FINITE-TIME BLOWUP OF SOLUTIONS TO SOME ACTIVATOR-INHIBITOR SYSTEMS

We study a dynamics of solutions to a system of reaction-diffusion equations modeling a biological pattern formation. This model has activatorinhibitor type nonlinearities and we show that it has solutions blowing up in a finite time. More precisely, in the case of absence of a diffusion of an activator, we show that there are solutions which blow up in a finite time at one point, only. This result holds true for the whole range of nonlinearity exponents in the considered activator-inhibitor system. Next, we consider a range of nonlinearities, where some space-homogeneous solutions blow up in a finite time and we show an analogous result for space-inhomogeneous solutions.


(Communicated by Hirokazu Ninomiya)
Abstract. We study a dynamics of solutions to a system of reaction-diffusion equations modeling a biological pattern formation. This model has activatorinhibitor type nonlinearities and we show that it has solutions blowing up in a finite time. More precisely, in the case of absence of a diffusion of an activator, we show that there are solutions which blow up in a finite time at one point, only. This result holds true for the whole range of nonlinearity exponents in the considered activator-inhibitor system. Next, we consider a range of nonlinearities, where some space-homogeneous solutions blow up in a finite time and we show an analogous result for space-inhomogeneous solutions.
1. Introduction. In biology, a pattern formation mechanism is one of the most interesting subjects to understand. Mathematical analysis of such a phenomenon begun by the seminal paper of Turing [25], where a notion of diffusion-driven instability was used to explain that a reaction between two chemicals with different diffusion rates may cause a destabilization of a spatially homogeneous state, thus leading to the formation of nontrivial spatial structure. Turing's idea can be demonstrated by using reaction-diffusion equations for two interacting morphogens (u, v): where ε, D are nonnegative constants. Many such models based on Turing's idea have been proposed and there are several mathematical results on corresponding reaction-diffusion systems when both diffusion coefficients ε, D are positive.
Recently, however, the diffusion-driven instability has been observed in models describing a coupling of cell-localized processes with a cell-to-cell communication via diffusion. Such models are of a form of systems consisting of a single ordinary differential equation coupled with a reaction-diffusion equation: such as in Refs. [8,12,13,18]. There are numerical simulations describing a formation of spatial patterns in such reaction-diffusion-ODE models, but there are few mathematical results on the existence and stability of stationary solutions (and their dependence on kinetics of such systems) and on a dynamics of non-stationary solutions. Our aim in this work is to contribute to the mathematical theory of system (1.2) with particular nonlinearities by showing that it has solutions which become unbounded (blow up) in a finite time.
In this work, we consider both systems (1.1) and (1.2) with the following nonlinearities (kinetics) Here, a, b, γ are positive constants, and the exponents p, q, r, s satisfy p > 1, q, r > 0 and s ≥ 0. This is a typical kinetics based on Turing's idea which leads to the so-called activator-inhibitor system. Reaction-diffusion equations (1.1) and (1.3) with both non-degenerate diffusion coefficients ε > 0 and D > 0 and under the assumption were proposed by Gierer and Meinhardt [14] and have been widely used to model various biological pattern formations. In this work, we consider these systems in a bounded domain Ω ⊂ R n with a sufficiently smooth boundary ∂Ω, supplemented with the Neumann boundary conditions and with nonnegative initial data. First, we deal with equations (1.2)-(1.3), hence there is no-diffusion of the activator u(x, t). We show in Theorem 2.1 below that, for the whole range of exponents p, q, r, s, there are solutions of the initial-boundary value problem for the considered reaction-diffusion-ODE system where the function u(x, t) blows up in a finite time and at one point.
Here, let us recall the work [17] which gives a complete description of entire dynamics of the kinetic system of ordinary differential equations associated with It turns out that this dynamics already exhibits various kinds of interesting behaviors including the convergence to the equilibria (0, 0) and ( Concerning the existence and boundedness of solutions to initial-boundary value problems for reaction-diffusion equations (1.1) and (1.3) with both ε > 0 and D > 0, there are several results, see, e.g., [22,11,7,23,3,26]. In particular, under the assumption (p − 1)/r < 2/(N + 2), Masuda and Takahashi [11] proved that all solutions exist for all t > 0 and are uniformly bounded in time in the case when the first equation in (1.1) and (1.3) is supplemented with a nontrivial "basic production term", see [24] for studies of stability properties of stationary solutions to such equations. The global existence result from [11] was extended by Li, Chen and Qin [7] to the case of all exponents satisfying p − 1 < r. Jiang [3] and the author of [23] independently obtained similar results on the existence and the boundedness of solutions to a general activator-inhibitor system including the one in (1.1) and (1.3). Other recent results on the global-in-time existence of solutions to system (1.1) and (1.3) can be found in Ref. [26].
On the other hand, the dynamics of solutions to the activator-inhibitor system (1.1) and (1.3) is far from being understood and one may quote here a few works, only. In [24], a phenomenon called a collapse of patterns has been studied and, in [4], a stability of a periodic solution has been proved. Moreover, a blowup of solutions to the corresponding shadow system (namely, when the second equation in (1.1) and (1.3) is replaced by a nonlocal equation obtained formally in the fast diffusion limit D → ∞) has been shown in [6].
Methods developed in this work appeared to be useful to study blowup phenomena in reaction-diffusion-ODE systems (1.2) with other types of nonlinearities. In particular, in our recent work [10], we apply them to another class of equations which appear in mathematical biology.
2. Formulation of results. Now, let us state precisely of our results. We consider the following system of reaction-diffusion equations with ε ≥ 0 and D > 0, where a, b, γ are nonnegative constants, and the nonlinearity exponents in (2.1)-(2.2) satisfy We consider this system in a bounded domain Ω ⊂ R n with a smooth boundary ∂Ω, supplemented with the initial data u 0 , v 0 ∈ C(Ω) such that Moreover, we impose the Neumann boundary conditions In the following, for simplicity of notation, we use the quantities In our first result, we show that the theory on the global-in-time existence of solutions to problem (2.1)-(2.5) with both ε > 0 and D > 0 developed in [11,7,23,3,26] is no longer valid if ε = 0. Thus, in the following, we consider the initial-boundary value problem for the reaction-diffusion-ODE system of the form Here, as before, Ω ⊂ R n is an arbitrary bounded domain with a smooth boundary and, without loss of generality, we assume that 0 ∈ Ω. Moreover, system (2.7)-(2.8) is rescaled in such a way so that the diffusion coefficient in equation (2.8) is equal to one. In the following theorem, we prove that if u 0 is sufficiently well concentrated around an arbitrary point x 0 ∈ Ω (here, for simplicity of notation, we choose x 0 = 0), if v 0 is a constant function, and if γ > 0 is sufficiently small then the corresponding solution to problem (2.7)-(2.10) blows up in a finite time T max > 0, without additional restrictions on the exponents in nonlinearities.
Theorem 2.1. Assume the nonlinearity exponents satisfy (2.3) and let T > 0 be arbitrary. Suppose that 0 ∈ Ω and • there exists a number is a certain number determined in the proof. Then the corresponding solution to problem (2.7)-(2.10) blows up at some T max ≤ T . Moreover, the following uniform estimates are valid Remark 2.2. (Diffusion-induced blowup) Let us emphasize one application of Theorem 2.1 in the range of exponents If an initial datum in (2.10) is constant (i.e. x-independent), the corresponding solution of problem (2.7)-(2.10) is also x-independent and global-in-time, see [7,17].
On the other hand, by Theorem 2.1, there are nonconstant initial conditions, such that the corresponding solution to (2.7)-(2.10) with small γ > 0 blows up at one point in a finite time. This is another example of an initial-boundary value problem for reaction-diffusion equations, where a diffusion in one equation induces a blowup of solutions. First example of one reaction-diffusion equation coupled with one ODE, where some solutions blow up due to a diffusion, appeared in the paper by Morgan [16] and another example can be found in Ref. [2]. The term "diffusioninduced blowup" was introduced by Mizoguchi et al. [15] who proved a blowup of solutions to certain system of reaction-diffusion equations with nonzero diffusion coefficients in both equations for which space homogeneous solutions are global in time. Another system of reaction-diffusion equations with such a property was discovered by Pierre and Schmitt [19,20]. A discussion of other initial-boundary value problems with a diffusion-induced blowup, as well as several references, can be found in the survey paper [1] as well as in the monograph [21, Ch. 33.2].
Remark 2.4. If ε = 0 and D > 0, Theorem 2.3 is a particular case of Theorem 2.1 (however, with a different proof). By this remark, we would like to emphasize that Theorem 2.3 provides conditions for a blowup of solutions to problem (2.1)-(2.5) with both non degenerate diffusion coefficients ε > 0 and D > 0. This result is related to the one in Ref. [17] on a blowup of solutions to the kinetic system (1.5) with nonlinearity exponents satisfying r/(p − 1) < 1 and with initial data satisfying estimates more-or-less as the one in (2.15). Theorem 2.3 asserts that nonlinearities in system (2.1)-(2.5) cause an analogous blow up of space inhomogeneous solutions of this initial boundary value problem, however, for a smaller range of nonlinearity exponents (i.e. those satisfying first inequality in (2.14)).
Both Theorems 2.1 and 2.3 are proved in the following section.

Blowup of solutions in a finite time.
To show that some solutions to problems (2.1)-(2.5) and (2.7)-(2.10) blow up in a finite time, we first notice that if u(x, t), v(x, t) is their solution, then the functions u(x, t)e at and v(x, t)e bt satisfy the following boundary-value problem  If T max < +∞, then sup t∈[0,Tmax) u(·, t) ∞ = +∞.
Proof. The proof of the existence and the uniqueness of solutions to problem (3.1)-(3.4) is quite standard. Here, it suffices to apply the theory reported in the monograph [22] and in the work [11]. It is clear that, as long as both functions u and v are positive, they satisfy the inequalities u t ≥ ε∆u and v t ≥ D∆v in the domain Ω together with the Neumann boundary conditions. Thus, for ε > 0 and D > 0, this solution satisfies estimates (3.6) by a comparison principle for parabolic equations. For ε = 0, we have u t (x, t) ≥ 0 for all x ∈ Ω and t ∈ (0, T max ), hence u(x, t) increases on [0, T max ) for each fixed x ∈ Ω. Thus, u(x, t) satisfies the estimate in (3.6), again. We refer the reader to [9, Lemma 3.4] for a similar reasoning in the case of another reactiondiffusion-ODE system. Now, we show that an upper bound for v(x, t) leads to the blowup of u(x, t) in a finite time.
(3.7) Then u(x, t) blows up at certain T max ≤ T .
Proof. Applying the comparison principle to equation (3.1) either for parabolic equations if ε > 0 or for ordinary differential equations if ε = 0, we obtain the estimate The functionū 1 may be computed explicitly: (3.10) Recalling the definition of the number R 0 in (3.7), we obtain thatū 1 (t) blows up at t = T , which due to inequality (3.8) implies that T max ≤ T .  (3.14) where the functions f (t) and g(t) are defined in (3.5). First, we recall a classical result on the Hölder continuity of solutions to the inhomogeneous heat equation.
where e tD∆ t≥0 is the semigroup of linear operators on L (Ω) generated by D∆ in a bounded domain with a smooth boundary, and supplemented with the Neumann boundary conditions. There exist numbers β ∈ (0, 1) and C = C > 0 depending on for all x, y ∈ Ω. From now on, we deal with problem (3.11)-(3.14). It follows from Lemma 3.2 that it suffices to estimate the function v(x, t) from above to obtain a blowup of u(x, t) in a finite time. Now, we prove that such an upper bound for v(x, t) is a consequence of a certain a priori estimate imposed on u(x, t).  u(x, t) and v(x, t) be a solution to problem (3.11)-(3.14). Suppose that there is a number such that, a priori, the following inequality holds true Then, there is an explicit number C 0 > 0 (see equation (3.23) below) such that for Proof. We use the following integral formulation of equation (3.12) Here, we recall the following well-known estimates for the heat semigroup which are valid for all t > 0, D > 0, and all w 0 ∈ L ∞ (Ω): and for each ∈ [1, ∞], with a constant C = C( , n, D, Ω) independent of w 0 and of t, see e.g. [22, p. 25]. Now, we compute the L ∞ -norm of equation (3.19). Using inequalities (3.20) and (3.21), the lower bound of v in (3.6) as well as the a priori assumption on u in (3.17) we obtain the estimate where the constant g 1,T is defined in (2.6). Here, we choose > n/2 to have n/(2 ) < 1, which leads to the equality Moreover, we assure that < n(p − 1)/(αr) or, equivalently, that α r/(p − 1) < n to have |x| − αr p−1 ∈ L (Ω). Such a choice of ∈ [1, ∞) is always possible because max{1, n/2} < n(p − 1)/(αr) under our assumptions on α in (3.16).
Next, we apply Lemma 3.3 to show the Hölder continuity of v(x, t).
We are ready to prove a result on the one-point blowup of solutions to the reaction-diffusion-ODE problem (3.11)-(3.14).
Proof of Theorem 2.1. Let (u(x, t), v(x, t)) be a solution to the modified problem (3.11)-(3.14). By Lemmas 3.2 and 3.4, it suffices to show the a priori estimate (3.24) under the assumption that γ > 0 is sufficiently small. Let T > 0 be a number such that inequality (2.12) holds true. By assumption (2.11), we have 0 < u 0 (x) < |x| − α p−1 for all x ∈ Ω, hence, by a continuity argument, inequality (3.24) is satisfied on a certain initial time interval. Suppose a contrario that there exists T 1 ∈ 0, min{T max , T } such the solution of problem Now, for a given function v(x, t), we solve equation (3.11) with respect to u(x, t) to obtain (3.27) We are going to use this explicit formula for u(x, t) and the Hölder regularity of v(x, t) from Lemma 3.3 to obtain a contradiction with equality (3.26). First, notice that assumption (2.11) can be written as u 0 (x) 1−p ≥ 2|x| α + u 0 (0) 1−p . Thus, we may estimate the denominator of the fraction in (3.27) using this assumption as follows (3.28) By the definition of T max and due to formula (3.27), we immediately obtain for all t ∈ [0, T max ). (3.29) Next, we use our hypotheses (3.25) and (3.26) implying estimate (3.18) and the Hölder continuity of v(x, t) established in Lemma 3.5, as well as the lower bound of v(x, t) in (3.6), to find constants C > 0 and α ∈ (0, 1), satisfying also (3.16), such that for all (x, τ ) ∈ Ω × [0, T 1 ]. Hence, since T 1 ≤ T , we obtain the following lower bound This inequality for t = T 1 contradicts our hypothesis (3.26).
Thus, estimate (3.24) holds true on the whole interval [0, min{T max , T }). Then, by Lemma 3.4, the function v(x, t) is bounded from above by a constantv 0 + γC 0 < R 0 , provided γ > 0 is sufficiently small. Finally, Lemma 3.2 implies that u(x, t) blows up at certain T max ≤ T .
Proof of Theorem 2.3. Let us finally prove our second blowup result.