary null fluxes conditions imply the absence of jumps discontinuities at the inner vertexes, for the component u of a stationary solution.

    For general networks and the parameters ai and bi in the same range as above, it is easy to show that, for any fixed mass, a stationary solution constant on the whole network exists and the constant values of the densities are determined by the mass. In this case we also obtain a uniqueness result: in the set of stationary solutions with small density u in H1-norm and fixed mass of cells, the constant state on the network is the unique element.

    Finally we study the large time behaviour of global solutions on general networks, when the ratio between ai and bi is constant. We consider initial data with fixed small mass, which are small perturbations of the constant state on the network with the same mass, then we prove that such state is the asymptotic profile of the solutions corresponding to the data. So, we point out that, for small global solutions to our problem, the discontinuities at the inner nodes vanish when t goes to infinity, since their asymptotic profiles are continuous functions on the whole network.

    The study of the asymptotic behaviour provide informations about the evolution of a small mass of individuals moving on a network driven by chemotaxis: suitable initial distributions of individuals and chemoattractant, for large time evolve towards constant distributions on the network, preserving the mass of individuals.

    We recall that the stability of the constant solutions to this system, considered on bounded interval in R, is studied in [11] and stationary solutions and asymptotic behaviour for a linear system of uncoupled conservation laws on network are studied in [19].

    Finally, in [3] the authors introduce a numerical scheme to approximate the solutions to the problem (2.1); in that paper transmission conditions are set for the Riemann invariants of the hyperbolic part of the system, w±i=12(ui±vi), and are equivalent to our ones for some choices of the transmission coefficients. The tests presented there, in the case of acyclic graph and dissipative transmission coefficients, show an asymptotic behaviour of the solutions which agrees with our theoretical results.

    The paper is organized as follows. In Section 2 we give the statement of the problem and, in particular, we introduce the transmission conditions, while in Section 3 we prove the results about existence and uniqueness of stationary solutions. The last section is devoted to study the asymptotic behaviour of solutions; the results obtained in this section constitute the sequel and the development of the result of existence of global solutions in [12] and the proofs are based on the same techniques and use simple modifications of the a priori estimates obtained in [12].

    We consider a finite connected graph G=(V,A) composed by a set V of n nodes (or vertexes) and a set A of m oriented arcs, A={Ii:iM={1,2,...,m}}.

    Each node is a point of the plane and each oriented arc Ii is an oriented segment joining two nodes.

    We use ej, jJ, to indicate the external vertexes (or boundary vertexes) of the graph, i.e. the vertexes belonging to only one arc, and by Ii(j) the external arc outgoing or incoming in the external vertex ej.

    Moreover, we use Nν, νP, to denote the inner nodes; for each of them we consider the set of incoming arcs Aνin={Ii:iIν} and the set of the outgoing ones Aνout={Ii:iOν}; let Mν=IνOν.

    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 contain cycles: for each couple of nodes, there exists a unique path with no repeated arcs connecting them (an example of acyclic graph is in Fig. 1).

    Figure 1. 

    Example of acyclic network; the highlighted arcs form the path linking the nodes N4 and N5.

    .

    Each arc Ii is considered as a one dimensional interval (0,Li). A function f defined on A is a m-tuple of functions fi, iM, each one defined on Ii; fi(Nν) denotes fi(0) if Nν is the initial point of the arc Ii and fi(Li) if Nν is the end point, and similarly for f(ej).

    We set Lp(A):={f:fiLp(Ii),iM}, Hs(A):={f:fiHs(Ii),iM} and

    f2:=iMfi2, fHs:=iMfiHs.

    We consider the evolution of the following one-dimensional problem on the graph G

    {tui+λixvi=0,tvi+λixui=uixϕiβivi,t0, xIi, iM,tϕi=Dixxϕi+aiuibiϕi, (2.1)

    where ai0,λi bi,Di,βi>0.

    We complement the system with the initial conditions

    ui0,vi0H1(Ii), ϕi0H2(Ii)  for iM ; (2.2)

    the boundary conditions at each outer point ej are the null flux conditions

    vi(j)(ej,t)=0,t>0, jJ, (2.3)
    ϕi(j)x(ej,t)=0t>0, jJ. (2.4)

    In addition, at each inner node Nν we impose the following transmission conditions for the unknown ϕ

    {Diϕix(Nν,t)=jMνανij(ϕj(Nν,t)ϕi(Nν,t)),    iIν, t>0,Diϕix(Nν,t)=jMνανij(ϕj(Nν,t)ϕi(Nν,t)),  iOν, t>0,ανij0, ανij=ανji  for all i,jMν, (2.5)

    which imply the continuity of the flux at each node, for all t>0,

    iIνDiϕix(Nν,t)=iOνDiϕix(Nν,t).

    For the unknonws v and u we impose the transmission conditions

    {λivi(Nν,t)=jMνKνij(uj(Nν,t)ui(Nν,t)), iIν, t>0,λivi(Nν,t)=jMνKνij(uj(Nν,t)ui(Nν,t)),   iOν, t>0,Kνij0, Kνij=Kνji  for all i,jMν .  (2.6)

    These conditions ensure the conservation of the flux of the density of cells at each node Nν, for t>0,

    iIνλivi(Nν,t)=iOνλivi(Nν,t),

    which corresponds to the conservation of the total mass

    iMIiui(x,t)dx=iMIiu0i(x)dx,

    i.e. no death nor birth of individuals occours during the observation.

    Motivations for the constraints on the coefficients in the transmission conditions can be found in [12].

    Finally, we impose the following compatibility conditions

    ui0,vi0,ϕi0 satisfy conditions (2.3)-(2.6) for all iM . (2.7)

    Existence and uniqueness of local solutions to problem (2.1)-(2.7),

    u,vC([0,T];H1(A))C1([0,T];L2(A)),ϕC([0,T];H2(A))C1([0,T];L2(A)) 

    are achieved in [12] by means of the linear contraction semigroup theory coupled with the abstract theory of nonhomogeneous and semilinear evolution problems; in fact, the transmission conditions (2.5) and (2.6) allows us to prove that the linear differential operators in (2.1) are m-dissipative and then, to apply the Hille-Yosida-Phillips Theorem (see [4]). The existence of global solutions when the initial data are small in (H1(A))2×H2(A) norm is proved [12] too; this result holds under the further assumption

     for all νP, for some kMν,Kνik0 for all iMν,ik . (2.8)

    In this section we approach the question of existence and uniqueness of stationary solutions of problem (2.1)-(2.8), with fixed mass

    μ:=iMIiui(x)dx0,

    in the case of an acyclic network (see Section 2). We look for stationary solutions (u,v,ϕ)(H1(A))2×H2(A).

    Obviously, the flux v of a stationary solution has to be constant on each arc and has to be null on the external arcs; in the case of acyclic networks, the boundary and transmission conditions (2.3), (2.6) force it to be null on each arc. In order to prove this fact we consider an internal arc Ij and its initial node Nμ; we consider the set

    Q={νP:Nν is linked to Nμ by a path not covering Ij} 

    (see Fig. 1: if, for example, j=9 then μ=4, Q={1,2,3,5} and the arcs in bold type form the path which links the nodes N5 and N4).

    At each node the conservation of the flux of the density of cells, stated in Section 2, holds; then

    νQ{μ}(iIνλivi(Nν)iOνλivi(Nν))=0 .

    Since, for all iM, vi(x) is constant on Ii and vi(x)=0 if Ii is an external arc, the above equality reduces to

    vj(Nμ)=0 ;

    then vj(x)=0 for all xIj.

    The previous result implies that stationary solutions must have the form (u,0,ϕ), where u and ϕ have to verify the system

    {λiuix=uiϕix,Diϕixx+biϕi=aiui,xIi, iM, t>0,  (3.1)

    with the boundary condition at each outer point ej, jJ,

    ϕi(j)x(ej,t)=0t>0, (3.2)

    and the transmission conditions, at each inner node Nν,

    jMνKνij(uj(Nν)ui(Nν))=0,iMν, (3.3)
    Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iIν,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iOν. (3.4)

    For each fixed inner node Nν, let kMν be the index in condition (2.8) and let consider the transmission relations, for iMν, ik,

    0=jMν,jiKνij(uj(Nν)ui(Nν))=jMν,ji,kKνij(uj(Nν)uk(Nν))(jMν,jiKνij)(ui(Nν)uk(Nν)) ; (3.5)

    the assumptions on Kνkj in (2.8) ensure that the matrix of the coefficients of this linear system in the unknowns (uj(Nν)uk(Nν)), jk, is non singular (if k=1 it is immediate to check that it has strictly dominant diagonal). Then the condition (3.3) can be rewritten as

    uj(Nν)=uk(Nν) for all jMν.

    Now we fix μ00 and we look for stationary solutions such that

    iMIiui(x)dx=μ0; (3.6)

    notice that for the evolution problem, the quantity iMIiui(x,t)dx is preserved for all t0, thanks to the transmission conditions (2.6).

    Integrating the first equation in (3.1) we can rewrite problem (3.1)-(3.6) as the following elliptic problem on network:

    Find C=(C1,C2,...,Cm) and ϕH2(A) such that

    {Diϕixx+biϕi=aiuixIi,iM,ui(x)=Ciexp(ϕi(x)λi)xIi,iM,ϕi(j)x(ej)=0,jJ,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iIν,νP,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iOν,νP,Cjexp(ϕj(Nν)λj)=Ciexp(ϕi(Nν)λi),i,jMν,νP,iMCiIiexp(ϕi(x)λi)dx=μ0. (3.7)

    We consider the linear operator A:D(A)L2(A),

    D(A)={ϕH2(A):(3.2),(3.4) hold },A(ϕ)={Diϕixx+biϕi}iM ; (3.8)

    then the equation in (3.7) and the boundary and transmission conditions for ϕ can be written as

    Aϕ=F(ϕ,C), (3.9)

    where, for iM, Fi(ϕ(x),C)=aiCiexp(ϕi(x)λi).

    We are going to prove the existence and uniqueness of solutions to the problem (3.7) by using the Banach Fixed Point Theorem; in order to do this we need some preliminary results about the linear equation

    Aϕ=F(f,Cf), (3.10)

    where fH2(A) is a given function, Cf=(Cf1,...,Cfm) and Cfi are non-negative given real constants.

    The existence and uniqueness of the solution ϕH2(A) to the above problem (for a general FL2(A) and a general network) is showed in [12], by Lax-Milgram theorem, in the proof of Proposition 4.1; here, we need to prove some properties holding for the solution in the case of acyclic graphs, under suitable assumptions on f and Cfi.

    The transmission conditions (2.5) imply the following equality which will be useful in the next proofs:

    iMIiDi(ϕi(x)ϕix(x))x dx=νP(iIνDiϕi(Nν)ϕix(Nν)iOνDiϕi(Nν))ϕix(Nν))=νP ijMνανijϕi(Nν)(ϕj(Nν)ϕi(Nν)) =12νP ijMνανij(ϕj(Nν)ϕi(Nν))2 . (3.11)

    Let |A|:=iM|Ii| and g:=max{gi,iM}.

    Lemma 3.1. Let G=(V,A) be an acyclic network, let fH2(A) and let Cfi be non-negative real numbers, for iM. Then the solution ϕ to problem (3.8), (3.10) is non-negative. Moreover, if

    iMCfiIiexp(fi(x)λi)dx=μ0, (3.12)

    then

    ϕx2max{ai}iMmin{Di}iMμ0; (3.13)

    if (3.12) holds and

    fx2max{ai}iMmin{Di}iMμ0, (3.14)

    then there exists a quantity Kμ0=Kμ0(ai,bi,Di,λi,|A|,μ0), depending only on the parameters appearing in brackets, infinitesimal when μ0 goes to zero, such that

    ϕW2,1(A),ϕH2(A)Kμ0 . (3.15)

    Proof. Let consider a function ΓC1(R), strictly increasing in (0,+), and let Γ(y)=0 for y0; following standard methods for the proofs of the maximum principle for elliptic equations and setting Fi(x)=Fi(f(x),Cf), we obtain

    iMIi(Di(ϕix(x)Γ(ϕi(x)))xDiΓ(ϕi(x))ϕ2ix(x)+biϕi(x)Γ(ϕi(x))Fi(x)Γ(ϕi(x))) dx=0 .

    As regard to the first term, we can argue as in (3.11), taking into account the properties of Γ,

    iMIiDi(Γ(ϕi)ϕix)x=12νP ijMνανij(ϕj(Nν)ϕi(Nν))(Γ(ϕj(Nν))Γ(ϕi(Nν)))0 ; (3.16)

    the above inequality and the non-negativity of Fi imply that

    iMbiIiϕi(x)Γ(ϕi(x))dx0,

    so that, thanks to the properties of Γ, we can conclude that ϕi(x)0 for all iM.

    By integration of the equation (3.10), taking into account (3.4) and (3.2), we obtain

    iMbiIiϕi(x)dx=iMIiFi(f(x),Cf)dx (3.17)

    which implies

    ϕ1max{ai}min{bi}μ0 . (3.18)

    In order to obtain (3.13), first we notice that, if Ij is an external arc, then the following inequality holds

    |Djϕjx(x)|IjDj|ϕjyy(y)|dyIj(bjϕj(y)+Cfjajexp(fj(y)λj))dy.

    Then we consider an internal arc Ij and its initial node Nμ and the sets

    Q={νP:Nν is linked to Nμ by a path not covering Ij},
    S={iM:Ii is incident with Nl for some lQ} 

    (see Fig. 1: if, for example, j=9, then μ=4, Q={1,2,3,5}, S={1,2,3,4,5,6,8, 10,11,12}); at each node the conservation of the flux, stated in Section 2 as a consequence of the transmission conditions, holds; then

    νQ{μ}(iIνDiϕix(Nν)iOνDiϕix(Nν))=0 .

    Let x be a point on the arc Ij (see Fig. 1 for j=9, μ=4) and Ixj be the part of Ij which connects Nμ and x; then, using the above equality and the boundary conditions (2.4), we have

    |Djϕjx(x)|=|Djϕjx(x)+νQ{μ}(iIνDiϕix(Nν)iOνDiϕix(Nν))||iSIiDiϕiyy(y)dy+IxjDjϕjyy(y)dy||iSIi(biϕi(y)Cfiaiexp(fi(y)λi))dy+Ixj(bjϕj(y)Cfjajexp(fj(y)λj))dy|. (3.19)

    Then Djϕjx2μ0max{ai}iM for all jM and we obtain (3.13) which implies

    ϕx12max{ai}iMmin{Di}iM|A|μ0, (3.20)

    and

    ϕx22max{ai}iMmin{Di}iM|A|12μ0 ; (3.21)

    moreover, by Sobolev embedding theorem, we obtain

    ϕK1μ0, (3.22)

    where K1=K1(ai,bi,Di,|A|) is a suitable constant.

    The estimates for the function ϕxx follow by using the equation (3.10); first, using (3.12) and (3.17), we obtain

    ϕxx12max{ai}iMmin{Di}iMμ0 ; (3.23)

    then, using (3.11), we have

    iMD2ibiIiϕixx2(x) dxiMFibiIiFi(x) dxmax{aiFi}min{bi}μ0 

    and the embedding of W1,1(Ii) in L(Ii) gives

    iMIiϕixx2(x) dxK2(1+fx)μ20, (3.24)

    where K2=K2(ai,bi,Di,λi) is a suitable constant.

    Finally, the inequalities (3.18), (3.20)-(3.24) imply the inequalities (3.15) in the claim.

    Now we can prove the following theorem.

    Theorem 3.1. Let G=(V,A) be an acyclic network. There exists ϵ>0 such that, if 0μ0ϵ, then problem (2.1)-(2.8) has a unique stationary solution satisfying (3.6); the solution has the form

    (Ciexp(ϕi(x)λi),0,ϕi(x))  iM,

    where ϕi(x)0 and Ci are nonnegative constants such that uj(Nν)=ui(Nν) for all νP, i,jMν.

    Proof. First we notice that, if a stationary solution (u,v,ϕ) satisfying (3.6) exists, then u is non-negative, since the costants Ci in (3.7) must have the same sign, so that they have to be non-negative to satisfy the condition μ00; arguing as in the proof of Lemma 3.1 we prove that ϕ is non-negative too. If μ0>0 then u and ϕ are positive functions.

    We are going to use a fixed point technique. Given ϕ0D(A), we want to define a function u0(x) on the network, such that, for iM,

    u0i(x)=Cϕ0iexp(ϕ0i(x)λi),

    where the constants Cϕ0i satisfy the following linear system composed by the last conditions in (3.7)

    Cϕ0jexp(ϕ0j(Nν)λj)=Cϕ0iexp(ϕ0i(Nν)λi),i,jMν,νP, (3.25)
    iMCϕ0iIiexp(ϕ0i(x)λi)dx=μ0 . (3.26)

    The system (3.25), (3.26) has a unique solution; actually, since the network has no cycles, the system (3.25) has 1 solutions Cϕ0,α=(α,αδ2,αδ3,...,αδm), αR, where δi are suitable coefficients, and the condition (3.26) determines the value of α.

    In order to give an explicit expression for the coefficients δi we consider an arc, I1, and we define

    u01(x):=αexp(ϕ01(x)λ1) .

    Let Nμ one of the extreme points of I1, then we define the function u0 on the other arcs which are incident with Nμ in such a way to verify the equalities in (3.25) for the node Nμ,

    u0j(x):=αexp(ϕ01(Nμ)λ1)exp(ϕ0j(Nμ)λj)exp(ϕ0j(x)λj)  for all jMμ,j1;

    i.e. we set Cϕ0j=αexp(ϕ01(Nμ)λ1)exp(ϕ0j(Nμ)λj), jMμ,j1.

    This procedure can be iterated at each node reached by one of the arcs Ij, jMμ, and at the other extreme point of I1, if it is an internal arc, and so on, defining in this way the function u0 on each arc of the network. Notice that this construction is possible since there are no cycles in the graph. The function u can be expressed, on each arc of the network, as follows (if it is the case, renumbering in suitable way the arcs and the nodes): let consider the path from the outer point e1 to an inner node Nh1, composed from the arcs Ii, i=1,...,h1, (passing through the vertexes Ni, i=1,...,h1), and let Ih be an arc incident with the node Nh1, not belonging to the path (see Fig. 2 where h = 5 and the highlighted arcs form the path); following the procedure described before, after setting

    Eh(ϕ0):=Πi=1,...,h1exp(ϕ0i(Ni)λi)Πi=1,...,h1exp(ϕ0i+1(Ni)λi+1),
    Figure 2. 

    Example: the highlighted arcs form the path from the outer point e1 to the inner node N4 and I5 is an arc incident with N4, not belonghing to the path.

    .

    we define

    u0h(x):=αEh(ϕ0)exp(ϕ0h(x)λh) .

    The quantity α is fixed in such a way to verify the last condition in (3.7),

    αiMEi(ϕ0)Iiexp(ϕ0i(x)λi)dx=μ0,

    so that, for all iM,

    u0i(x)=Cϕ0iexp(ϕ0i(x)λi),  Cϕ0i:=μ0 Ei(ϕ0) jMEj(ϕ0)Ijexp(ϕ0j(x)λj)dx . (3.27)

    Let G be the operator defined in D(A) such that ϕ1:=G(ϕ0) is the solution of problem (3.10) where f=ϕ0 and Cfi=Cϕ0i for iM,

    Aϕ1=F(Cϕ0,ϕ0) ;

    let Kμ0=Kμ0(ai,bi,Di,λi,|A|,μ0) be the quantity in Lemma 3.1, and let

    Bμ0:={ϕD(A):ϕ0,ϕx2max{ai}iMmin{Di}iMμ0,ϕH2Kμ0 }

    equipped with the distance d generated by norm of H2(A); (Bμ0,d) is a complete metric space. From the lemma we know that solutions to problem (3.8)-(3.12) have to belong to Bμ0, then G(Bμ0)Bμ0; next we are proving that, if μ0 is small enough, then G is a contraction in Bμ0.

    We consider ϕ0,¯ϕ0Bμ0 and the corresponding u0,¯u0 and ϕ1,¯ϕ1; using the equation satisfied by ϕ0 and ¯ϕ0, for all iM we can write

    biIi(ϕ1i(x)¯ϕ1i(x))2,dx+DiIi(ϕ1ix(x)¯ϕ1ix(x))2,dxDiIi((ϕ1ix(x)¯ϕ1ix(x))(ϕ1i(x)¯ϕ1i(x)))xdx=aiIi(u0i(x)¯u0i(x))(ϕ1i(x)¯ϕ1i(x))dx ; (3.28)

    using (3.11), from (3.28) we infer that

    iMϕi1¯ϕi1H2K(ai,bi,Di)iMu0i¯u0i2, (3.29)

    We set

    Jϕ0i:=Iiexp(ϕ0i(x)λi)dx,Eϕ0i(x):=exp(ϕ0i(x)λi)  ;

    we have

    |u0i(x)ˉu0i(x)|=μ0|εi(ϕ0)Eϕ0i(x)jMεj(ϕ0)Jϕ0jεi(ˉϕ0)Eˉϕ0i(x)jMεj(ˉϕ0)Jˉϕ0j|. (3.30)

    In order to treat the above quantity we have to consider that, for all gBμ0, Egi(x)1, Jgi|Ii| and there exists a constant K6=K6(Kμ0,λi), increasing with μ0, such that, for all iM

    maxIiEgi(x)K6,JgiK6|Ii|,
    |Eϕ0i(x)Eˉϕ0i(x)|K6|ϕ0i(x)ˉϕ0i(x)|,
    |Jϕ0iJˉϕ0i|K6Ii|ϕ0i(x)ˉϕ0i(x)|dx.

    The above inequalities can be used in (3.30) so that (3.29) implies

    iMϕi1¯ϕi1H2μ0K7(ai,bi,Di,Kμ0,|A|)iMϕ0i¯ϕ0iH1, (3.31)

    where K7 increases with μ0; hence, for μ0 small enough, G is a contraction on Bμ0 and we can use the Banach Fixed Point Theorem.

    Let ϕ be the unique fixed point of G in Bμ0 and let Cϕ=(Cϕ1,Cϕ2,...,Cϕm) where Cϕi, for iM, are computed as in (3.27); then (ϕ,Cϕ) is the unique solution to Problem (3.7) and the claim is proved.

    For any constant U0, the triple (U,0,aibiU) satisfies the equations in (2.1) on the arc Ii. Let Q be a real non-negative number; if aibi=Q for all iM, then the same triple (U,0,QU) satisfies the equations on each arc Ii and it is a stationary solution to the problem (2.1)-(2.8). Then, as a consequence of the previous theorem, we have the following proposition, with ϵ as in the theorem.

    Proposition 3.1. Let G=(V,A) be an acyclic network. If aibi=Q for all iM and 0μ0ϵ, then the unique stationary solution to problem (2.1)-(2.8), (3.6) is the constant solution (μ0|A|,0,Qμ0|A|).

    Remark 3.1. For general networks, when the value of aibi=Q on each arc, the stationary solution of Proposition 3.1 always exists. More precisely, if aibi=Q, in the class of the functions (u,v,ϕ) which are constant on each arc, the stationary solution (μ0|A|,0,Qμ0|A|) is the unique stationary solution with mass μ0; this fact is true without any restrictions on the value of μ0 and on the structure of the network. Actually, if we assume that u is constant on each arc, then, using the equations, we infer that, on each arc, ϕx(x) is constant too, hence ϕxx=0 and ϕ(x) is constant. Then v(x)=0 on each arc; hence, arguing as at the beginning of this section, we obtain that u is continuous on the network.

    In the next proposition we are going to prove that, in a set of small solutions, such stationary solution is the unique one with fixed mass μ0.

    Proposition 3.2. Let aibi=Q for all iM and let (u,v,ϕ)(H1(A))2×H2(A) be a stationary solution of problem (2.1)-(2.8), (3.6). There exists ϵ0>0, depending on λi,ai,bi,Di,βi,|A|, such that, if uH1ϵ0, then (u,v,ϕ)=(μ0|A|,0,Qμ0|A|).

    Proof. We set H:=uH1. The transmission conditions (2.6) imply that

    νP(iIνλiui(Nν)vi(Nν)iOνλiui(Nν)vi(Nν))0,

    so, by using the first two equations in (2.1), we obtain

    2iMβiIiv2i(x)dxiMuiIi(v2i(x)+ϕ2ix(x)) dx

    and

    iMλiIiu2ix(x) dxiMuiIi(u2ix(x)+ϕ2ix(x)) dx +iMβ2iλiIiv2i(x) dx ;

    the above inequalities implies the following one

    v22+ux22K0H(ϕx22+v22+ux22), (3.32)

    where K0 is a positive constant depending on the parameters λi,βi and the Sobolev embedding costant.

    The transmission conditions (2.5) imply that

    νP(iIνDiϕi(Nν)ϕix(Nν)iOνDiϕi(Nν)ϕix(Nν))0 ;

    moreover, the assumption (2.8) imply that, for each νP, for suitable coefficients θνij and suitable kMν,

    uj(Nν)=uk(Nν)+iMν,ikθνijvi(Nν) for all jMν,

    (see Lemma 5.9 in [12]); then, by the last equation in (2.1), arguing as in the proof of Proposition 5.8 in [12], we obtain

    ϕx22+ϕxx22K1(v22+ux22), (3.33)

    where K1 is a positive constant depending on the parameters Di,ai,bi,θνij.

    By inequalities (3.32) and (3.33) we deduce the following one

    v22+ux22K0(1+K1)H(v22+ux22),

    which, for small H, implies v2,ux2=0.

    In the cases when aibi depends on the arc in consideration, stationary solutions with the component u constant on each arc, can be inadmissible. As we showed before, v should be zero, u should be constant on the whole network and ϕ should be constant on each arc,

    ui(x)=μ0|A|,   ϕi(x)=aibiμ0|A|,iM .

    Therefore the transmission conditions, for each νP,

    jMνανijμ0|A|(ajbjaibi)=0,iMν,

    are constraints on the relations between the parameters of the problem which have to hold if the constant stationary solution exists.

    For example, in the case of two arcs, if b2a2b1a1 (and 0<μ0ϵ), the stationary solution can not be constant on the arcs, since the trasmission condition at the node,

    α11μ0|A|(b2a2b1a1)=0,

    cannot be satisfied.

    Hence, in the cases when aibi depends on the arc in consideration, if (u,v,ϕ) is the stationary solution in Theorem 3.1, then u is a continuous function on all the network but it is not constant on each arc.

    In this section we are going to show that the constant stationary solutions previously introduced, provide the asymptotic profiles for a class of solutions to problem (2.1)-(2.8). We recall that existence and uniqueness of global solutions

    u,vC([0,+);H1(A))C1([0,+);L2(A)),ϕC([0,+);H2(A))C1([0,+);L2(A)),ϕxH1(A×(0,+)), (4.1)

    to such problem is proved in [12], when the initial data are sufficiently small in (H1(A))2×H2(A) norm and the following condition holds

    aibi=Q  for all iM ; (4.2)

    in particular it is proved that the functional F defined by

    F2T(u,v,ϕ):=iM(supt[0,T]ui(t)2H1+supt[0,T]vi(t)2H1+supt[0,T]ϕix(t)2H1)+T0(ux(t)22+v(t)2H1+vt(t)22+ϕx(t)2H1+ϕxt(t)22) dt (4.3)

    is uniformly bounded for T>0.

    Here and below we use the notations

    fi(t)2:=fi(,t)L2(Ii),  fi(t)Hs:=fi(,t)Hs(Ii) .

    Now we assume (4.2), we fix ¯μ0 and we consider the constant stationary solution, (¯u,0,¯ϕ), to problem (2.1)-(2.8), such that ¯u|A|=¯μ; moreover let (˜u0,˜v0,˜ϕ0)(H1(A))2×H2(A) be a small perturbation of (¯u,0,¯ϕ), i.e., setting u0:=˜u0¯u,v0:=˜v0,ϕ0:=˜ϕ0¯ϕ , the (H1(A))2×H2(A) norm of (u0,v0,ϕ0) is bounded by a suitable small ϵ0>0.

    If (˜u,˜v,˜ϕ) is the solution to problem (2.1)-(2.8) with initial data (˜u0,˜v0,˜ϕ0) and u:=˜u¯u,v:=˜v,ϕ:=˜ϕ¯ϕ , then (u,v,ϕ) is solution to the system

    {tui+λixvi=0tvi+λixui=(ui+¯u)xϕiβivixIi,t0,iM,tϕi=Dixxϕi+aiuibiϕi, (4.4)

    complemented with the conditions (2.2)-(2.8) and initial data (u0,v0,ϕ0) defined above.

    The existence and uniqueness of local solutions to this problem can be achieved by means of semigroup theory, following the method used in[12], with little modifications.

    On the other hand, if we assume that ¯u is suitably small, the method used in that paper to obtain the global existence result in the case of small initial data can be used here too, modifying the estimates in order to treat the further term in the second equation and then using the smallness of ¯u.

    Below we list a priori estimates holding for the solutions to problem (4.4), (2.2)-(2.7); we don't give the proofs since they are equal to those in [12], in Section 5, except for easy added calculations to treat the term ¯uϕix.

    Proposition 4.1. Let (u,v,ϕ) be a local solution to problem (4.4), (2.2)-(2.7),

    u,vC([0,T];H1(A))C1([0,T];L2(A)),ϕC([0,T];H2(A))C1([0,T];L2(A)), ϕxH1(A.×(0,T)) ;

    then

    a)

    iM(sup[0,T]ui(t)22+sup[0,T]vi(t)22+βiT0vi(t)22dt)CiM(u0i22+v0i22)+CiM(sup[0,T]ui(t)H1+ˉu)To(ϕix(t)22+vi(t)22)dt;

    b)

    iM(sup[0,T]vix(t)22+sup[0,T]vit(t)22+T0vit(t)22,dt)C(v02H1+u02H1ϕ02H2)+CiM(sup[0,T]ui(t)H1+ˉu)T0(ϕixt(t)22+vit(t)22) dt+CiMsup[0,T]ϕx(t)H1T0(vit(t)22+vi(t)2H1)dt ;

    c)

    iMsup[0,T]uix(t)22CiM(sup[0,T]vit(t)22+sup[0,T]vi(t)22)+CiM(sup[0,T]ui(t)H1+ˉu)(sup[0,T]uix(t)22+sup[0,T]ϕix(t)22) ;

    d)

    iMT0uix(t)22,dtCiMT0(vit(t)22+vi(t)22)dt+CiM(sup[0,T]ui(t)H1+ˉu)T0(uix(t)22+ϕix(t)22)dt ;

    e)

    iMT0vix(t)22,dtCiM(v0i22+u0i2H1(1+ϕ0i2H1))+CiM(T0vit(t)22,dt+sup[0,T]vit(t)22)+CiM(sup[0,T]ui(t)H1+sup[0,T]ϕix(t)H1+ˉu)×T0(vi(t)2H1+ϕixt(t)22)dt ;

    f)

    iM(sup[0,T]ϕit(t)22+T0(ϕit(t)22+ϕitx(t)22) dt)CiM(ϕ0i2H2+u0i22+T0uit(t)22) ;

    g)

    iM(sup[0,T]ϕixx(t)22+sup[0,T]ϕix(t)22)CiM(sup[0,T]ϕit(t)22+sup[0,T]ui(t)22) ; 

    h) if (2.8) and (4.2) hold, then

    iMT0(ϕix(t)22+ϕixx(t)22) dtCiMT0(uix(t)22+vi(t)2H1+ϕit(t)22) dt,

    for suitable costants C.

    The estimates in the previous proposition allow to prove the following theorem about the existence of global solutions to problem (4.4), (2.2)-(2.8).

    Let FT(u,v,ϕ) be the functional defined in (4.3).

    Theorem 4.1. Let (4.2) hold. There exists ϵ0,ϵ1>0 such that, if

    ¯uϵ1,  u0H1,v0H1,ϕ0H2ϵ0,

    then there exists a unique global solution (u,v,ϕ) to problem (4.4), (2.2)-(2.8),

    u,vC([0,+);H1(A))C1([0,+);L2(A)),
    ϕC([0,+);H2(A))C1([0,+);L2(A)), ϕxH1(A×(0,+)) .

    Moreover, FT(u,v,ϕ) is bounded, uniformly in T.

    Proof. It is sufficient to prove that the functional FT(u,v,ϕ) is bounded, uniformly in T.

    We notice that each term in F2T(u,v,ϕ) is in the left hand side of one of the estimates in Proposition 4.1, therefore, arranging all the estimates, we can prove the following inequality

    F2T(u,v,ϕ)c1F20(u,v,ϕ)+c2¯uF2T(u,v,ϕ)+c3F3T(u,v,ϕ),

    taking into account also that, on the right hand side of the estimates, the quadratic terms (not involving initial data) which have not the coefficient ¯u, can be bounded by means of cubic ones.

    If ¯u is sufficiently small, the previous inequality implies

    F2T(u,v,ϕ)c4F20(u,v,ϕ)+c5F3T(u,v,ϕ)

    for suitable positive constants ci.

    It is easy to verify that, if y0 is a sufficiently small positive real number and h(y)=c5y3y2+c4y0 then there exists 0<¯y<23c5 such that h(y)>0 in [0,¯y) and h(y)<0 in (¯y,23c5].

    Then we can conclude that, if F0 is suitably small, then FT remains bounded for all T>0.

    The above result, in particular the uniform, in time, boundedness of the functional FT, allow us to prove the theorem below.

    Let (4.2) hold and let (¯u,0,¯ϕ) be the constant stationary solution to problem (2.1)-(2.8) such that ¯u|A|=¯μ; moreover, let C(A) be the set of the funcions f such that fiC(¯Ii) for iM.

    Theorem 4.2. Let (4.2) hold. There exist ϵ0,ϵ2>0 such that, if

    ¯uϵ2,iMIiu0(x)=¯μ,(u0¯u,v0,ϕ0¯ϕ)(H1)2×H2ϵ0,

    then problem (2.1)-(2.8) has a unique global solution (u,v,ϕ),

    u,vC([0,+);H1(A))C1([0,+);L2(A)),
    ϕC([0,+);H2(A))C1([0,+);L2(A)),

    and, for all iM,

    limt+ui(,t)¯uC(¯Ii),limt+vi(,t)C(¯Ii),limt+ϕi(,t)¯ϕC1(¯Ii)=0 .

    Proof. Let (u,v,ϕ) be the local solution to problem (2.1)-(2.8) and let

    ˆu:=uˉu,ˆv:=v,ˆϕ:=ϕˉϕ;

    we already noticed that (ˆu,ˆv,ˆϕ) is the local solution to system (4.4) complemented by the initial condition (u0¯u,v0,ϕ0¯ϕ) and the same boundary and transmission condition given by (2.3)-(2.8) for system (2.1).

    For suitable ϵ0,ϵ2 the assumptions of Theorem 4.1 are satisfied, then we obtain the uniform boundedness of the functional FT(ˆu,ˆv,ˆϕ), for T[0,+). Hence the set {ˆu(t),ˆv(t),ˆϕ(t)}t[0,+) is uniformly bounded in (H1(A))2×H2(A); thus, if we call Es the set of accumulation points of {ˆu(t),ˆv(t),ˆϕ(t)}ts in (C(A))2×C1(A), then Es is not empty and E:=s0Es.

    Let (U(x),V(x),Φ(x)) be such that, for a sequence tn+,

    limn+iMˆui(,tn)Ui()C(¯Ii)=0,limn+iMˆvi(,tn)Vi()C(¯Ii)=0,limn+iMˆϕi(,tn)Φi()C1(¯Ii)=0 . (4.5)

    In order to identify the limit functions we notice that iMIiUi(x)dx=0, since iMIiˆu(x,tn),dx=0 for all tn.

    Moreover, since ˆviH1(Ii×(0,+)) for all iM, if we set

    ωi(t):=ˆvi(t,)L2(Ii)

    then ωiH1((0,+)) and, as a consequence, limt+ωi(t)=0.

    As limn+ˆvi(,tn)2=Vi()2, we obtain V2=0.

    The same argument can be applied to the functions ˆϕix, for all iM, since they belongs to H1(Ii×(0,+)). Finally, it can be applied to the functions aiˆuibiˆϕi since ˆϕit,ˆϕixx,ˆuix,ˆϕixL2(Ii×(0,+)), thanks to the uniform boundedness of FT(ˆu,ˆv,ˆϕ) and to estimate f) in Proposition 4.1.

    As a consequence we have that

    Vi(x)=0,aiUi(x)biΦi(x)=0,Φi(x)=¯Φi, xIi,

    where ¯Φi are real numbers, so that the limit function is given by (biaiΦi,0,Φi) in each interval Ii, for all iM. It is easily seen that such function is a stationary solution to problem (2.1)-(2.8), which is constant in each arc Ii; in particular it verifies the transmission conditions since (ˆu,ˆv,ˆϕ) verifies them and the convergence result (4.5) holds.

    The condition iMIiUi(x)dx=0 and Remark 3.1 imply that Φi=0 for all iM, so that we can conclude that the unique possible limit function is (U(x),V(x),Φ(x))=(0,0,0); this fact proves the claimed convergence results.

    The main features of the present work are: a) the proof of the existence and uniqueness of stationary solutions with fixed small mass to problem (2.1)-(2.6) considered on acyclic networks; b) the proof of the stability of particular stationary solutions, the constant states on the whole network, when the their masses are small and the quantity aibi does not vary with the index i, for general networks. We can conclude that, in this range of parameters, although the transmission conditions do not impose continuity of the densities at the internal nodes, for suitable initial data the asymptotic profiles of the solutions are continuous functions, constant on the network.

    These results are useful in describing the large time behaviour of small masses of individuals moving on networks driven by chemotaxis.

    For the future, our aim is approaching the same questions when the system (2.1) is complemented by non-null fluxes conditions at the boundaries, which provide models for different situations at the outer nodes, in order to describe the features of the behaviour of cells moving along the arcs searching food. We notice that the condition vi(x)=0 for xIi, for all iM, which prevents the presence of jumps for the density u at the inner nodes, in this case is no longer necessary for stationary solutions on acyclic networks.



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