
Example of acyclic network; the highlighted arcs form the path linking the nodes
Let K=Q(√m) be an imaginary quadratic field with OK its ring of integers. Let π and β be an irreducible element and a nonzero element, respectively, in OK. In the authors' earlier work, it was proved for the cases, m≢1 (mod 4) and m≡1 (mod 4) that if π=αnβn+αn−1βn−1+⋯+α1β+α0=:f(β), where n≥1, αn∈OK∖{0}, α0,…,αn−1 belong to a complete residue system modulo β, and the digits αn−1 and αn satisfy certain restrictions, then the polynomial f(x) is irreducible in OK[x]. In this paper, we extend these results by establishing further irreducibility criteria for polynomials in OK[x]. In addition, we provide elements of β that can be applied to the new criteria but not to the previous ones.
Citation: Phitthayathon Phetnun, Narakorn R. Kanasri. Further irreducibility criteria for polynomials associated with the complete residue systems in any imaginary quadratic field[J]. AIMS Mathematics, 2022, 7(10): 18925-18947. doi: 10.3934/math.20221042
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Let K=Q(√m) be an imaginary quadratic field with OK its ring of integers. Let π and β be an irreducible element and a nonzero element, respectively, in OK. In the authors' earlier work, it was proved for the cases, m≢1 (mod 4) and m≡1 (mod 4) that if π=αnβn+αn−1βn−1+⋯+α1β+α0=:f(β), where n≥1, αn∈OK∖{0}, α0,…,αn−1 belong to a complete residue system modulo β, and the digits αn−1 and αn satisfy certain restrictions, then the polynomial f(x) is irreducible in OK[x]. In this paper, we extend these results by establishing further irreducibility criteria for polynomials in OK[x]. In addition, we provide elements of β that can be applied to the new criteria but not to the previous ones.
In this paper we consider a semilinear hyperbolic-parabolic system evolving on a finite planar network composed from nodes connected by
{∂tui+λi∂xvi=0,∂tvi+λi∂xui=ui∂xϕi−βivi,t≥0,x∈Ii,i=1,...,m;∂tϕi=Di∂xxϕi+aiui−biϕi, | (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
Starting from the Keller-Segel paper [18] in 1970 until now, a lot of articles have been devoted to PDE models in domains of
In [11] the Cauchy and the Neumann problems for the system in (1.1), respectively in
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
For general networks and the parameters
Finally we study the large time behaviour of global solutions on general networks, when the ratio between
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
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,
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
Each node is a point of the plane and each oriented arc
We use
Moreover, we use
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).
Each arc
We set
‖f‖2:=∑i∈M‖fi‖2, ‖f‖Hs:=∑i∈M‖fi‖Hs. |
We consider the evolution of the following one-dimensional problem on the graph
{∂tui+λi∂xvi=0,∂tvi+λi∂xui=ui∂xϕi−βivi,t≥0, x∈Ii, i∈M,∂tϕi=Di∂xxϕi+aiui−biϕi, | (2.1) |
where
We complement the system with the initial conditions
ui0,vi0∈H1(Ii), ϕi0∈H2(Ii) for i∈M ; | (2.2) |
the boundary conditions at each outer point
vi(j)(ej,t)=0,t>0, j∈J, | (2.3) |
ϕi(j)x(ej,t)=0t>0, j∈J. | (2.4) |
In addition, at each inner node
{Diϕix(Nν,t)=∑j∈Mνανij(ϕj(Nν,t)−ϕi(Nν,t)), i∈Iν, t>0,−Diϕix(Nν,t)=∑j∈Mνανij(ϕj(Nν,t)−ϕi(Nν,t)), i∈Oν, t>0,ανij≥0, ανij=ανji for all i,j∈Mν, | (2.5) |
which imply the continuity of the flux at each node, for all
∑i∈IνDiϕix(Nν,t)=∑i∈OνDiϕix(Nν,t). |
For the unknonws
{−λivi(Nν,t)=∑j∈MνKνij(uj(Nν,t)−ui(Nν,t)), i∈Iν, t>0,λivi(Nν,t)=∑j∈MνKνij(uj(Nν,t)−ui(Nν,t)), i∈Oν, t>0,Kνij≥0, Kνij=Kνji for all i,j∈Mν . | (2.6) |
These conditions ensure the conservation of the flux of the density of cells at each node
∑i∈Iνλivi(Nν,t)=∑i∈Oνλivi(Nν,t), |
which corresponds to the conservation of the total mass
∑i∈M∫Iiui(x,t)dx=∑i∈M∫Iiu0i(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 i∈M . | (2.7) |
Existence and uniqueness of local solutions to problem (2.1)-(2.7),
u,v∈C([0,T];H1(A))∩C1([0,T];L2(A)),ϕ∈C([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
for all ν∈P, for some k∈Mν,Kνik≠0 for all i∈Mν,i≠k . | (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:=∑i∈M∫Iiui(x)dx≥0, |
in the case of an acyclic network (see Section 2). We look for stationary solutions
Obviously, the flux
Q={ν∈P:Nν is linked to Nμ by a path not covering Ij} |
(see Fig. 1: if, for example,
At each node the conservation of the flux of the density of cells, stated in Section 2, holds; then
∑ν∈Q∪{μ}(∑i∈Iνλivi(Nν)−∑i∈Oνλivi(Nν))=0 . |
Since, for all
vj(Nμ)=0 ; |
then
The previous result implies that stationary solutions must have the form
{λiuix=uiϕix,−Diϕixx+biϕi=aiui,x∈Ii, i∈M, t>0, | (3.1) |
with the boundary condition at each outer point
ϕi(j)x(ej,t)=0t>0, | (3.2) |
and the transmission conditions, at each inner node
∑j∈MνKνij(uj(Nν)−ui(Nν))=0,i∈Mν, | (3.3) |
Diϕix(Nν)=∑j∈Mνανij(ϕj(Nν)−ϕi(Nν)),i∈Iν,Diϕix(Nν)=−∑j∈Mνανij(ϕj(Nν)−ϕi(Nν)),i∈Oν. | (3.4) |
For each fixed inner node
0=∑j∈Mν,j≠iKνij(uj(Nν)−ui(Nν))=∑j∈Mν,j≠i,kKνij(uj(Nν)−uk(Nν))−(∑j∈Mν,j≠iKνij)(ui(Nν)−uk(Nν)) ; | (3.5) |
the assumptions on
uj(Nν)=uk(Nν) for all j∈Mν. |
Now we fix
∑i∈M∫Iiui(x)dx=μ0; | (3.6) |
notice that for the evolution problem, the quantity
Integrating the first equation in (3.1) we can rewrite problem (3.1)-(3.6) as the following elliptic problem on network:
Find
{−Diϕixx+biϕi=aiuix∈Ii,i∈M,ui(x)=Ciexp(ϕi(x)λi)x∈Ii,i∈M,ϕi(j)x(ej)=0,j∈J,Diϕix(Nν)=∑j∈Mνανij(ϕj(Nν)−ϕi(Nν)),i∈Iν,ν∈P,Diϕix(Nν)=−∑j∈Mνανij(ϕj(Nν)−ϕi(Nν)),i∈Oν,ν∈P,Cjexp(ϕj(Nν)λj)=Ciexp(ϕi(Nν)λi),i,j∈Mν,ν∈P,∑i∈MCi∫Iiexp(ϕi(x)λi)dx=μ0. | (3.7) |
We consider the linear operator
D(A)={ϕ∈H2(A):(3.2),(3.4) hold },A(ϕ)={−Diϕixx+biϕi}i∈M ; | (3.8) |
then the equation in (3.7) and the boundary and transmission conditions for
Aϕ=F(ϕ,C), | (3.9) |
where, for
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
The existence and uniqueness of the solution
The transmission conditions (2.5) imply the following equality which will be useful in the next proofs:
∑i∈M∫IiDi(ϕi(x)ϕix(x))x dx=∑ν∈P(∑i∈IνDiϕi(Nν)ϕix(Nν)−∑i∈OνDiϕi(Nν))ϕix(Nν))=∑ν∈P ∑ij∈Mνανijϕi(Nν)(ϕj(Nν)−ϕi(Nν)) =−12∑ν∈P ∑ij∈Mνανij(ϕj(Nν)−ϕi(Nν))2 . | (3.11) |
Let
Lemma 3.1. Let
∑i∈MCfi∫Iiexp(fi(x)λi)dx=μ0, | (3.12) |
then
‖ϕx‖∞≤2max{ai}i∈Mmin{Di}i∈Mμ0; | (3.13) |
if (3.12) holds and
‖fx‖∞≤2max{ai}i∈Mmin{Di}i∈Mμ0, | (3.14) |
then there exists a quantity
‖ϕ‖W2,1(A),‖ϕ‖H2(A)≤Kμ0 . | (3.15) |
Proof. Let consider a function
∑i∈M∫Ii(−Di(ϕix(x)Γ(−ϕi(x)))x−DiΓ′(−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
∑i∈M∫IiDi(Γ(−ϕi)ϕix)x=−12∑ν∈P ∑ij∈Mνανij(ϕj(Nν)−ϕi(Nν))(Γ(−ϕj(Nν))−Γ(−ϕi(Nν)))≥0 ; | (3.16) |
the above inequality and the non-negativity of
∑i∈Mbi∫Iiϕi(x)Γ(−ϕi(x))dx≥0, |
so that, thanks to the properties of
By integration of the equation (3.10), taking into account (3.4) and (3.2), we obtain
∑i∈Mbi∫Iiϕi(x)dx=∑i∈M∫IiFi(f(x),Cf)dx | (3.17) |
which implies
‖ϕ‖1≤max{ai}min{bi}μ0 . | (3.18) |
In order to obtain (3.13), first we notice that, if
|Djϕjx(x)|≤∫IjDj|ϕjyy(y)|dy≤∫Ij(bjϕj(y)+Cfjajexp(fj(y)λj))dy. |
Then we consider an internal arc
Q={ν∈P:Nν is linked to Nμ by a path not covering Ij}, |
S={i∈M:Ii is incident with Nl for some l∈Q} |
(see Fig. 1: if, for example,
∑ν∈Q∪{μ}(∑i∈IνDiϕix(Nν)−∑i∈OνDiϕix(Nν))=0 . |
Let
|Djϕjx(x)|=|Djϕjx(x)+∑ν∈Q∪{μ}(∑i∈IνDiϕix(Nν)−∑i∈OνDiϕix(Nν))||∑i∈S∫IiDiϕiyy(y)dy+∫IxjDjϕjyy(y)dy||∑i∈S∫Ii(biϕi(y)−Cfiaiexp(fi(y)λi))dy+∫Ixj(bjϕj(y)−Cfjajexp(fj(y)λj))dy|. | (3.19) |
Then
‖ϕx‖1≤2max{ai}i∈Mmin{Di}i∈M|A|μ0, | (3.20) |
and
‖ϕx‖2≤2max{ai}i∈Mmin{Di}i∈M|A|12μ0 ; | (3.21) |
moreover, by Sobolev embedding theorem, we obtain
‖ϕ‖∞≤K1μ0, | (3.22) |
where
The estimates for the function
‖ϕxx‖1≤2max{ai}i∈Mmin{Di}i∈Mμ0 ; | (3.23) |
then, using (3.11), we have
∑i∈MD2ibi∫Iiϕixx2(x) dx≤∑i∈M‖Fi‖∞bi∫IiFi(x) dx≤max{ai‖Fi‖∞}min{bi}μ0 |
and the embedding of
∑i∈M∫Iiϕixx2(x) dx≤K2(1+‖fx‖∞)μ20, | (3.24) |
where
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
(Ciexp(ϕi(x)λi),0,ϕi(x)) i∈M, |
where
Proof. First we notice that, if a stationary solution
We are going to use a fixed point technique. Given
u0i(x)=Cϕ0iexp(ϕ0i(x)λi), |
where the constants
Cϕ0jexp(ϕ0j(Nν)λj)=Cϕ0iexp(ϕ0i(Nν)λi),i,j∈Mν,ν∈P, | (3.25) |
∑i∈MCϕ0i∫Iiexp(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
In order to give an explicit expression for the coefficients
u01(x):=αexp(ϕ01(x)λ1) . |
Let
u0j(x):=αexp(ϕ01(Nμ)λ1)exp(−ϕ0j(Nμ)λj)exp(ϕ0j(x)λj) for all j∈Mμ,j≠1; |
i.e. we set
This procedure can be iterated at each node reached by one of the arcs
Eh(ϕ0):=Πi=1,...,h−1exp(ϕ0i(Ni)λi)Πi=1,...,h−1exp(ϕ0i+1(Ni)λi+1), |
we define
u0h(x):=αEh(ϕ0)exp(ϕ0h(x)λh) . |
The quantity
α∑i∈MEi(ϕ0)∫Iiexp(ϕ0i(x)λi)dx=μ0, |
so that, for all
u0i(x)=Cϕ0iexp(ϕ0i(x)λi), Cϕ0i:=μ0 Ei(ϕ0) ∑j∈MEj(ϕ0)∫Ijexp(ϕ0j(x)λj)dx . | (3.27) |
Let
Aϕ1=F(Cϕ0,ϕ0) ; |
let
Bμ0:={ϕ∈D(A):ϕ≥0,‖ϕx‖∞≤2max{ai}i∈Mmin{Di}i∈Mμ0,‖ϕ‖H2≤Kμ0 } |
equipped with the distance
We consider
bi∫Ii(ϕ1i(x)−¯ϕ1i(x))2,dx+Di∫Ii(ϕ1ix(x)−¯ϕ1ix(x))2,dx−Di∫Ii((ϕ1ix(x)−¯ϕ1ix(x))(ϕ1i(x)−¯ϕ1i(x)))xdx=ai∫Ii(u0i(x)−¯u0i(x))(ϕ1i(x)−¯ϕ1i(x))dx ; | (3.28) |
using (3.11), from (3.28) we infer that
∑i∈M‖ϕi1−¯ϕi1‖H2≤K(ai,bi,Di)∑i∈M‖u0i−¯u0i‖2, | (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)∑j∈Mεj(ϕ0)Jϕ0j−εi(ˉϕ0)Eˉϕ0i(x)∑j∈Mεj(ˉϕ0)Jˉϕ0j|. | (3.30) |
In order to treat the above quantity we have to consider that, for all
maxIiEgi(x)≤K6,Jgi≤K6|Ii|, |
|Eϕ0i(x)−Eˉϕ0i(x)|≤K6|ϕ0i(x)−ˉϕ0i(x)|, |
|Jϕ0i−Jˉϕ0i|≤K6∫Ii|ϕ0i(x)−ˉϕ0i(x)|dx. |
The above inequalities can be used in (3.30) so that (3.29) implies
∑i∈M‖ϕi1−¯ϕi1‖H2≤μ0K7(ai,bi,Di,Kμ0,|A|)∑i∈M‖ϕ0i−¯ϕ0i‖H1, | (3.31) |
where
Let
For any constant
Proposition 3.1. Let
Remark 3.1. For general networks, when the value of
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
Proposition 3.2. Let
Proof. We set
∑ν∈P(∑i∈Iνλiui(Nν)vi(Nν)−∑i∈Oνλiui(Nν)vi(Nν))≥0, |
so, by using the first two equations in (2.1), we obtain
2∑i∈Mβi∫Iiv2i(x)dx≤∑i∈M‖ui‖∞∫Ii(v2i(x)+ϕ2ix(x)) dx |
and
∑i∈Mλi∫Iiu2ix(x) dx≤∑i∈M‖ui‖∞∫Ii(u2ix(x)+ϕ2ix(x)) dx +∑i∈Mβ2iλi∫Iiv2i(x) dx ; |
the above inequalities implies the following one
‖v‖22+‖ux‖22≤K0H(‖ϕx‖22+‖v‖22+‖ux‖22), | (3.32) |
where
The transmission conditions (2.5) imply that
−∑ν∈P(∑i∈IνDiϕi(Nν)ϕix(Nν)−∑i∈OνDiϕi(Nν)ϕix(Nν))≥0 ; |
moreover, the assumption (2.8) imply that, for each
uj(Nν)=uk(Nν)+∑i∈Mν,i≠kθνijvi(Nν) for all j∈Mν, |
(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
‖ϕx‖22+‖ϕxx‖22≤K1(‖v‖22+‖ux‖22), | (3.33) |
where
By inequalities (3.32) and (3.33) we deduce the following one
‖v‖22+‖ux‖22≤K0(1+K1)H(‖v‖22+‖ux‖22), |
which, for small
In the cases when
ui(x)=μ0|A|, ϕi(x)=aibiμ0|A|,i∈M . |
Therefore the transmission conditions, for each
∑j∈Mνανijμ0|A|(ajbj−aibi)=0,i∈Mν, |
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
α11μ0|A|(b2a2−b1a1)=0, |
cannot be satisfied.
Hence, in the cases when
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,v∈C([0,+∞);H1(A))∩C1([0,+∞);L2(A)),ϕ∈C([0,+∞);H2(A))∩C1([0,+∞);L2(A)),ϕx∈H1(A×(0,+∞)), | (4.1) |
to such problem is proved in [12], when the initial data are sufficiently small in
aibi=Q for all i∈M ; | (4.2) |
in particular it is proved that the functional
F2T(u,v,ϕ):=∑i∈M(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
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
If
{∂tui+λi∂xvi=0∂tvi+λi∂xui=(ui+¯u)∂xϕi−βivix∈Ii,t≥0,i∈M,∂tϕi=Di∂xxϕi+aiui−biϕi, | (4.4) |
complemented with the conditions (2.2)-(2.8) and initial data
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
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
Proposition 4.1. Let
u,v∈C([0,T];H1(A))∩C1([0,T];L2(A)),ϕ∈C([0,T];H2(A))∩C1([0,T];L2(A)), ϕx∈H1(A.×(0,T)) ; |
then
a)
∑i∈M(sup[0,T]‖ui(t)‖22+sup[0,T]‖vi(t)‖22+βi∫T0‖vi(t)‖22dt)≤C∑i∈M(‖u0i‖22+‖v0i‖22)+C∑i∈M(sup[0,T]‖ui(t)‖H1+ˉu)∫To(‖ϕix(t)‖22+‖vi(t)‖22)dt; |
b)
∑i∈M(sup[0,T]‖vix(t)‖22+sup[0,T]‖vit(t)‖22+∫T0‖vit(t)‖22,dt)≤C(‖v0‖2H1+‖u0‖2H1‖ϕ0‖2H2)+C∑i∈M(sup[0,T]‖ui(t)‖H1+ˉu)∫T0(ϕixt(t)‖22+‖vit(t)‖22) dt+C∑i∈Msup[0,T]‖ϕx(t)‖H1∫T0(‖vit(t)‖22+‖vi(t)‖2H1)dt ; |
c)
∑i∈Msup[0,T]‖uix(t)‖22≤C∑i∈M(sup[0,T]‖vit(t)‖22+sup[0,T]‖vi(t)‖22)+C∑i∈M(sup[0,T]‖ui(t)‖H1+ˉu)(sup[0,T]‖uix(t)‖22+sup[0,T]‖ϕix(t)‖22) ; |
d)
∑i∈M∫T0‖uix(t)‖22,dt≤C∑i∈M∫T0(‖vit(t)‖22+‖vi(t)‖22)dt+C∑i∈M(sup[0,T]‖ui(t)‖H1+ˉu)∫T0(‖uix(t)‖22+‖ϕix(t)‖22)dt ; |
e)
∑i∈M∫T0‖vix(t)‖22,dt≤C∑i∈M(‖v0i‖22+‖u0i‖2H1(1+‖ϕ0i‖2H1))+C∑i∈M(∫T0‖vit(t)‖22,dt+sup[0,T]‖vit(t)‖22)+C∑i∈M(sup[0,T]‖ui(t)‖H1+sup[0,T]‖ϕix(t)‖H1+ˉu)×∫T0(‖vi(t)‖2H1+‖ϕixt(t)‖22)dt ; |
f)
∑i∈M(sup[0,T]‖ϕit(t)‖22+∫T0(‖ϕit(t)‖22+‖ϕitx(t)‖22) dt)≤C∑i∈M(‖ϕ0i‖2H2+‖u0i‖22+∫T0‖uit(t)‖22) ; |
g)
∑i∈M(sup[0,T]‖ϕixx(t)‖22+sup[0,T]‖ϕix(t)‖22)≤C∑i∈M(sup[0,T]‖ϕit(t)‖22+sup[0,T]‖ui(t)‖22) ; |
h) if (2.8) and (4.2) hold, then
∑i∈M∫T0(‖ϕix(t)‖22+‖ϕixx(t)‖22) dt≤C∑i∈M∫T0(‖uix(t)‖22+‖vi(t)‖2H1+‖ϕit(t)‖22) dt, |
for suitable costants
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
Theorem 4.1. Let (4.2) hold. There exists
¯u≤ϵ1, ‖u0‖H1,‖v0‖H1,‖ϕ0‖H2≤ϵ0, |
then there exists a unique global solution
u,v∈C([0,+∞);H1(A))∩C1([0,+∞);L2(A)), |
ϕ∈C([0,+∞);H2(A))∩C1([0,+∞);L2(A)), ϕx∈H1(A×(0,+∞)) . |
Moreover,
Proof. It is sufficient to prove that the functional
We notice that each term in
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
If
F2T(u,v,ϕ)≤c4F20(u,v,ϕ)+c5F3T(u,v,ϕ) |
for suitable positive constants
It is easy to verify that, if
Then we can conclude that, if
The above result, in particular the uniform, in time, boundedness of the functional
Let (4.2) hold and let
Theorem 4.2. Let (4.2) hold. There exist
¯u≤ϵ2,∑i∈M∫Iiu0(x)=¯μ,‖(u0−¯u,v0,ϕ0−¯ϕ)‖(H1)2×H2≤ϵ0, |
then problem (2.1)-(2.8) has a unique global solution
u,v∈C([0,+∞);H1(A))∩C1([0,+∞);L2(A)), |
ϕ∈C([0,+∞);H2(A))∩C1([0,+∞);L2(A)), |
and, for all
limt→+∞‖ui(⋅,t)−¯u‖C(¯Ii),limt→+∞‖vi(⋅,t)‖C(¯Ii),limt→+∞‖ϕi(⋅,t)−¯ϕ‖C1(¯Ii)=0 . |
Proof. Let
ˆu:=u−ˉu,ˆv:=v,ˆϕ:=ϕ−ˉϕ; |
we already noticed that
For suitable
Let
limn→+∞∑i∈M‖ˆui(⋅,tn)−Ui(⋅)‖C(¯Ii)=0,limn→+∞∑i∈M‖ˆvi(⋅,tn)−Vi(⋅)‖C(¯Ii)=0,limn→+∞∑i∈M‖ˆϕi(⋅,tn)−Φi(⋅)‖C1(¯Ii)=0 . | (4.5) |
In order to identify the limit functions we notice that
Moreover, since
ωi(t):=‖ˆvi(t,⋅)‖L2(Ii) |
then
As
The same argument can be applied to the functions
As a consequence we have that
Vi(x)=0,aiUi(x)−biΦi(x)=0,Φi(x)=¯Φi, x∈Ii, |
where
The condition
The main features of the present work are:
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
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