Mass conservative reaction–diffusion systems describing cell polarity

A reaction–diffusion system with mass conservation modeling cell polarity is considered. A range of the parameters is found where the ω‐limit set of the solution is spatially homogeneous, containing constant stationary solution as well as possible nontrivial spatially homogeneous orbit.

Given the sufficiently smooth nonlinearity f = f (u, v), standard theory allows the existence of a unique local-in-time classical solution (u, v) = (u(·, t), v(·, t)) to (1), as it can be seen in other studies, [1][2][3][4][5][6][7] we also refer the interested reader to Pierre's monograph 8 for a more complete presentation of the history of such systems. The solution (u, v) has the following total mass conservation property: The class of models that we are going to study, were proposed in Otsuji et al. 9 to describe cell polarity. The proposed mechanism shall separate different species inside the cell according to their diffusion coefficients, that is, slow and fast diffusions shall localize the species near the membrane and in the cytosol, respectively. Three kinds of molecules are interacting. Each one of them has two phases, active and inactive, which are characterized by slow and fast diffusions, respectively. The model problem (1) focuses on these two phases of a single species, ignoring the interactions between the other species.
The model shall allow Turing pattern, 10 which is the appearance of spatially inhomogeneous stable stationary states induced by diffusion. In their study, Otsuji et al. 9 suggest the following three models for this purpose: , where a, b, , 1 , and 2 are positive constants.
In their study, Mori et al. 11 suggested where b 1 , , , k 0 , k are positive constants (see Mori et al. 11 ). System (1) with the above reaction term will be referred in the following as the fourth model and is the main topic of study in this paper. The results of this paper can be directly applied to more general reaction terms of the type The main characteristics of the system are the following: • Quasi positivity for provides positivity for (u, v): Therefore, the solution is nonnegative, provided that nonnegative initial data are given. • Mass conservative reaction-diffusion system: For the global existence and uniform-in-time bounds of nonnegative classical solutions to this system in all space dimension, we refer the interested reader to Theorem 1.1 in Fellner et al. 12 Actually, Fellner et al. 12 consider an even more general class of systems where the reaction terms might have a (slightly super-) quadratic growth.
In this paper, we give an answer to the natural question which rises next about the asymptotic behavior of the solution and whether it converges to the equilibrium. Namely, is the solution to this fourth model (4) asymptotically spatially homogeneous or do we have a Turing paradigm (stable nonconstant stationary state under the local enhancement and long-range inhibition)?
This work is organized as follows: in Section 2, we summarize what has been done in the previous relevant models. In Section 3, we present and prove some of the key features of the fourth model, and we state our main Theorem 5. In Section 4, we prove our main result.

REVIEW OF THE PREVIOUS WORK
In the first and the second models, the stationary state is described by the elliptic eigenvalue problem with nonlocal term, with the eigenvalue associated with the total mass that is conserved in time. The stationary state has a variational functional J, while there is a Lyapunov functional L(u, v) for the nonstationary problem. This Lyapunov functional is reduced to the stationary variational functional, if the total mass of (u, v) is prescribed. This remarkable structure, called semi-unfolding minimality, induces dynamical stability of the local minimizer of J. We will briefly revisit what has already been done for these models.

First model
the first model takes the form Henceforth, C i , i = 1, 2, … , 9 denote positive constants independent of t. Since this h = h(u) is a smooth function of u ∈ R satisfying if 0 ≤ (u 0 , v 0 ) = (u 0 (x), v 0 (x)) ∈ C 2 (Ω) 2 , for simplicity, from here on, we denote X = C 2 (Ω) 2 , then problem (6) admits a unique classical solution (u, v) = (u(·, t), v(·, t)) uniformly bounded, and global-in-time (Theorem 1.1 in Fellner et al. 12 ). Therefore, the orbit  = {(u(·, t), v(·, t))} t≥0 is compact in X, and hence, the -limit set defined by is nonempty, compact, and connected. With the system (6) transforms into for w 0 = Du 0 + v 0 . In the stationary state, we have and therefore, this w is a constant denoted byw.w is prescribed by the initial value using (5): We thus obtain The set of stationary solutions to (6), denoted by E , is thus defined in accordance with in (5), that is, (u, v) ∈ E if and only if u = u(x) is a solution to (11) for = 1 − D, and v =w − Du, wherew is a constant defined bȳ By exploiting the above observations, Morita and Ogawa 7 and Morita 5 studied the spectral analysis of the stationary solution. The purpose of Latos and Suzuki 3 was to study the previous results from the point of view of global dynamics. In fact, with the use of the Lyapunov functional, they showed the existence of a global-in-time solution to (6) in X = C 2 (Ω) 2 with compact orbit. The following theorem is proven by the existence of the Lyapunov functional to (6): Remark 1. From the result 12 established later, the restriction on the space dimension N = 1, 2, 3, in Latos and Suzuki, 3 is excluded for the compactness of . This extension is also valid to the second model described below.
The problem (11) has a variational structure. Thus, u = u(x) is a solution if and only if J ′ (u) = 0, where for Q ′ (u) = q(u). Then we obtain the dynamical stability of local minimizers of this functional. (12), and put Then this stationary solution (u * , w * ) to (9), for (u 0 , w 0 ) satisfying (10), is dynamically stable in H 1 then it holds that for the solution (u, w) = (u(·, t), w(·, t)) to (9).

Second model
By letting if where (15) is thus compact in X = C 2 (Ω) 2 , and hence, the -limit set defined by is nonempty, compact, and connected. First, total mass conservation arises in the form of Second, there is a Lyapunov functional defined by satisfying dL dt Third, in the stationary state of (15), the component w = w(x) is spatially homogeneous similarly, denoted by w =w ∈ R. Hence, it holds thatw by (16). Plugging (19) into the first equation of (15), we see that the stationary state of (9) is reduced to a single equation concerning z = z(x), that is, This problem is the Euler-Lagrange equation corresponding to the variational functional Thus, the set of stationary solutions is associated with in (16), denoted by E . We say that Then we obtain the following results similarly. (21), and put Then the stationary solution (z * , w * ) to (15) for the solution (z, w) = (z(·, t), w(·, t)) to (15).
Remark 2. We note the following facts. First, the local minimizer in Theorems 2 and 4 may be degenerate. Second, there is a correspondence between the Morse index of the linearized operator around the stationary solution (u * , z * ) or (z * , w * ) and that of u * or z * as a critical point of the variational functional J . This property is called the spectral comparison, and a result in this direction is obtained in Latos et al. 4 for the second model.

THE MODEL AND THE RESULT
We skip the third model because it does not satisfy the quasi-positivity. Hence in this work we consider the fourth model, (1) for where > 0. One can consider the more general reaction term used in Holmes and Edelstein-Keshet, 13 in the argument below. Putting we obtain f(u, v) = va(u) − u and Therefore, this model is reduced to The nonlinearity a(u) in (23) is not so wild. If it is a contact denoted by a > 0, the system (25) is linear, but a special form of the first model. Hence the stationary state is reduced to There is a unique spatially homogeneous solution to (26), that is, The linearized operator around this u * is given by Using the eigenvalues and eigenfunctions of −Δ under the Neumann boundary condition, we see that this L is nondegenerate always. Thus there is no Turing pattern in this case, that is, in the case when a(u) is a constant. We can actually confirm the linearized stability of this spatially homogeneous stationary solution (u * , v * ) to (25) for v * satisfying = u * + v * . In fact, this linearized equation takes the form Using the eigenvalues and eigenfunctions of −Δ under the Neumann boundary condition again, we see that all the eigenvalues of the linearized operator are real and negative. Hence, (u * , v * ) is asymptotically stable. In spite of these simple profiles of the solution for the case that a(u) = a > 0 is a constant, the global dynamics of (25) for (23) is not subject to a Lyapunov functional.
To confirm this property, we take a look at the stationary problem to (25): By the argument in the previous section, the function w = Du+v in (28) is a constant denoted byw, which is determined by (29):w Therefore, the system (28) is reduced to We see that (30) admits no variational functional unless a(u) is a constant as in (26). Therefore, no Lyapunov functional is expected in the nonstationary problem (25).
The first observation is the existence of a unique spatially homogeneous stationary solution to (25).

Proposition 1. For every > 0, let G be the set of solutions
for

f = f(u, v) defined by (22). Then G is composed of at least one and at most three elements.
Proof. Equality (31) is equivalent to where Thus, u * is a zero of a qubic polynomial, and the number of G is at most three. Since on the other hand, there is u * ∈ (0, ) satisfying (32). Hence, we obtain the result.
Remark 3. There is a case that G is composed of three elements. In the case described in Morita and Shinjo, 6 two of them are stable, and the other is unstable, as stationary solutions to the ordinary differential equation (50) given below.
Put (u * , v * ) = (u * ( ), v * ( )) in Proposition 1. The linearized operator around the solution u * = u * ( ) to (30) is given by We examine the degeneracy of this L in accordance with the eigenvalues 0 = 1 < 2 ≤ … → ∞ and the eigenfunctions j in || || 2 = 1 for = 1, 2, … . Where is the eigenfunction of −Δ with Neumann boundary conditions. First, for 1 = 0, it holds that 1 = constant and this condition is equivalent to although the possible bifurcated object is spatially homogeneous. Second, for j , j ≥ 2, it holds that ∫ Ω = 0, and the above degeneracy condition is reduced to Then, there is a chance of a spatially inhomogeneous bifurcation. From the above analysis, our main target is revealed. We want to prove that when D ≫ 1 is the case, in relation to , the solution (u, v) is asymptotically spatially homogeneous. The region that this holds cannot be the entire one because of the possible spatially inhomogeneous bifurcation of stationary states suggested above. Our result in the paper is the following theorem valid under the technical assumption Recall that 2 > 0 is the second eigenvalue of −Δ under the Neumann boundary condition and a 1 = b( + k 0 ) is the upper bound of a(u) in (24). Note that = 1 − D > 0 is not assumed in the following theorem and inequality (34) above and inequality (35) below are consistent if is sufficiently large.

Theorem 5. Assume (34). There exists a constant
for the solution (u, v) = (u(x, t), v(x, t)) to (25) for (23), wherē The -limit set (u 0 , v 0 ) defined by (8) satisfies for some (u * , v * ) ∈ G and So far, spatially homogenization in reaction-diffusion system under the Neumann boundary condition with fast diffusion has been studied in several contexts. In Conway et al., 14 this property is shown under the presence of the invariant region. Even without this property, the local theory 15 assures that spatially homogeneous compact attractor is also an attractor of the corresponding spatially inhomogeneous system. Both results are significantly based on the theory of dynamical systems. In these works, the exponential convergence rate to the spatially homogeneous part is assured, which our method did not reach. Theorem 5, however, assures that any solution exhibits asymptotic spatial homogeneity under the cost of large in this fourth model.
Since wave-propagation phenomena are reported in numerical simulations, 11,13 we can suspect some dynamics inside (u 0 , v 0 ) for the general case. In accordance with the conclusion (37), there is a possibility of (u 0 , v 0 ) ≠ {(u * ( ), v * ( ))}, as this -limit set contains nontrivial spatially homogeneous orbit of (25). See the final remark of the present paper. Concluding the present section, we refer to Henry 16 for fundamental concepts on the dynamical systems, -limit sets and LaSalle's principle used below.

PROOF OF THEOREM 5
Using w = Du + v, we transform (25) into where Lemma 6. The solution (u, w) to (39) satisfies the estimate: where (· , ·) is the usual inner product in L 2 (Ω) andw Proof. Recalling by (39). By Hence, there arises ( , and therefore, it holds that Then we obtain (40) because of the positivity of (− Δ) −1 . Since we get by (40). Here, a result on the boundedness of v = w − Du follows from the second equation of (25). In the following lemma, C 4 is a constant independent of v 0 . Actually, we use the semigroup estimate to bound ||v(· , t)|| 2 above by ||v 0 || 1 + ||u 0 || 1 , where the conditions t ≥ 1 and N ≤ 3 are required.
we argue as in Latos et al., 17 recalling a 0 = b > 0 in (24). First, we apply the comparison theorem to deduce where Δ is provided with the homogeneous Neumann boundary condition. Second, the semigroup estimate 18 is applied to the right-hand side of (44). It follows that provided that t ≥ 1 and N ≤ 3.