
Example of acyclic network; the highlighted arcs form the path linking the nodes
This paper approaches the question of existence and uniqueness of stationary solutions to a semilinear hyperbolic-parabolic system and the study of the asymptotic behaviour of global solutions. The system is a model for some biological phenomena evolving on a network composed by a finite number of nodes and oriented arcs. The transmission conditions for the unknowns, set at each inner node, are crucial features of the model.
Citation: Francesca R. Guarguaglini. 2018: Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network, Networks and Heterogeneous Media, 13(1): 47-67. doi: 10.3934/nhm.2018003
[1] | Francesca R. Guarguaglini . Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks and Heterogeneous Media, 2018, 13(1): 47-67. doi: 10.3934/nhm.2018003 |
[2] | Raul Borsche, Axel Klar, T. N. Ha Pham . Nonlinear flux-limited models for chemotaxis on networks. Networks and Heterogeneous Media, 2017, 12(3): 381-401. doi: 10.3934/nhm.2017017 |
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[5] | Avner Friedman . PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7(4): 691-703. doi: 10.3934/nhm.2012.7.691 |
[6] | James Nolen . A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6(2): 167-194. doi: 10.3934/nhm.2011.6.167 |
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[10] | Jérôme Coville, Nicolas Dirr, Stephan Luckhaus . Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks and Heterogeneous Media, 2010, 5(4): 745-763. doi: 10.3934/nhm.2010.5.745 |
This paper approaches the question of existence and uniqueness of stationary solutions to a semilinear hyperbolic-parabolic system and the study of the asymptotic behaviour of global solutions. The system is a model for some biological phenomena evolving on a network composed by a finite number of nodes and oriented arcs. The transmission conditions for the unknowns, set at each inner node, are crucial features of the model.
In this paper we consider a semilinear hyperbolic-parabolic system evolving on a finite planar network composed from nodes connected by
$
\left\{ ∂tui+λi∂xvi=0,∂tvi+λi∂xui=ui∂xϕi−βivi,t≥0,x∈Ii,i=1,...,m;∂tϕi=Di∂xxϕi+aiui−biϕi, \right.$
|
(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
$\Vert f\Vert_2: = \sum\limits_{i \in \mathcal M}\Vert f_i\Vert_2, \ \Vert f\Vert_{H^s}: = \sum\limits_{i \in \mathcal M} \Vert f_i\Vert_{H^s}.$ |
We consider the evolution of the following one-dimensional problem on the graph
$
\left\{ ∂tui+λi∂xvi=0,∂tvi+λi∂xui=ui∂xϕi−βivi,t≥0, x∈Ii, i∈M,∂tϕi=Di∂xxϕi+aiui−biϕi, \right.$
|
(2.1) |
where
We complement the system with the initial conditions
$ u_{i0}, v_{i0}\in H^1(I_i), \ \phi_{i0}\in H^2(I_i) \ \textrm{ for } i\in\mathcal{M}\ ;$ | (2.2) |
the boundary conditions at each outer point
$ v_{i(j)}(e_j, t) = 0, \;\;\;\; t > 0, \ j\in\mathcal J, $ | (2.3) |
$ {\phi_{i(j)}}_x(e_j, t) = 0 \;\;\;\; t > 0 , \ j\in \mathcal J .$ | (2.4) |
In addition, at each inner node
$\left\{Diϕix(Nν,t)=∑j∈Mνανij(ϕj(Nν,t)−ϕi(Nν,t)), i∈Iν, t>0,−Diϕix(Nν,t)=∑j∈Mνανij(ϕj(Nν,t)−ϕi(Nν,t)), i∈Oν, t>0,ανij≥0, ανij=ανji for all i,j∈Mν, \right.$
|
(2.5) |
which imply the continuity of the flux at each node, for all
$\sum\limits_{i \in {\mathcal{I}^\nu }} {{D_i}} {\phi _{{i_x}}}({N_\nu },t) = \sum\limits_{i \in {\mathcal{O}^\nu }} {{D_i}} {\phi _{{i_x}}}({N_\nu },t)\;.$ |
For the unknonws
$\left\{−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 . \right.\
$
|
(2.6) |
These conditions ensure the conservation of the flux of the density of cells at each node
$ \sum\limits_{i \in {\mathcal{I}^\nu }} {\lambda _i}{v_i}(N_\nu, t) = \sum\limits_{i \in {\mathcal{O}^\nu }}{\lambda _i}{v_i}(N_\nu, t), $ |
which corresponds to the conservation of the total mass
$ \sum\limits_{i \in \mathcal{M}} {\int_{{I_i}} {{u_i}(x,t)\;dx} } = \sum\limits_{i \in \mathcal{M}} {\int_{{I_i}} {{u_{0i}}(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
$ u_{i0}, v_{i0}, \phi_{i0}\textrm{ satisfy conditions (2.3)-(2.6) for all } i\in\mathcal{M}\ . $ | (2.7) |
Existence and uniqueness of local solutions to problem (2.1)-(2.7),
$u, v \in C([0, T];H^1(\mathcal A))\cap C^1([0, T];L^2(\mathcal A)), \phi \in C([0, T];H^2(\mathcal A))\cap C^1([0, T];L^2(\mathcal 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
$ \textrm{ for all }\nu\in\mathcal P, \textrm{ for some } k\in\mathcal{M}^\nu, K_{ik}^\nu\neq 0 \textrm{ for all } i\in\mathcal{M}^\nu, i\neq 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
$ \mu : = \sum\limits_{i \in \mathcal{M}} {\int_{{I_i}} {{u_i}(x)\;dx} } \ge 0, $ |
in the case of an acyclic network (see Section 2). We look for stationary solutions
Obviously, the flux
$\mathcal Q = \{ \nu\in \mathcal P: N_\nu \textrm{ is linked to } N_\mu \textrm{ by a path not covering } I_j\} \ $ |
(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
$ \sum\limits_{\nu\in\mathcal Q\cup \{\mu\}} \left( \sum\limits_{i\in I^\nu} {\lambda _i}{v_i}(N_\nu) -\sum\limits_{i\in O^\nu} {\lambda _i}{v_i}(N_\nu)\right) = 0\ .$ |
Since, for all
$v_j(N_\mu) = 0\ ;$ |
then
The previous result implies that stationary solutions must have the form
$\left\{ λiuix=uiϕix,−Diϕixx+biϕi=aiui, \right.x\in {{I}_{i}},\ i\in \mathcal{M},\ t > 0,\text{ } $
|
(3.1) |
with the boundary condition at each outer point
$ {\phi_{i(j)}}_x(e_j, t) = 0 \;\;\;\; t > 0 , $ | (3.2) |
and the transmission conditions, at each inner node
$ \sum\limits_{j \in {\mathcal{M}^\nu }} {K_{ij}^\nu ({u_j}({N_\nu }) - {u_i}({N_\nu }))} = 0,\;\;\;\;i \in {\mathcal{M}^\nu }, $ | (3.3) |
$ \label{stcf}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
$ u_j(N_\nu) = u_k(N_\nu)\;\;\;\; \textrm{ for all } j\in\mathcal{M}^\nu.$ |
Now we fix
$ \sum\limits_{i \in \mathcal{M}} {\int_{{I_i}} {{u_i}(x)dx} } = {\mu _0}\;; $ | (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
$\left\{−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.
\right.
$
|
(3.7) |
We consider the linear operator
$ \label{A2}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
$ A \phi = F (\phi, 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
$A \phi = F (f, C^f), $ | (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
$ \sum\limits_{i \in \mathcal{M}} C_i^f \int_{{I_i}}\exp{ \left( \frac {f_i(x)} {\lambda_i}\right)} dx = \mu_0 , $ | (3.12) |
then
$\Vert \phi_{x}\Vert_{\infty} \leq \frac {2 \max \{a_i\}_{i\in\mathcal{M}}}{\min \{D_i\}_{i\in\mathcal{M}}} \mu_0 ; $ | (3.13) |
if (3.12) holds and
$ \Vert f_{x}\Vert_{\infty} \leq \frac {2 \max\{a_i\}_{i\in\mathcal{M}} }{\min\{D_i\}_{i\in\mathcal{M}} }\mu_0 , $ | (3.14) |
then there exists a quantity
$ \label{12} \Vert \phi \Vert_{W^{2, 1}\;\;(\mathcal{A})}, \Vert \phi \Vert_{H^2(\mathcal{A})} \leq K_{\mu_0}\ . $ | (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
$\sum\limits_{i \in \mathcal{M}} b_i\int_{{I_i}} {\phi _i}(x) \Gamma(-{\phi _i}(x)) dx \geq 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
$\sum\limits_{i \in \mathcal{M}} b_i\int_{{I_i}} {\phi _i}(x) dx = \sum\limits_{i \in \mathcal{M}} \int_{{I_i}} F_i(f(x), C^f) dx $ | (3.17) |
which implies
$ \Vert \phi\Vert_1 \leq \frac {\max\{a_i\}}{\min\{b_i\}} \mu_0\ .$ | (3.18) |
In order to obtain (3.13), first we notice that, if
$|{D_j}{\phi _j}_x(x)| \le \int_{{I_j}} {{D_j}} \left| {{\phi _j}_{yy}(y)} \right|dy \le \int_{{I_j}} {\left( {{b_j}{\phi _j}(y) + C_j^f{a_j}\exp \left( {\frac{{{f_j}(y)}}{{{\lambda _j}}}} \right)} \right)} dy\;.$ |
Then we consider an internal arc
$\mathcal Q = \{ \nu\in \mathcal P: N_\nu \textrm{ is linked to } N_\mu \textrm{ by a path not covering } I_j\} , $ |
$\mathcal S = \{ i\in \mathcal M: I_i \textrm{ is incident with } N_l \textrm{ for some }l \in\mathcal Q\}\ $ |
(see Fig. 1: if, for example,
$ \sum\limits_{\nu\in\mathcal Q\cup \{\mu\}} \left( \sum\limits_{i\in I^\nu} D_i{\phi _{{i_x}}}(N_\nu) -\sum\limits_{i\in O^\nu} D_i{\phi _{{i_x}}}(N_\nu)\right) = 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
$ \Vert \phi_x\Vert_1 \leq \frac {2 \max \{a_i\}_{i\in\mathcal{M}}}{\min \{D_i\}_{i\in\mathcal{M}}} |\mathcal A| \mu_0 , $ | (3.20) |
and
$ \Vert \phi_{x}\Vert_2 \leq \frac {2 \max \{a_i\}_{i\in\mathcal{M}}}{\min \{D_i\}_{i\in\mathcal{M}}} |\mathcal{A}|^{\frac 1 2} \mu_0\ ; $ | (3.21) |
moreover, by Sobolev embedding theorem, we obtain
$\Vert \phi\Vert_{\infty} \leq K_1 \mu_0, $ | (3.22) |
where
The estimates for the function
$ \Vert \phi_{xx}\Vert_1 \leq \frac{2 \max\{a_i\}_{i\in\mathcal{M}}}{\min\{D_i\}_{i\in\mathcal{M}}}\mu_0\ ; $ | (3.23) |
then, using (3.11), we have
$\sum\limits_{i \in \mathcal{M}} \frac {D_i^2}{b_i} \int_{{I_i}}{\phi _{{i_{xx}}}}^2(x) \ dx \leq \sum\limits_{i \in \mathcal{M}} \frac {\Vert F_i\Vert_{\infty}}{b_i} \int_{{I_i}} F_i(x)\ dx \leq \frac { \max \{a_i \Vert F_i\Vert_{\infty}\}}{\min\{b_i\}}\mu_0\ $ |
and the embedding of
$ \sum\limits_{i \in \mathcal{M}} \int_{{I_i}}{\phi _{{i_{xx}}}}^2(x) \ dx \leq K_2 (1+\Vert {f}_x\Vert_\infty) \mu_0^2, $ | (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
$\left(C_i {\exp \left( {\frac{{{\phi _i}(x)}}{{{\lambda _i}}}} \right)}, 0, {\phi _i}(x)\right)\ \ i\in \mathcal{M}, $ |
where
Proof. First we notice that, if a stationary solution
We are going to use a fixed point technique. Given
$u_i^0(x) = C^{\phi^0}_i \exp \left(\frac {\phi_i^0(x)} {\lambda_i}\right), $ |
where the constants
$ C^{\phi^0}_j \exp \left( {\frac{{\phi _j^0\left( {{N_\nu }} \right)}}{{{\lambda _j}}}} \right) = C^{\phi^0}_i\exp \left( {\frac{{\phi _i^0\left( {{N_\nu }} \right)}}{{{\lambda _i}}}} \right) , \;\;\;\; i, j \in\mathcal M^\nu, \;\;\;\; \nu\in\mathcal P, $ | (3.25) |
$\sum\limits_{i \in \mathcal{M}} C^{\phi^0}_i\int_{{I_i}} \exp \left( {\frac{{\phi _i^0\left( x \right)}}{{{\lambda _i}}}} \right) dx = \mu_0\ . $ | (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
$u^0_1(x): = \alpha \exp\left(\frac {\phi_1^0(x)} {\lambda_1}\right)\ .$ |
Let
$u^0_j(x): = \alpha \exp\left(\frac {\phi_1^0(N_\mu)}{ \lambda_1}\right)\exp\left( -\frac {\phi_j^0(N_\mu)}{\lambda_j}\right)\exp\left(\frac {\phi_j^0(x)} {\lambda_j}\right) \ \textrm{ for all } j\in\mathcal M^\mu, j\neq 1;$ |
i.e. we set
This procedure can be iterated at each node reached by one of the arcs
$ \mathcal E_h(\phi^0) : = \frac {\Pi_{i = 1, ..., h-1} \exp \left( \frac {\phi^0_i(N_i)} {{{\lambda _i}}} \right) } {\Pi_{i = 1, ..., h-1} \exp \left( \frac {\phi^0_{i+1}(N_i)} {\lambda_{i+1}} \right) }, $ |
we define
$u^0_h(x) : = \alpha\mathcal E_h(\phi^0) \exp\left(\frac { \phi^0_h(x)} {\lambda_h} \right) \ . $ |
The quantity
$\alpha\sum\limits_{i \in \mathcal{M}} \mathcal E_i(\phi^0)\int_{{I_i}} \exp \left( {\frac{{\phi _i^0\left( x \right)}}{{{\lambda _i}}}} \right) dx = \mu_0, $ |
so that, for all
$ u^0_i(x) = C_i^{\phi^0} \exp\left(\frac {\phi_i^0(x)} {\lambda_i}\right), \ \ C_i^{\phi^0}: = \mu_0 \frac {\ \mathcal E_i(\phi^0) } { \ \sum\limits_{j \in \mathcal{M}} \mathcal E_j(\phi^0) \int_{I_j} \exp \left( {\frac{{\phi _j^0\left( x \right)}}{{{\lambda _j}}}} \right) dx \ }. $ | (3.27) |
Let
$A\phi^1 = F(C^{\phi^0}, \phi^0)\ ;$ |
let
$B_{\mu_0}: = \{ \phi\in D(A) : \phi\geq 0, \Vert \phi_{x}\Vert_{\infty} \leq \frac {2 \max \{a_i\}_{i\in\mathcal{M}}}{\min \{D_i\}_{i\in\mathcal{M}}} \mu_0, \Vert \phi \Vert_{H^2} \leq K_{ \mu_0}\ \}$ |
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
$\sum\limits_{i \in \mathcal{M}} \Vert {{\phi _i}}^1 -\overline {{\phi _i}}^1\Vert_{H^2} \leq K(a_i, b_i, D_i) \sum\limits_{i \in \mathcal{M}} \Vert u^0_i -\overline {u}_i^0\Vert_{2} , $ | (3.29) |
We set
$ J_i^{\phi^0}: = \int_{I_i}\exp \left( {\frac{{\phi _i^0\left( x \right)}}{{{\lambda _i}}}} \right) dx, \;\;\;\;E_i^{\phi^0}(x): = \exp \left( {\frac{{\phi _i^0\left( x \right)}}{{{\lambda _i}}}} \right)\ \ ;$ |
we have
$ \left| {u_i^0(x) - \bar u_i^0(x)} \right| = {\mu _0}\left| {\frac{{\;{\varepsilon _i}({\phi ^0})E_i^{{\phi ^0}}(x)\;}}{{\;\sum\limits_{j \in \mathcal{M}} {{\varepsilon _j}({\phi ^0})J_j^{{\phi ^0}}} \;}} - \frac{{\;{\varepsilon _i}({{\bar \phi }^0})E_i^{{{\bar \phi }^0}}(x)\;}}{{\;\sum\limits_{j \in \mathcal{M}} {{\varepsilon _j}({{\bar \phi }^0})J_j^{{{\bar \phi }^0}}} \;}}} \right|. $ | (3.30) |
In order to treat the above quantity we have to consider that, for all
$ \mathop {\max }\limits_{{I_i}} E_i^g(x) \le {K_6},\;\;\;\;J_i^g \le {K_6}\left| {{I_i}} \right|, $ |
$ \left| {E_i^{{\phi ^0}}(x) - E_i^{{{\bar \phi }^0}}(x)} \right| \le {K_6}\left| {\phi _i^0(x) - \bar \phi _i^0(x)} \right|, $ |
$ \left| {J_i^{{\phi ^0}} - J_i^{{{\bar \phi }^0}}} \right| \le {K_6}\int_{{I_i}} {\left| {\phi _i^0(x) - \bar \phi _i^0(x)} \right|{dx}} . $ |
The above inequalities can be used in (3.30) so that (3.29) implies
$ \sum\limits_{i \in \mathcal{M}} \Vert {{\phi _i}}^1 -\overline {{\phi _i}}^1\Vert_{H^2} \leq \mu_0 K_7(a_i, b_i, D_i, K_{\mu_0}, |\mathcal{A}|) \sum\limits_{i \in \mathcal{M}} \Vert \phi_i^0 -\overline {\phi}_i^0\Vert_{H^1} , $ | (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
$ \sum\limits_{\nu\in\mathcal P}\left( \sum\limits_{i\in\mathcal I^\nu} {\lambda _i} {u_i}(N^\nu){v_i}(N^\nu) - \sum\limits_{i\in\mathcal O^\nu} {\lambda _i} {u_i}(N^\nu){v_i}(N^\nu)\right)\geq 0, $ |
so, by using the first two equations in (2.1), we obtain
$2 \sum\limits_{i \in \mathcal{M}} \beta_i \int_{{I_i}} v_i^2(x)dx \leq \sum\limits_{i \in \mathcal{M}} \Vert u_i\Vert_{\infty} \int_{{I_i}} (v_i^2(x) +\phi^2_{i_x}(x) ) \ dx $ |
and
$ \sum\limits_{i \in \mathcal{M}} {\lambda _i}\int_{{I_i}} u^2_{i_x}(x) \ dx \leq \sum\limits_{i \in \mathcal{M}} \Vert u_i\Vert_{\infty} \int_{{I_i}} (u_{i_x}^2(x) +\phi^2_{i_x}(x) ) \ dx \ + \sum\limits_{i \in \mathcal{M}} \frac {\beta_i^2}{{\lambda _i}} \int_{{I_i}} v^2_i(x)\ dx \ ;$ |
the above inequalities implies the following one
$ \Vert v\Vert_2^2 + \Vert u_x\Vert_2^2\leq K_0 H \left( \Vert \phi_x\Vert_2^2 + \Vert v\Vert_2^2 +\Vert u_x\Vert_2^2 \right), $ | (3.32) |
where
The transmission conditions (2.5) imply that
$ -\sum\limits_{\nu\in\mathcal P}\left( \sum\limits_{i\in\mathcal I^\nu} D_i{\phi _i}(N^\nu){\phi _{{i_x}}}(N^\nu) - \sum\limits_{i\in\mathcal O^\nu} D_i {\phi _i}(N^\nu){\phi _{{i_x}}}(N^\nu)\right)\geq 0\ ;$ |
moreover, the assumption (2.8) imply that, for each
$ u_j(N_\nu) = u_k(N_\nu)+\sum\limits_{i\in \mathcal{M}^\nu, i\neq k} \theta^\nu_{ij} {v_i}(N_\nu) \;\;\;\;\textrm{ for all } j\in\mathcal{M}^\nu, $ |
(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
$ \Vert \phi_x\Vert_2^2 + \Vert \phi_{xx}\Vert_2^2\leq K_1 \left( \Vert v\Vert_2^2 +\Vert u_x\Vert_2^2 \right), $ | (3.33) |
where
By inequalities (3.32) and (3.33) we deduce the following one
$ \Vert v\Vert_2^2 + \Vert u_x\Vert_2^2\leq K_0(1+K_1) H \left( \Vert v\Vert_2^2 +\Vert u_x\Vert_2^2 \right), $ |
which, for small
In the cases when
$ {u_i}(x) = \frac {\mu_0}{|\mathcal{A}|} , \ \ \ \phi_i(x) = \frac{a_i}{b_i} \frac{\mu_0}{|\mathcal{A}|}, \;\;\;i\in\mathcal{M}\ .$ |
Therefore the transmission conditions, for each
$ \sum\limits_{j\in\mathcal{M}^\nu}\alpha^\nu_{ij} \frac{\mu_0}{|\mathcal{A}|} \left( \frac {a_j} {b_j}-\frac {a_i} {b_i}\right) = 0 , \;\;\;i\in\mathcal{M}^\nu , $ |
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
$ \alpha_{11} \frac{\mu_0}{|\mathcal{A}|} \left( \frac {b_2} {a_2}-\frac {b_1} {a_1}\right) = 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
$\label {Q} \frac {a_i}{b_i} = Q\ \ \textrm{for all }i\in \mathcal{M}\ ;$ | (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
$\Vert f_i(t)\Vert_2: = \Vert f_i(\cdot, t)\Vert_{L^2(I_i)}, \ \ \Vert f_i(t)\Vert_{H^s}: = \Vert f_i(\cdot, t)\Vert_{H^s(I_i)} \ . $ |
Now we assume (4.2), we fix
If
$
\left\{ ∂tui+λi∂xvi=0∂tvi+λi∂xui=(ui+¯u)∂xϕi−βivix∈Ii,t≥0,i∈M,∂tϕi=Di∂xxϕi+aiui−biϕi, \right.$
|
(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
$\overline {u}\leq \epsilon_1, \ \ \Vert u_0\Vert_{H^1}, \Vert v_0\Vert_{H^1}, \Vert \phi_{0}\Vert_{H^2}\leq \epsilon_0 , $ |
then there exists a unique global solution
$u, v \in C([0, +\infty);H^1(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})) , $ |
$\phi\in C([0, +\infty);H^2(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})), \ \phi_{x}\in H^1\left( \mathcal A\times (0, +\infty)\right) \ .$ |
Moreover,
Proof. It is sufficient to prove that the functional
We notice that each term in
$F_T^2(u, v, \phi)\leq c_1 F_0^2(u, v, \phi) +c_2 \overline {u} F^2_T(u, v, \phi) +c_3 F_T^3(u, v, \phi), $ |
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
$F_T^2(u, v, \phi)\leq c_4 F_0^2(u, v, \phi) +c_5 F_T^3(u, v, \phi)$ |
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
$\overline {u} \leq \epsilon_2, \;\;\;\sum\limits_{i \in \mathcal{M}}\int_{{I_i}} u_0(x) = \overline \mu, \;\;\;\Vert( u_0-\overline {u}, v_0, \phi_0-\overline {\phi})\Vert_{(H^1)^2\times H^2} \leq \epsilon_0, $ |
then problem (2.1)-(2.8) has a unique global solution
$u, v \in C([0, +\infty);H^1(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})) , $ |
$\phi\in C([0, +\infty);H^2(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})), $ |
and, for all
$ \lim\limits_{t\to+\infty}\Vert {u_i}(\cdot, t)-\overline {u}\Vert_{C(\overline I_i)}, \lim\limits_{t\to+\infty}\Vert {v_i}(\cdot, t)\Vert_{C(\overline I_i)}, \lim\limits_{t\to+\infty}\Vert {\phi _i}(\cdot, t)-\overline {\phi}\Vert_{C^1(\overline I_i)} = 0\ . $ |
Proof. Let
$ \hat u: = u - \bar u,\;\hat v: = v,\;\hat \phi : = \phi - \bar \phi ; $ |
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
$\omega_i(t): = \Vert \hat v_i(t, \cdot)\Vert_{L^2(I_i)}$ |
then
As
The same argument can be applied to the functions
As a consequence we have that
$ V_i(x) = 0, \;\;\;\; a_i U_i(x)- b_i \Phi_i(x) = 0, \;\;\;\; \Phi_i(x) = \overline \Phi_i, \;\;\;\; \ x\in I_i, $ |
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
1. | Francesca Romana Guarguaglini, Marco Papi, Flavia Smarrazzo, Local and global solutions for a hyperbolic–elliptic model of chemotaxis on a network, 2019, 29, 0218-2025, 1465, 10.1142/S021820251950026X | |
2. | Yuta Ishii, Multi-peak solutions for the Schnakenberg model with heterogeneity on star shaped graphs, 2023, 446, 01672789, 133679, 10.1016/j.physd.2023.133679 | |
3. | Francesca R. Guarguaglini, Roberto Natalini, Vanishing viscosity approximation for linear transport equations on finite star-shaped networks, 2021, 21, 1424-3199, 2413, 10.1007/s00028-021-00688-0 | |
4. | Yafeng Li, Chunlai Mu, Qiao Xin, Global existence and asymptotic behavior of solutions for a hyperbolic‐parabolic model of chemotaxis on network, 2022, 45, 0170-4214, 6739, 10.1002/mma.8204 |
Example of acyclic network; the highlighted arcs form the path linking the nodes
Example: the highlighted arcs form the path from the outer point