ON A MODEL OF TARGET DETECTION IN MOLECULAR COMMUNICATION NETWORKS

. This paper is concerned with a target-detection model using bio-nanomachines in the human body that is actively being discussed in the ﬁeld of molecular communication networks. Although the model was originally proposed as spatially one-dimensional, here we extend it to two dimensions and analyze it. After the mathematical formulation, we ﬁrst verify the solvability of the stationary problem, and then the existence of a strong global-in-time solution of the non-stationary problem in Sobolev–Slobodetski˘ı space. We also show the non-negativeness of the non-stationary solution.

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 where and hereafter I ≡ (0, 1) and R + ≡ (0, ∞). It was proposed as a simple version of a model proposed by Okaie et al. [24] [25] that was similar to the Keller-Segel model [17] but with a variant target concentration: u, v, w t=0 = u 0 , v 0 , w 0 on I.
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.
3. Formulation. 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 C b (x, t), C a (x, t) and C r (x, t), respectively, are represented as follows.
with boundary and initial conditions For the sake of simplicity, we introduced the notation u = C b , C a , C r T above.
Here, t is time; x is the location of the materials on the tissue surface; V a and V r are positive constants denoting attractant and repulsive coefficients, respectively; a 1 (x) = g a h(x)/(h(x) + K a ), a 2 (x) = g r K r /(h(x) + K r ), with h(x) being the target concentration; ∇ is the two-dimensional gradient; n, is the outer unit normal to Ω; and K a (or K r ) is the positive constant standing for the target concentration leading to the half maximum attractant (or repellent) production rate, respectively. The notation g a (or g r ) is also a positive constant representing the maximum attractant (or repellent) production rate, respectively; and the positive constant k a (or k r ) is the decay rate of the attractant (or repellent), respectively. Note that in this type of formulation, C b is a probability density so that C b ≥ 0 and Ω C b dx = 1 must always be satisfied, while C a and C r are the concentrations, and satisfy only C τ ≥ 0 (τ = a, r). For instance, we can take C r t=0 = 0 as was the case in [24].
With the term of classical Keller-Segel equation, this corresponds to the case when the sensitivity function χ(u) = u. This type of model is used to formulate the angiogenesis model (see, e.g., [8]). Friedman and Tello [8] studied a system of equations close to ours: There, the authors imposed the assumption g(p, w) = φ(p, w)h(p, w), which satisfies φ > 0 and h(p 1 , w 1 ) = h(p 2 , w 2 ) with some constants (p i , w i ) (i = 1, 2) satisfying 0 ≤ p 1 < p 2 and w 1 < w 2 . Together with some other assumptions, they showed the unique existence of a global-in-time solution in C 2+β, 2+β 2 (Ω ∞ ). They also showed the asymptotic stability of the solution when the initial data is sufficiently close to the stationary solution. However, their assumptions are not satisfied in general in our case, except for a special case (which will be stated later).
The arguments in this direction are expounded on by Guarguaglini and Natalini [9] [10] with a more general setting: With some assumptions on F , G, φ and ψ, Guarguaglini and Natalini [9] obtained the global-in-time weak solution to this system with small data and the spatial dimension of 2 or more. They also discussed the case when the first equation contains the gradient of the first unknown variable: By the change of variable, this case is reduced to (3.3): with µφ (c) = φ(c)χ(c). Guarguaglini and Natalini [9] considered an example, which is close to ours: holds with some other assumptions, the a-priori estimate is obtained on which the existence of a weak global-in-time solution follows. Since φ(c) = φ 0 e c/µ with some constant φ 0 in our case, the conditions above are not satisfied in our case since χ(c) ≡ 1. In the paper by Iwasaki, Yang and Nakano [15], the authors provided the existence of a stationary solution u( They also showed the stability of u(x), that is, the convergence of the solution of (3.1)-(3.2) to that of (3.4) through numerical simulations under specific parameter values.
To extract the mathematical essence, we introduce the notation 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: They also discussed the bifurcation from the stationary solution. Therein, they showed that the system (3.6) is reduced to an equation: with u(x) = φ(v(x), λ). (See also, [13].) In our case, we can reduce our problem to this form by taking v(x) ≡ V a C a − V r C r , k c = 0, and g(u, v) , and the result partially gives us an insight. But we will have more general results, which will be stated in Section 4. For the equation similar to (3.5), there have been some arguments in the past. Struwe and Tarantello [29] discussed the problem where Ω is a two-dimensional torus. By using variational techniques and mountainpass solution, they showed the existence of a nontrivial solution to (3.8) for λ ∈ (8π, 4π 2 ). The studies by Wang and Wei [33] and Senba and Suzuki [28] extended these results to more general cases. They studied the following problem: As in Schaaf [27], they translated (3.9) by using u = σe v into the form: 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 (3.10) They found the global-in-time solutions to (3.10) for d ≥ 2 with very general settings on χ(c). They showed that for d ≥ 3, there exists a global-in-time solution under the smallness of initial data, whereas for d = 2, without smallness of initial data. Their method is based on changing the form of unknown variables, and then the energy estimate of the p-norm of new variables. Although it seems simple and widely applicable, it does not apply to our case. The reason is in our case, C a is represented as: which will be shown later as (7.3). Similar representation holds for C r also. Therefore, we cannot assume the essential boundedness of them as in case of [3]. In their estimate, they made use of the boundedness of sup 0≤c≤ c0 ∞ φ 2 (c)χ(c)c m , which no longer holds in our case. Sugiyama et al. [30] studied the system They obtained the global-in-time weak solution to (3.11) in Besov spaces under the smallness of initial data for n ≥ 3, and local-in-time solution for n ≥ 1. Ahn and Kang [1] studied the system (3.12) They classified the approaches to (3.12) according to the range of λ, among which the case of λ = 1 is close to ours. Indeed, in that case, they transformed the problem into the form: (3.13) in case Ω = R d , with decay at infinity of unknown variables. They implied that the blow-up of the solution to (3.13) in the Sobolev space occurs. Fontelos et al. [7] studied the following system in one-dimensional case.
They transformed (3.14) into a simpler one: After further change of variables, they considered Then, they adopted the change of variables c = e −V and u = P e V . However, this approach is not applicable to our case, for in our case, k a = k r in general. Even if k a = k r holds, since C a and C r take the form of (7.3) again, which prevents us from assuming the boundedness of c.
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) from the original problem, and consider the problem concerning 4. Notations. In the following, let G be an arbitrary open set in R 2 , and let T > 0, and G T ≡ G × (0, T ). For a set S in general, S denotes its closure and ∂S its boundary.
Hereafter, C l (G) (l ∈ N) stands for a set of functions defined on G ⊂ R 2 that have l-th order continuous derivatives.
By C r+α (G) with a non-negative integer r and α ∈ (0, 1), we mean the Banach space of functions from C r (G), whose rth derivatives satisfy the Hölder condition with exponent α, i.e. the space of functions with the finite norm By the notation C l,1 (G) (l ∈ N), we denote a set of functions defined on G that have l-th order Lipschitz continuous derivatives. We also use similar notations to represent the regularity of domain boundaries. L 2 (G) means a set of squareintegrable functions defined on G, equipped with the norm The inner product in L 2 (G) is defined by where z stands for the complex conjugate of z ∈ C.
Analogously, we define By · Lp(G) , we denote the usual L p -norm with 1 < p ≤ +∞ on G: For simplicity, we hereafter denote the L p -norm of a function f over Ω (Ω is the region where we consider the problem) merely by |f | p . Especially, we denote the L 2 -norm of a function f over Ω merely as |f | when it is obvious.

MOLECULAR COMMUNICATION NETWORKS 643
The set of functions with vanishing initial data, , is defined as [18]: We also introduce We also define a function space Its norm is defined as . For simplicity, we shall use the following notations later.
The norms of these spaces are denoted as · W (l) and so forth.
The norms of vector and product spaces are defined in the usual manner.

5.
Main results. 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.
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 1dimensional case. There is little modification due to the difference of dimension. In his argument, however, he derived the linear differential equation of log C b from the beginning. Since it is not clear whether C b > 0 holds or not in advance, we need to somewhat modify the proof.
Recall (3.4) is equivalent to (3.5), and further introduce Φ = ∇ψ there. Then, we have This and the fact Φ = 0 lead to Thus, following the arguments by Iwasaki [14], we arrive at the unique existence of a solution to (3.4), We first assume that there exists a point x 1 ∈ Ω such that holds. Then, it satisfies ∇C b (x 1 ) = 0 and ∇ 2 C b (x 1 ) > 0. Otherwise, there exists a neighborhood U 1 ⊂ Ω of x 1 and x 1 ∈ U 1 , such that the same situation holds at x 1 . This comes from (3.4) 5 .
Under the assumption of the lemma, the first issue and (5.3) yield Next, assume that there exists a point x 2 ∈ Γ, such that holds. We note that the case n · ∇C b (x 2 ) = 0 has been discussed in the argument above. This time n · ∇C b < 0 on x = x 2 . The boundary condition (3.4) 2 reads From the assumption, this yields n · ∇C b > 0 on x = x 2 , a contradiction. Thus, we can conclude that C b (x) ≥ 0 on Ω.
Next, we state the existence theorem of the solution to (3.17).
Theorem 5.3. In addition to the assumptions in Theorem 5.1, let l ∈ (1/2, 1). We also assume: iv) the following inequality is satisfied: where c Ω is a constant dependent on the size of Ω, (v) and the following inequality is satisfied with a certain c 52 > 0: In addition, let the compatibility condition up to order 1 [18] be satisfied. Then, problem (3.17) has a unique solution, (Ω ∞ ) also.
In order to prove Theorem 5.3, we first consider the linear problem. Next, we make use of the multiplicative inequalities and fixed point theorem to verify the existence of the global-in-time solution. These discussions are held in the next section. We also show The proof of Theorem 5.4 is provided in Section 7.
6. Proof of solvability. In this section, we discuss the solvability of (3.17). In the following, we use general positive constant c.
6.1. Linear problem. 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.
We have Theorem 6.1.
Let Ω be bounded, Γ ∈ C 2,1 , l ∈ (1/2, 1), and assume conditions (i)- In addition, let the compatibility conditions up to order 1 be satisfied. That is, Then, there exists a unique solution u = ( (Ω ∞ ) to (6.1) In order to prove Theorem 6.1, we apply the Fourier transform with respect to t to unknown functions extended into the region t < 0 by zero [2]: for a function f in general. After applying this transform to (6.1), we further substitute τ = −iλ to obtain We substitute (6.4) 2 and (6.4) 3 into (6.4) 1 , and then reduce the problem to one concerningĈ b (x, −iλ).
Note thatF 1 (x, −iλ) = R e −λt F 1 (x, t) dt is the Fourier transform of e −σ0t F 1 with respect to σ 1 . Therefore, from the assumption is finite for σ 0 > 0. Thus,F 1 (x, −iλ) has an analytic extension onto the region Re λ > 0. Similar facts hold for other functions in (6.5). Below, we write λ = σ 0 + iσ 1 , and D (+) ≡ z ∈ C Re z > 0 . In order to define the term generalized solution to (6.5), we first introduce the following operator.
Proof. Since the assumption (6.2) in Theorem 6.1 implies that Ω C b (x, t) dx = 0 ∀t > 0 if it exists in W 1 2 (Ω), we seek for the solution satisfying it. Observe that thanks to the Poincaré inequality. Next, note that when σ 0 ≥ 0, Elementary calculations make us to obtain (6.8), so we omit the detail here. Now, (6.8) yields Then, the assumption (iv) in Theorem 5.3 implies This is sufficient for the following estimate to hold: which is equivalent to (6.7).
Next, we state the following lemma.
where we have applied the trace and Sobolev embedding theorems. Hereafter, ε is an arbitrary small positive number and C ε , a positive constant non-decreasingly dependent on ε. Note that the Poincaré inequality holds for u ∈ W 1 2 (Ω). This yields Thus, it is clear that u is equivalently zero if we replaceF 1 andF 2 with zero.
Next, we show the existence of a generalized solution to (6.5). Before that, we prepare an inner product: As is easily seen, Re(G 1 (x, λ)) ≥ ∃c 63 holds under the assumption (v) of Theorem 5.3. Thus, we have . By virtue of the Cauchy-Schwartz inequality, it is also seen that Thus, [u, u] is equivalent to the norm of W 1 2 (Ω). Next, we define Then, it is easily observed that Thus, the Riesz representation theorem enables us to represent I 1 in the form of the scalar product: (6.10) This operator K is a bounded map on W 1 2 (Ω) in virtue of the following. Kv 2 , it is represented in the form of the scalar product: (6.11) We also state the following lemma.
Lemma 6.5. The operator K defined above is a compact operator in W 1 2 (Ω).
Proof. Let {v m } m denote a sequence of elements in W 1 2 (Ω), satisfying v m W 1 2 (Ω) ≤ c 67 . Then, the sequence {Kv m } m are also uniformly bounded. Thanks to the Rellich-Kondrachov theorem, W 1 2 (Ω) is compactly embedded into L 2 (Ω). Therefore, there are subsequences of {v m } and {Kv m } that converge strongly in L 2 (Ω). By virtue of the equality , and therefore, {Kv m } converges strongly in W 1 2 (Ω). 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 η ∈ W 1 2 (Ω), it is equivalent to the abstract equation in W 1 2 (Ω): Since K is a linear and compact operator, we can apply Fredholm's theorems. Especially, the first statement of it guarantees that (6.13) has a solution if the homogeneous equation has only a trivial solution w = 0. But such w is a solution to (6.5) withF 1 =F 2 = 0, since (6.14) is equivalent to the integral identity [w + Kw, η] = 0, which is simply the identity L[w, η] = 0. Lemma 6.4 states that w = 0, and therefore (6.14) certainly has a solution. Since (6.13) is equivalent to (6.12), and consequently to (6.6), this is the desired solution. Now we are in a position to state Proposition 6.1. Let us assume the same assumptions as in Theorem 6.1. Then, for each λ ∈ D (+) , there exists a unique solutionĈ b (x, −iλ) ∈ W 3+l 2 (Ω) to (6.5) satisfying .

(6.15)
Proof. Since the existence of the generalized solution in W 1 2 (Ω) has been established before, we limit ourselves to the regularity discussions.
Due to the Paley-Wiener theorem, e −σ0t F 1 = 0 for t < 0. The Plancherel theorem yields Since the right-hand side is finite due to the assumption, if we make σ 0 tend to zero, it tends to F 1 Similarly, the L 2 -norm of λ converges as σ 0 → 0, whose limit is estimated from above by . Thus, the right-hand side of (6.15) is finite for each λ ∈ D (+) , and we apply the Lebesgue convergence theorem, to obtain (Ω)), and e −σ0t C b → C b as σ 0 → 0 in this space.
Next, define u (0) as a solution to This u (0) actually satisfies the desired features. Now, we rewrite the problem (6.16) for new variable u (1) ≡ u − u (0) : (Ω ∞ ), then, using our method of constructing (1) ] at t = 0. Thus, the right-hand side of (6.19) belongs to . Let A 0 be a solution operator of Theorem 6.1 for the linear problem with zero initial data. Then, by virtue of Theorem 6.1, if (Ω ∞ ). In order to show the solvability of (6.20), let us define a map and show that it has a fixed point, assuming that with a sufficiently small δ 0 > 0. By our choice of u (0) , we have  (Ω ∞ ) has been established.
Next, we argue the uniqueness of the solution in the similar line with Beale [2]. Let us assume that there exists a number T 1 > 0 and another solution u (2) ∈ W 3+l, 3+l 2 2 (Ω T1 ) to (3.17), and define If T 2 < T 1 , we may set T 2 = 0 and, thereby, we assume u (2) (Ω T 0 ) < c 618 δ 0 , where c 618 and δ 0 are as before.
Thus, we have A w − w = 0 on t ∈ (0, T 0 ). From the uniqueness of the solution of the linear problem, we then observew = w on t ∈ (0, T 0 ). Replacingw by w in (6.28) and using (6.26), we then have . Take δ 0 so small that c 620 c 622 δ 0 < 1, and then, we have w = 0 on t ∈ (0, T 0 ). This means u and u (2) coincide on t ∈ (0, T 0 ), which contradicts the original assumption. Thus, we obtain the uniqueness of the solution to (3.17).
Finally, it is easily seen that Ω C b (x, t) dx = 0, by by virtue of the boundary condition. This completes the proof of Theorem 5.3.

7.
Non-negativeness of the solution. In this section, we argue the nonnegativeness of solution C b to (3.1)-(3.2) to prove Theorem 5.4. We use the socalled Stampacchia truncation method (see, for instance, [6] and [11]). To do that, we divide C b into its positive and negative parts: for each t > 0. Let us multiply (3.1) 1 by C (−) b , and integrate with respect to x over Ω (−) (t). We show Indeed, by Green's theorem and the boundary condition (3.2) 1 , we observe and therefore, we have Noting that C (−) b (x, t) = 0 on Ω\Ω (−) (t), we arrive at (7.1). Next, taking into mind C dx.
Thereby, we have Finally, we further modify the first term of the right-hand side in (7.1). Note that since From these, we have 1 2 Then, by virtue of the Young's inequality, the right-hand side is estimated as (Ω (−) (t)) .
By noting that C b (x, t) = 0, i.e., C b (x, t) ≥ 0 for each x ∈ Ω, t > 0. By virtue of the Sobolev embedding theorem, this holds in the pointwise sense. Finally, from (3.1) 2 , we have C a (x, t) = C a0 e −kat + a 1 (x) t 0 e −ka(t−τ ) C b (x, τ ) dτ. (7.3) This and assumption (iii) of Theorem 5.3 yield C a (x, t) ≥ 0 on Ω ∞ . The fact C r (x, t) ≥ 0 is derived in a similar manner. This completes the proof of Theorem 5.4.

8.
Conclusion. 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-Slobodetskiȋ space. We also showed the nonnegativeness of the non-stationary solution. We will tackle the stability arguments in the near future.