Citation: Syed Ahtsham Ul Haq Bokhary, Zill-e-Shams, Abdul Ghaffar, Kottakkaran Sooppy Nisar. On the metric basis in wheels with consecutive missing spokes[J]. AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400
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Molecular communication, which is dramatically different from conventional communication, is attracting the attention of a number of researchers these days. Numerous interdisciplinary discussions are taking place among researchers in a variety of fields such as biology, mathematics, statistics, information theory, and so forth. Researchers in molecular communication are also closely interacting with those in another active research area, namely, nanonetworks and body area networks.
Molecular communication is also expected to possess various applications for engineering fields, such as bio-, medical, industrial and communications engineering [21]. Among them, the most important application in the medical field is drug delivery and intercellular therapy [21]. In those processes, a bio-nanomachine detects some target area, such as tumor cells in the human body, and releases drug molecules in an appropriate location, with a suitable volume and timing. Nowadays, many theoretical frameworks concerning the method of target detection in molecular communication networks are being proposed, and implementation and experiments are taking place in laboratories.
One of the key factors for carrying signals and materials in molecular communication is molecule diffusion in the medium. The early research in this direction was carried out by Einolghozati et al. [4][5] and is widely applied in the arguments for the coding and encoding methods in molecular communication networks these days. Another key factor in molecular communication is that it utilizes cooperation between bio-nanomachines to achieve a certain objective. For instance, Nakano et al. [22] recently proposed a model of target detection that imposes two different functionals on bio-nanomachines: leader and follower. The leader nanomachines search for a target in the human body and release attractant molecules upon detecting it. Follower nanomachines move according to the concentration gradient of the attractant toward the source of it, and then release drug molecules.
Nakano's model utilizes the chemotaxis of bio-nanomachines. Investigations in this direction can also be found in several papers [24][25], and these activated the modeling of the behavior of bio-nanomachines based on the Keller-Segel model [17]. The discussions by Nakano et al. [22] were limited to those of agent-based simulations and experiments. Recently, Iwasaki, Yang and Nakano [15] proposed a theoretical study based on arguments by Okaie and other authors [24][25]. In these models, both the attractant and repellent exist to effectively deliver drug molecules. The attractant plays the role of absorbing bio-nanomachines closer to the target cells, while the repellent plays the role of diffusing bio-nanomachines over an area and enabling them to search for the target over a broader area. Iwasaki, Yang and Nakano focused on the temporal behaviors of the concentration of bio-nanomachines, attractant and repellent.
The proposed model was a coupling of a one-dimensional reaction-diffusion-type partial differential equation and two ordinary differential equations.
It is also to be noted that the proposed model in that paper [15] was based on the variant model of the Keller-Segel equation, but the diffusion terms in attractant and repellent equations are neglected. After showing the existence of a positive stationary solution, Iwasaki numerically verified its stability. However, few theoretical analyses concerning the proposed model were conducted in that research. Besides, their models were limited to only the one-dimensional case.
In this paper, we extend their formulation to the two-dimensional. Although higher dimensions (such as the third dimension) can be discussed in a similar framework, two dimensions seem appropriate since bio-nanomachines move along the tissue surface. We rigorously discuss the well-posedness of the model. More concretely, we discuss the following issues:
(ⅰ) the unique existence of a positive stationary solution,
(ⅱ) the global-in-time solvability of the non-stationary problem under the smallness of data,
(ⅲ) and the non-negativeness of the non-stationary solution.
Our argument in this paper directly applies to the one-dimensional case. The remainder of this paper is organized as follows: In the next section, we give an overview of the existing mathematical arguments. In Section 3, we formulate the problem and, in Section 4, we introduce the notations used throughout this paper. The main results of this paper are stated in Section 5, followed by their proofs in Sections 6 and 7. The final section briefly concludes this paper.
There exist a number of works concerning molecular communication networks these days, and the following arguments are limited to those concerning the theoretical modeling of target detection in the human body. Recently, Nakano et al. [22] proposed a mathematical model that describes the temporal behavior of attractant molecules. In it, they proposed that two functionals are imposed on bio-nanomachines, which they call the leaders and followers. They showed the effectiveness of the proposed method through numerical simulations. They also clarified the situations in which their proposed model works better than the conventional method, but few mathematically rigorous discussions have been obtained.
Following these works, Iwasaki, Yang and Nakano [15] proposed a mathematical model that describes non-diffusion-based mobile molecular communication networks. They focused only on the temporal behavior of the concentration of the attractant, repellent, and bio-nanomachines under the assumption that the concentration of the target is time invariant. A similar model was discussed in a previous paper [14], including the existence and uniqueness of the solution, and the stability of the stationary solution. It reads
{∂u∂t=∂2∂x2(au+αu2)−μ∂∂x[u∂∂x(T(x)u)],inI×R+,∂u∂x=0on∂I×(0,∞),u|t=0=(u0,v0,w0)onI, | (2.1) |
where and hereafter
{∂u∂t=a1∂2u∂x2−∂∂x[u(∂∂xχ1(v)−∂∂xχ2(w))],∂v∂t=a2∂2v∂x2+g1T(x,t)u−dv,∂w∂t=a3∂2w∂x2+g2u−hwinI×(0,∞),∂u∂x=∂v∂x=∂w∂x=0on∂I×(0,∞),(u,v,w)|t=0=(u0,v0,w0)onI. |
Iwasaki [14] proved the global-in-time solvability of (2.1), and argued the stability of the stationary solution by constructing the Lyapunov function. However, he did not discuss the convergence rate as time tends to infinity. We also point out that, although the model in that research admits constructing a global-in-time solution without the smallness of the initial data, the method does not apply to the model studied in this paper. For other arguments concerning the model by Okaie et al. [24][25], see the review by Iwasaki [14] and the references therein.
The model discussed in this paper is a coupling of the reaction-diffusion equation and ordinary differential equations. Recently, Marciniak-Czochra et al. [31] studied the nonstability of such systems under certain conditions. That seems meaningful since the reaction-diffusion equation reflects the denovo patterns or the Turing instability.
On the other hand, Iwasaki, Yang and Nakano [15] numerically indicated the stability of a stationary solution of their model. It seems correct under some assumptions through the analysis of the corresponding eigenvalue problem.
Since the models presented so far arise from the Keller-Segel model, we will give a brief overview of the mathematical arguments concerning the Keller-Segel equations. A huge number of contributions concerning the mathematical arguments of Keller-Segel equations and its variations exists, and therefore, we limit ourselves to the arguments that closely relate ours.
Schaaf [27] studied the stationary solution to the Keller-Segel equation under the general non-linearity, and reduced the problem to a scalar equation by using the bifurcation technique. She also provided a criterion for bifurcation of the solution. Osaki and Yagi [26] provided a global-in-time solution of the classical one-dimensional Keller–Segel equation.
Kang [16] investigated the existence and stability of a spike solution in the asymptotic limit of a large mass. They also studied the global-in-time existence of a solution to a reduced version of the Keller–Segel equation. The latter part is conducted by using the energy method.
Thorough surveys are provided by Horstmann [13] and the references therein.
In this section, we formulate the problem to be discussed in this paper. From Iwasaki, Yang and Nakano [15], the temporal behavior of the concentrations of bio-nanomachines, attractant, and repellent in one-dimensional space, denoted as
{∂Cb∂t=Db∇2Cb−∇⋅{Cb(Va∇Ca−Vr∇Cr)}∂Ca∂t=a1(x)Cb(x,t)−kaCa(x,t),∂Cr∂t=a2(x)Cb(x,t)−krCr(x,t)inΩ | (3.1) |
with boundary and initial conditions
{n⋅{Db∇Cb−Cb(Va∇Ca−Vr∇Cr)}=0onΓ≡∂Ω,u(x,0)=u0(x)≡(Cb0(x),Ca0(x),Cr0(x))TonΩ. | (3.2) |
For the sake of simplicity, we introduced the notation
Note that in this type of formulation,
With the term of classical Keller–Segel equation, this corresponds to the case when the sensitivity function
{∂p∂t=∇⋅[∇p−pχ(w)∇w],∂w∂t=g(p,w)inΩ,∂p∂n−pχ(w)∂w∂n=0onΓ. |
There, the authors imposed the assumption
The arguments in this direction are expounded on by Guarguaglini and Natalini [9] [10] with a more general setting:
{∂∂t(ϕ(c)s)=∇⋅(ϕ(c)∇s)+F(s,c),∂c∂t=G(s,c)inΩ,s(x,t)=ψ(x,t)=0onΓ,(s,c)|t=0=(s0(x),c0(x)) | (3.3) |
With some assumptions on
{∂u∂t=μ△u−∇⋅(uχ(c)∇c)+f(u,c),∂c∂t=g(u,c)inΩ,s(x,t)=ψ(x,t)=0onΓ,(s,c)|t=0=(s0(x),c0(x)) |
By the change of variable, this case is reduced to (3.3):
s=uϕ(c),F(s,c)=f(ϕ(c)s,c),G(s,c)=g(ϕ(c)s,c) |
with
{∂s∂t=△u+∇s⋅∇cϕ(c)−ϕ′(c)ϕ(c)s(αϕ(c)s−βc),∂c∂t=αϕs−βcinΩ,s(x,t)=ψ(x,t)=0onΓ,(s,c)|t=0=(s0(x),c0(x)) |
Here, if
supc∈[0,+∞)cϕ(c)≤L,cϕ(c)≤K2c2+K1 |
holds with some other assumptions, the a-priori estimate is obtained on which the existence of a weak global-in-time solution follows. Since
{Db∇2¯Cb−∇⋅{¯Cb(Va∇¯Ca−Vr∇¯Cr)}=0,a1(x)¯Cb(x)−ka¯Ca(x)=0,a2(x)¯Cb(x)−kr¯Cr(x)=0inΩ,n⋅{Db∇¯Cb−¯Cb(Va∇¯Ca−Vr∇¯Cr)}=0onΓ,∫Ω¯Cb(x)dx=1. | (3.4) |
They also showed the stability of
To extract the mathematical essence, we introduce the notation
{∇⋅Φ=0,a1(x)¯Cb(x)−ka¯Ca(x)=0,a2(x)¯Cb(x)−kr¯Cr(x)=0inΩ,n⋅Φ=0onΓ,∫Ω¯Cb(x)dx=1, | (3.5) |
We will study (3.5) later.
As for the stationary solution, we again mention the discussion by Friedman and Tello [8], in which they obtained only a constant stationary solution under some assumptions. This is a special case of ours.
Schaaf [27] first discussed the one-dimensional stationary solution to the Keller–Segel system:
{∇{k1(u,v)∇u−k2(u,v)∇v}=0,kc△v+g(u,v)=0inΩ,∂u∂n=∂v∂n=0on∂Ω. | (3.6) |
They also discussed the bifurcation from the stationary solution. Therein, they showed that the system (3.6) is reduced to an equation:
kc△v+g(ϕ(v(x),λ),v)=0, | (3.7) |
with
△u−λ(eu∫Ωeudx−1|Ω|)=0, | (3.8) |
where
{∂u∂t=∇⋅(∇u−u∇v),τ∂v∂t=△v−av+uinΩ,∂u∂ν=∂v∂ν=0on∂Ω,(u,v)|t=0=(u0(x),v0(x))onΩ. | (3.9) |
As in Schaaf [27], they translated (3.9) by using
{△u−βu+λ(eu∫Ωeudx−1|Ω|)=0inΩ,∂u∂ν=0on∂Ω. |
However, the studies in this direction concern the stationary solution to the elliptic problem, and are not applicable to our case.
Consider the coupled system of chemotaxis equation and ODE in the Keller–Segel literature. Corrias et al. [3] thoroughly studied the system
{∂n∂t=△n−∇⋅[nχ(c)∇c],∂c∂t=−cmninΩ⊂Rd,t>0,(n,c)|t=0=(n0,c0)onΩ. | (3.10) |
They found the global-in-time solutions to (3.10) for
Ca(x,t)=Ca0e−kat+a1(x)∫t0e−ka(t−τ)Cb(x,τ)dτ, |
which will be shown later as (7.3). Similar representation holds for
{∂u∂t=△u−∇⋅(u∇vv),∂v∂t=uvλinRn,t>0,(u,v)|t=0=(a(x),b(x))onRn. | (3.11) |
They obtained the global-in-time weak solution to (3.11) in Besov spaces under the smallness of initial data for
{∂u∂t=△u−χ∇⋅(u∇(logw)),∂w∂t=uwλinRn,t>0,∂u∂ν=∂w∂ν=0on∂Ω,(u,w)|t=0=(a(x),b(x))onΩ. | (3.12) |
They classified the approaches to (3.12) according to the range of
{∂u∂t=△u−χ∇⋅(u∇v),∂v∂t=uinRn,t>0,∂u∂ν=∂v∂ν=0on∂Ω,(u,v)|t=0=(u0(x),v0(x))onΩ, | (3.13) |
in case
{ηt=D1ηxx−D1[η(lnτ1(c,f))x]x,∂v∂t=−λ1vη1+λ2v,∂c∂t=−λ1vη1+λ2v,∂f∂t=−λ3f(fM−f)η−λ4cf1+λ5finx∈(0,1),t>0, | (3.14) |
They transformed (3.14) into a simpler one:
{ηt=D1ηxx−D1[η(γ1θxθ−γ2fxf)]x,∂f∂t=λ3fη−λ4θfinx∈(0,1),t>0,ηx−η(γ1θxθ−γ2fxf)=0onx=0,1,t>0. | (3.15) |
After further change of variables, they considered
{Pt=(Px+VxP)x,Vt=P−ginx∈(0,1),t>0,Px+VxP=0onx=0,1,t>0,P(x,0)=P0(x),V(x,0)=V0(x)onx∈(0,1). | (3.16) |
Then, they adopted the change of variables
The previous paper of the author [12] provides the local-in-time solvability of (3.1)–(3.2) in one-dimensional space but here we will go further. In this paper, we study the well-posedness of (3.1)–(3.2). Since we wish to consider the global-in-time solvability around the stationary solution, we first subtract
˜u(x,t)≡(˜Cb,˜Ca,˜Cr)T≡u(x,t)−ˉu(x). |
It reads
{∂˜Cb∂t=Db∇2˜Cb−∇⋅{˜Cb∇(Va¯Ca−Vr¯Cr)}−∇⋅{¯Cb∇(Va˜Ca−Vr˜Cr)}−∇⋅{˜Cb∇(Va˜Ca−Vr˜Cr)},∂˜Ca∂t=a1(x)˜Cb(x,t)−ka˜Ca(x,t),∂˜Cr∂t=a2(x)˜Cb(x,t)−kr˜Cr(x,t)inΩ,n⋅{Db∇˜Cb−˜Cb∇(Va¯Ca−Vr¯Cr)−¯Cb∇(Va˜Ca−Vr˜Cr)−˜Cb∇(Va˜Ca−Vr˜Cr)}onΓ,˜u(x,0)=˜u0(x)onΩ, | (3.17) |
where
In the following, let
Hereafter,
By
|u|(r+α)G=∑k≤r|Dku|G+[Dru](α)G, |
where
|u|G=supx∈G|u(x)|,[u](α)G=supx,y∈G|u(x)−u(y)||x−y|α. |
By the notation
‖f‖L2(G)≡(∫G|f(x)|2dx)12. |
The inner product in
(f1,f2)≡∫Ωf1(x)¯f2(x)dx, |
where
Analogously, we define
(f1,f2)Γ≡∫Γf1(x)¯f2(x)dx. |
By
‖f‖Lp(G)≡{(∫G|f(x)|pdx)1p(p∈[1,+∞)),esssupx∈G|f(x)|(p=∞). |
For simplicity, we hereafter denote the
By
{‖f‖2˙Wr2(G)=∑|α|=r‖Dαf‖2L2(G)=∑|α|=r∫G|Dαf(x)|2dxif r is an integer, ‖f‖2˙Wr2(G)=∑|α|=[r]∫G∫G|Dαf(x)−Dαf(y)|2|x−y|2+2{r}dxdyif r is a non-integer, r=[r]+{r},0<{r}<1. |
Next, for arbitrary
Wr,r22(GT)≡Wr,02(GT)⋂W0,r22(GT), |
whose norms are defined by
‖f‖2Wr,r22(GT)=∫T0‖f(⋅,t)‖2Wr2(G)dt+∫G‖f(x,⋅)‖2Wr22(0,T)dx≡‖f‖2Wr,02(GT)+‖f‖2W0,r22(GT). |
The set of functions with vanishing initial data,
∘Wr,r22(GT)={f∈Wr,r22(GT)|∂kf∂tk|t=0=0(k=0,1,2,…,[r2])}. |
We also introduce
˜Wr,r22(GT)={f∈Wr,r22(GT)|∫Gf(x,t)dx=0}. |
We also define a function space
W1∞(G)≡{u∈L∞(G)|∂u∂xj∈L∞(G)(j=1,2)}. |
Its norm is defined as
‖u‖W1∞(G)=‖u‖L∞(G)+‖∇u‖L∞(G). |
For simplicity, we shall use the following notations later.
W(l)≡Wl,l22(Ω∞)×Wl+12,l2+142(Γ∞),∘W(l)≡∘Wl,l22(Ω∞)×∘Wl+12,l2+142(Γ∞). |
The norms of these spaces are denoted as
For a Banach space
|f|Lp(a,b;B)≡{(∫ba‖f(t)‖pBdt)1p(p∈[1,+∞)),esssupa≤t≤b‖f(t)‖B(p=∞). |
The norms of vector and product spaces are defined in the usual manner.
In this section, we state the main results of this paper. Detailed proofs are provided in Sections 6 and 7. Before discussing the solvability of the non-stationary problem, we first argue the solvability of the stationary problem.
Theorem 5.1. Assume that
(i)
(ii)
Then, there exists the unique solution
Proof. The first case falls into the case of Friedman and Tello. [8], and their result guarantees that there exists only a constant solution to (3.4).
Our argument basically follows the one by Iwasaki [14] applied to the 1- dimensional case. There is little modification due to the difference of dimension. In his argument, however, he derived the linear differential equation of
Recall (3.4) is equivalent to (3.5), and further introduce
{∇⋅Φ=0,a1(x)¯Cb(x)−ka¯Ca(x)=0,a2(x)¯Cb(x)−kr¯Cr(x)=0inΩ,n⋅Φ=0onΓ,∫Ω¯Cb(x)dx=1, | (5.1) |
and, consequently,
{∇2ψ=0inΩ,∂ψ∂n=0onΓ. | (5.2) |
It is clear that
Now, since
Such a point does exist by virtue of Lemma 5.2 below. Thus, if there exists a non-zero solution to (3.4), it has to satisfy
This and the fact
log¯Cb−F¯Cb=const. |
Thus, following the arguments by Iwasaki [14], we arrive at the unique existence of a solution to (3.4), and the positiveness of
Lemma 5.2. Let us assume
Proof. Substitute (3.4)
(1−F¯Cb)∇2¯Cb−¯Cb(∇¯Cb⋅∇F)−F|∇¯Cb|2−2¯Cb∇¯Cb⋅∇F−¯C2b∇2F=0. | (5.3) |
We first assume that there exists a point
¯Cb(x1)=minx∈Ω¯Cb(x)<0 |
holds.
Then, it satisfies
Under the assumption of the lemma, the first issue and (5.3) yield
∇2¯Cb(x1)<0, |
a contradiction.
Next, assume that there exists a point
¯Cb(x2)=minx∈¯Ω¯Cb(x)<0 |
holds. We note that the case
This time
(1−¯CbF)(n⋅∇¯Cb)−(¯Cb)2(n⋅∇F)=0onΓ. |
From the assumption, this yields
Next, we state the existence theorem of the solution to (3.17).
Theorem 5.3. In addition to the assumptions in
(i)
(ii)
(iii)
(iv) the following inequality is satisfied:
(Va‖a1‖W1∞(Ω)ka+Vr‖a2‖W1∞(Ω)kr)|¯Cb|∞<Db−cΩ|∇{(Vaa1−Vra2)¯Cb}|∞, |
where
(v) and the following inequality is satisfied with a certain
Db−a1Va¯Cb(x)ka>c52∀x∈Ω. |
In addition, let the compatibility condition up to order
Then, problem
˜u(x,t)=(˜Cb,˜Ca,˜Cr)T∈W3+l,3+l22(Ω∞), |
that satisfies
In order to prove
Theorem 5.4. Let
The proof of Theorem 5.4 is provided in Section 7.
In this section, we discuss the solvability of (3.17). In the following, we use general positive constant
In this subsection, we first prove the solvability of the linear problem associated with (3.17), and then Theorem 5.3. We first consider the following linearized problem with vanishing initial data.
{∂˜Cb∂t−Db∇2˜Cb+∇⋅{˜Cb(Va∇¯Ca−Vr∇¯Cr)}+∇⋅{¯Cb(Va∇˜Ca−Vr∇˜Cr)}=F1,∂˜Ca∂t=a1(x)˜Cb(x,t)−ka˜Ca(x,t),∂˜Cr∂t=a2(x)˜Cb(x,t)−kr˜Cr(x,t)inΩ,n⋅{Db∇˜Cb−˜Cb∇(Va¯Ca−Vr¯Cr)−¯Cb∇(Va˜Ca−Vr˜Cr)}=F2onΓ,˜u(x,0)=0att=0. | (6.1) |
We have
Theorem 6.1. Let
∫ΩF1(x,t)dx+∫ΓF2(s,t)ds=0∀t>0. | (6.2) |
In addition, let the compatibility conditions up to order
n⋅{Db∇Cb0−Cb0(Va∇Ca0−Vr∇Cr0)}=0. |
Then, there exists a unique solution
‖˜u‖W3+l,3+l22(Ω∞)≤c61(‖F1‖W1+l,1+l22(Ω∞)+‖F2‖W32+l,34+l22(Γ∞)). | (6.3) |
In order to prove
ˆf(τ)≡∫Re−iτtf(t)dt, |
for a function
{λˆCb(x,−iλ)−Db∇2ˆCb(x,−iλ)+∇⋅{ˆCb(x,−iλ)(Va∇¯Ca−Vr∇¯Cr)}+∇⋅{¯Cb(Va∇ˆCa(x,−iλ)−Vr∇ˆCr(x,−iλ))}=ˆF1(x,−iλ),λˆCa(x,−iλ)=a1(x)ˆCb(x,−iλ)−kaˆCa(x,−iλ),λˆCr(x,−iλ)=a2(x)ˆCb(x,−iλ)−krˆCr(x,−iλ)inΩ,n⋅{Db∇ˆCb(x,−iλ)−ˆCb(x,−iλ)∇(Va¯Ca−Vr¯Cr)−¯Cb∇(VaˆCa(x,−iλ)−VrˆCr(x,−iλ))}=ˆF2(x,−iλ)onΓ. | (6.4) |
We substitute (6.4)
{ˆLˆCb≡∇⋅{G1(x,λ)∇ˆCb(x,−iλ)−G2(x,λ)ˆCb(x,−iλ)}−λˆCb(x,−iλ)=−ˆF1(x,−iλ)inΩ,ˆBˆCb≡n⋅(G1(x,λ)∇ˆCb(x,−iλ)−G2(x,λ)ˆCb(x,−iλ))=ˆF2(x,−iλ)onΓ, | (6.5) |
where
G1(x,λ)=Db−¯Cb(a1Vaλ+ka−a2Vrλ+kr),G2(x,λ)=(Va∇¯Ca−Vr∇¯Cr)+¯Cb(Va∇a1λ+ka−Vr∇a2λ+kr). |
Note that
∫R|ˆF1(⋅,−iλ)|2dσ1=∫R|e−σ0tF1(⋅,t)|2dt |
is finite for
Below, we write
L[u,η]≡∫Ω{G1(x,λ)∇u−G2(x,λ)u}⋅∇η(x)dx+λ∫Ωu(x)η(x)dx, |
for
Definition 6.2. A function
L[u,ˉη]=l(η)≡−∫ΩˆF1(x,−iλ)ˉη(x)dx+∫ΓˆF2(s,−iλ)ˉη(s)ds∀λ∈D(+)≡{z∈C|Re(z)>0},∀η(x)∈W12(Ω). | (6.6) |
First, we state
Lemma 6.3. Under the assumptions of
minxRe|G1(x,λ)|>cΩRe|G2(λ)|∞>0∀λ∈D(+). | (6.7) |
Proof. Since the assumption (6.2) in Theorem 6.1 implies that
|∫Ω(G2(x,λ)⋅∇ˆCb)ˆCb(x,λ)dx|≤|G2(λ)|∞|ˆCb(−iλ)||∇ˆCb(−iλ)|≤cΩ|G2(λ)|∞|∇ˆCb(−iλ)|2, |
thanks to the Poincaré inequality. Next, note that when
Re(1λ+ka)∈(0,k−1a],Re(1λ+kr)∈(0,k−1r]. | (6.8) |
Elementary calculations make us to obtain (6.8), so we omit the detail here. Now, (6.8) yields
|Vaa1Re(1λ+ka)−Vra2Re(1λ+kr)|∞≤Va|a1|∞ka+Vr|a2|∞kr,|∇{Vaa1Re(1λ+ka)−Vra2Re(1λ+kr)}|∞≤Va|∇a1|∞ka+Vr|∇a2|∞kr. |
Then, the assumption (ⅳ) in Theorem 5.3 implies
|¯Cb|∞‖{Vaa1Re(1λ+ka)−Vra2Re(1λ+kr)}‖W1∞(Ω)≤Db−cΩ|∇{(Vaa1−Vra2)¯Cb}|∞ |
This is sufficient for the following estimate to hold:
minx|¯Cb(x)||{Vaa1Re(1λ+ka)−Vra2Re(1λ+kr)}|+cΩ‖¯Cb(x)‖∞|∇{Vaa1Re(1λ+ka)−Vra2Re(1λ+kr)}|∞≤Db−cΩ|∇{(Vaa1−Vra2)¯Cb}|∞, |
which is equivalent to (6.7).
Next, we state the following lemma.
Lemma 6.4. Under the assumptions of
Proof. From the definition of
c62|∇u|2+σ0|u|2≤L[u,ˉu]=l(u)≤|ˆF1(−iλ)||u|+‖ˆF2(−iλ)‖L2(Γ)‖u‖L2(Γ)≤ε|∇u|2+Cε(|ˆF1(−iλ)|2+‖ˆF2(−iλ)‖2L2(Γ)), |
where we have applied the trace and Sobolev embedding theorems. Hereafter,
(c62−ε)|∇u|2+σ0|u|2≤Cε(|ˆF1(−iλ)|2+‖ˆF2(−iλ)‖2L2(Γ)). | (6.9) |
Thus, it is clear that
Next, we show the existence of a generalized solution to (6.5). Before that, we prepare an inner product:
[v,w]=Db∫ΩRe(G1(x,λ))2∑j=1∂v∂xj¯∂w∂xjdx+σ0∫Ωv(x)¯w(x)dx. |
As is easily seen,
[u,u]≥c63Db|∇u|2+σ0|u|2≥c64‖u‖2W12(Ω). |
By virtue of the Cauchy-Schwartz inequality, it is also seen that
[u,u]≤c65‖u‖2W12(Ω). |
Thus,
Next, we define
I1(u,η)≡−∫ΩG2(x,λ)u(x)¯∇η(x) dx. |
Then, it is easily observed that
|I1(u,η)|≤c66|u|‖η‖W12(Ω) |
Thus, the Riesz representation theorem enables us to represent
I1[u,η]=[Ku,η]. | (6.10) |
This operator
‖Kv‖2W12(Ω)≤c−164[Kv,Kv]=c−164|I1(v,Kv)|=c−164|∫ΩG2(x,λ)v(x)¯Kv(x)dx|≤c66|v|‖Kv‖W12(Ω). |
Similarly, since
|l(η)|≤(|ˆF1(−iλ)|+‖ˆF2(−iλ)‖L2(Γ))‖η‖W12(Ω), |
it is represented in the form of the scalar product:
l(η)=[Fu,η]. | (6.11) |
We also state the following lemma.
Lemma 6.5. The operator
Proof. Let
[Kvl−Kvm,Kvl−Kvm]=I1(vl−vm,Kvl−Kvm), |
it is clear that
[Kvl−Kvm,Kvl−Kvm]=|I1(vl−vm,Kvl−Kvm)|≤c266|vl−vm|2, |
and therefore,
Now, (6.6) enables us to re-formulate (6.5) in the following form.
[u+Ku,η]=[F,η]. | (6.12) |
Since (6.12) should be satisfied for all
u+Ku=F. | (6.13) |
Since
w+Kw=0 | (6.14) |
has only a trivial solution
[w+Kw,η]=0, |
which is simply the identity
Now we are in a position to state
Proposition 6.1. Let us assume the same assumptions as in
‖ˆCb(−iλ)‖W3+l2(Ω)+|λ|3+l2|ˆCb(−iλ)|≤c68[‖ˆF1(−iλ)‖W1+l2(Ω)+‖ˆF2(−iλ)‖W32+l2(Γ)+|λ|1+l2{|ˆF1(−iλ)|+‖ˆF2(−iλ)‖W122(Γ)}]. | (6.15) |
Proof. Since the existence of the generalized solution in
Under the assumptions
From the relationship
|λ||u|2≤|∫Ω{G1(x,λ)∇u−G2(x,λ)u}⋅∇u(x)dx|+|l(u)|≤c69(|∇u|2+|u|2)+|F1|2+|F2|2. |
Thus, together with (6.9), we have the desired estimate. Similarly, if we assume
Finally, applying the molifier and interpolation argument [20] leads us to the desired results.
Now we prove Theorem 6.1. In virtue of (6.15),
If we fix
Thanks to (6.15), we thus have
Due to the Paley-Wiener theorem,
∫R‖ˆF1(⋅,σ1−iσ0)‖2W1+l2(Ω)dσ1=∫+∞0‖e−σ0tF1(⋅,t)‖2W1+l2(Ω)dt |
Since the right-hand side is finite due to the assumption, if we make
Thus, the right-hand side of (6.15) is finite for each
Cb∈L2(R+;W3+l2(Ω)), |
and
Similarly, the inverse Fourier transform of
e−σ0tCb→Cb |
in
The uniqueness of the solution is obvious thanks again to the coerciveness of
Next, we consider the nonlinear problem (3.17). Before proceeding to the detailed arguments, we prepare some lemmas. The following lemma is well known (see, for instance, [32]).
Lemma 6.6. Let
‖fg‖Wk+l,k+l22(Ω∞)≤c610‖f‖Wm+l,m+l22(Ω∞)‖g‖Wk+l,k+l22(Ω∞). |
Our problem (3.17) is described as
A˜u=F[˜u], | (6.16) |
where
L˜u=(Db∇2˜Cb−∇⋅{˜Cb∇(Va¯Ca−Vr¯Cr)}−∇⋅{¯Cb∇(Va˜Ca−Vr˜Cr)},a1(x)˜Cb(x,t)−ka˜Ca(x,t),a2(x)˜Cb(x,t)−kr˜Cr(x,t))T,F1[˜u]=−∇⋅{˜Cb∇(Va˜Ca−Vr˜Cr)},F2[˜u]=n⋅{˜Cb∇(Va˜Ca−Vr˜Cr)}|Γ. |
From Lemma 6.6, we have
‖F[˜u]‖W(1+l)≤c611‖˜u‖2W3+l,3+l22(Ω∞). | (6.17) |
for
‖F[˜u(1)]−F[˜u(2)]‖W(1+l)≤c612‖˜u(1)−˜u(2)‖W3+l,3+l22(Ω∞)×(‖˜u(1)‖W3+l,3+l22(Ω∞)+‖˜u(2)‖W3+l,3+l22(Ω∞)), | (6.18) |
for
‖˜u(0)‖W3+l,3+l22(Ω∞)≤c613‖˜u0‖W2+l2(Ω). |
This is achieved by a well-known method as follows (see, for instance, Theorem IV.4.3 in [18], p.298). Now, define
{∂u∗∂t−∇2u∗=0inR2∞,u∗|t=0=ϕ1−∇2˜u0onR2, |
Next, define
{∂˜u(0)∂t−∇2˜u(0)=u∗inR2∞,˜u(0)|t=0=˜u0onR2. |
This
A˜u(1)=F[˜u(0)+˜u(1)]−A˜u(0). | (6.19) |
If
˜u(1)=A−10[F[˜u(0)+˜u(1)]−A˜u(0)], | (6.20) |
then the above assumption is satisfied, for
M[˜u(1)]≡A−10[F[˜u(0)+˜u(1)]−A˜u(0)], |
and show that it has a fixed point, assuming that
‖˜u0‖W2+l2(Ω)≤δ0 |
with a sufficiently small
‖˜u(0)‖W3+l,3+l22(Ω∞)≤c613δ0 |
and
‖F[˜u(0)+˜u(1)]‖W(1+l)≤c615(δ20+‖˜u(1)‖2W3+l,3+l22(Ω∞)). | (6.21) |
By combining this with the boundedness of
‖M[˜u(1)]‖W3+l,3+l22(Ω∞)≤c616(‖˜u(1)‖2W3+l,3+l22(Ω∞)+δ20+δ0). | (6.22) |
Thus, if we take
‖M[˜u(1)]−M[˜u(2)]‖W3+l,3+l22(Ω∞)≤c617δ0‖˜u(1)−˜u(2)‖W3+l,3+l22(Ω∞). | (6.23) |
Thus, if we take
Next, we argue the uniqueness of the solution in the similar line with Beale [2]. Let us assume that there exists a number
T2≡sup{t1>0|˜u(2)(t)=˜u(t)for0<t<t1}. |
If
Introduce
‖˜w‖W3+l,3+l22(ΩT0)≤‖˜u‖W3+l,3+l22(ΩT0)+‖˜u(2)‖W3+l,3+l22(ΩT0). | (6.24) |
From the assumption, the second term of the right-hand side in (6.24) is estimated by
‖˜u‖W3+l,3+l22(ΩT0)≤‖˜u(0)‖W3+l,3+l22(ΩT0)+‖˜u(1)‖W3+l,3+l22(ΩT0)≤c613δ0+2c616δ0≡c619δ0. | (6.25) |
By using the extension by reflection [20], we are able to extend
‖˜w‖W3+l,3+l22(Ω∞)≤c620‖˜w‖W3+l,3+l22(ΩT0), | (6.26) |
with a certain
On the other hand, if we introduce
‖˘w‖W3+l,3+l22(Ω∞)≤c621(‖˜u‖W3+l,3+l22(Ω∞)+‖˜u(2)‖W3+l,3+l22(Ω∞))×‖˜w‖W3+l,3+l22(Ω∞). | (6.27) |
By virtue of (6.25)–(6.27), we then have
‖˘w‖W3+l,3+l22(Ω∞)≤c622δ0‖˜w‖W3+l,3+l22(ΩT0) | (6.28) |
with a certain
Thus, we have
‖˜w‖W3+l,3+l22(Ω∞)≤c620‖˜w‖W3+l,3+l22(ΩT0)<c620c622δ0‖˜w‖W3+l,3+l22(ΩT0). |
Take
Finally, it is easily seen that
ddt∫Ω˜Cb(x,t)dx=∫Ω∂˜Cb∂t(x,t)dx=∫Ω[Db∇2˜Cb−∇⋅{˜Cb(Va∇¯Ca−Vr∇¯Cr)}−∇⋅{¯Cb(Va∇˜Ca−Vr∇˜Cr)}]dx=0, |
by virtue of the boundary condition. This completes the proof of Theorem 5.3.
In this section, we argue the non- negativeness of solution
Cb(x,t)=C(+)b(x,t)−C(−)b(x,t), |
where
Ω(−)(t)≡{x∈Ω|Cb(x,t)<0} |
for each
∫ΩC(−)b∇⋅{Db∇Cb−Cb∇(VaCa−VrCr)}dx=−∫Ω(−)(t)∇C(−)b⋅{Db∇Cb−Cb∇(VaCa−VrCr)}dx. | (7.1) |
Indeed, by Green's theorem and the boundary condition (3.2)
∫Ω(−)(t)∇⋅[C(−)b{Db∇Cb−Cb∇(VaCa−VrCr)}]dx=0, |
and therefore, we have
∫Ω(−)(t)∇C(−)b⋅{Db∇Cb−Cb∇(VaCa−VrCr)}dx+∫Ω(−)(t)C(−)b∇⋅{Db∇Cb−Cb∇(VaCa−VrCr)}dx=0. |
Noting that
0=ddt∫ΩC(+)bC(−)bdx=ddt∫Ω(−)(t)C(+)bC(−)bdx=limε→0ε−1(∫Ω(−)(t+ε)−∫Ω(−)(t))C(+)bC(−)b|∂Ω(−)(t)dx+∫Ω(−)(t)∂∂t(C(+)bC(−)b)dx=∫Ω(−)(t)∂C(+)b∂tC(−)bdx=∫Ω∂C(+)b∂tC(−)bdx. |
Thereby, we have
∫Ω∂Cb∂tC(−)bdx=−∫Ω∂C(−)b∂tC(−)bdx=−12ddt∫Ω|C(−)b|2dx. |
Finally, we further modify the first term of the right-hand side in (7.1). Note that since
∫Ω(−)(t)∇⋅(Cb∇C(−)b)dx=0, |
we have
−∫Ω(−)(t)∇C(−)b⋅∇Cbdx=∫Ω(−)(t)Cb∇2C(−)bdx=−∫Ω(−)(t)C(−)b∇2C(−)bdx=∫Ω(−)(t)|∇C(−)b|2dx. |
From these, we have
12ddt∫Ω(−)(t)|C(−)b(t)|2dx+∫Ω(−)(t)|∇C(−)b(t)|2dx=∫Ω(−)(t)C(−)b∇C(−)b⋅∇(VaCa−VrCr)dx | (7.2) |
Then, by virtue of the Young's inequality, the right-hand side is estimated as
|∫Ω(−)(t)C(−)b∇C(−)b⋅∇(VaCa−VrCr)dx|≤c71(‖∇Ca(t)‖L∞(Ω(−)(t))+‖∇Cr(t)‖L∞(Ω(−)(t)))×(ε‖∇C(−)b(t)‖2L2(Ω(−)(t))+Cε‖C(−)b(t)‖2L2(Ω(−)(t))). |
By noting that
ddt‖C(−)b(t)‖2L2(Ω(−)(t))+‖∇C(−)b(t)‖2L2(Ω(−)(t))≤c71‖C(−)b(t)‖2L2(Ω(−)(t)). |
By noting that
Ca(x,t)=Ca0e−kat+a1(x)∫t0e−ka(t−τ)Cb(x,τ)dτ. | (7.3) |
This and assumption (ⅲ) of
In this paper, we provided the global-in-time solvability of the two-dimensional non-stationary problem of a target detection model in a molecular communication network in Sobolev–Slobodetskiĭ
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