Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space

In this paper, we consider solutions to a 
Cauchy problem for a parabolic-elliptic system in two dimensional 
space. This system is a simplified version of a chemotaxis model, and is 
also a model of self-interacting particles. 
 
The behavior of solutions to the problem closely depends on the 
 $L^1$-norm of the solutions. If the quantity is larger than $8\pi$, the 
 solution blows up in finite time. If the quantity is smaller than the 
 critical mass, the solution exists globally in time. In the critical 
 case, infinite blowup solutions were found. 
 
In the present paper, we direct our attention to radial solutions to the 
 problem whose $L^1$-norm is equal to $8\pi$ and find bounded and 
 unbounded oscillating solutions.

1. Introduction. In this paper, we consider radially symmetric solutions to the Cauchy problem for a parabolic-elliptic system where u 0 is a non-negative L 1 -function. This system is a simplified version of a chemotaxis model (see [4,9]). The model was introduced to describe the aggregation of cellular slime molds. In the model, the function u denotes the density of cells and the function v denotes the concentration of a chemoattractant secreted by themselves. The system is also a model of self-interacting particles (see [1]). Here, the function u is the density of particles interacting with themselves through the potential v.
It is well known that the behavior of solutions to (1), (2) and (3) closely depends on the quantity Λ = R 2 u 0 (x)dx. Actually, radial solutions to (1), (2) and (3) blow 1862 YŪKI NAITO AND TAKASI SENBA up in finite time if Λ > 8π and the condition Λ < 8π is a sufficient condition for the time-global existence and the boundedness of radial solutions (see [2]). This proposition holds also true without symmetry, if (see [3,5]). Under a weaker assumption than (4), Ogawa and Nagai [11] showed the global existence and the boundedness. Recently, Nagai [10] showed that solutions exist globally in time and that these solutions converge to 0 without the assumption (4).
If T < ∞, we say that the solution blows up in finite time. If T = ∞, we say that the solution blows up in infinite time.
In the critical case Λ = 8π, radial solutions exist globally in time (see [2]). Under the assumption (4), solutions also globally in time without symmetry (see [3]). The system has stationary solutions in the critical case. Some solutions converge to a stationary solution (see [2]) and some solutions are unbounded (see [3]). In [2] the existence of oscillating solutions was conjectured.
For b > 0, putting we see that (u b , log u b − log 8b) are stationary solutions to (1) and (2) and that the corresponding mass distribution functions are also stationary solutions to (6) and (7) with Λ = 8π. In [2] a stability of stationary solutions M b was shown in the sense of L 1 loc norm. Since the stability does not suffice to show the existence of oscillating solutions, we show a stronger stability (see Section 3). By using the stability, we construct oscillating solutions to (6), (7) and (8). Furthermore, we must deduce that the corresponding solution (u, v) to (1), (2) and (3) oscillates by using the parabolic regularity argument. That is the difference between our proof and the one in [12].
We find a solution whose ω-limit set includes a number of stationary solutions.

YŪKI NAITO AND TAKASI SENBA
We deduce from (13) and (14) that and that the initial data of the solutions do not satisfy (4). This means that solutions mentioned in Theorem 1.2 blow up in infinite time and that these solutions are different from blowup solutions mentioned in [3]. For solutions mentioned in Theorem 1.1 and any stationary solutions u b with b ∈ E, there exists a sequence {t n } n≥1 ⊂ (0, ∞) such that lim n→∞ t n = ∞ and that lim In the functional space L ∞ (R 2 ), the orbit {u(·, t)} t≥0 passes through any neighborhood of the stationary solution u b infinitely many times. This means that these solutions are bounded and oscillate among stationary solutions . Similarly, the solutions mentioned in Theorem 1.2 are unbounded and oscillate among stationary solutions {u bj } j≥1 , since the set Oscillation of solutions is caused by the stability of stationary solutions, the continuity of solutions with respect to initial functions and the asymptotic behavior of initial functions at |x| = ∞. We deduce from Proposition 2 and the parabolic regularity that lim dx for any sufficiently large r ∈ (0, ∞).
This means that the limit lim t→∞ u(·, t) is decided by the behavior of the initial function u 0 at |x| = ∞. On the other hand, we deduce from the continuity of solutions with respect to initial functions that Then, if the initial function u 0 satisfies that |x|<r u 0 (x)dx = |x|<r u a (x)dx for r ∈ (0, L), |x|<r u d (x)dx for r ≥ L for a sufficiently large L, the solution u stays near the stationary solution u a for a long time and finally converges to the stationary solution u d . Therefore, if |x|<r u 0 (x)dx suitably oscillates between |x|<r u a (x)dx and |x|<r u d (x)dx at r = ∞, the solution u(·, t) oscillates between two stationary solutions u a and u d . That is the reason why the solution oscillates.
This mechanism appears in the proof of Proposition 5 directly. This indicates that the dynamics of solutions to (1), (2) and (3) are complicated. The behavior is a complete contrast to the one of positive solutions to the so-called Keller-Segel system in a bounded domain of R 2 with zero Neumann boundary conditions. Actually, any positive solution to the problem converges to a stationary solution as time tends to infinity, if the solution exits globally in time and remains uniformly bounded (see [6]). This paper is organized as follows: In Section 2, we describe the well-posedness of solutions, some integral transformations and a stability of stationary solutions.
In Section 3, we show the continuity of solutions with respect to initial data. In Section 4, we show a special version of Theorem 1.1. The proof is simpler than the proofs of Theorems 1.1 and 1.2, and is essentially the same as those of Theorems 1.1 and 1.2. In Section 5, we describe the proofs of Theorems 1.1 and 1.2.
2. Well-posedness, integral transformations and stability of stationary solutions. The following result is an immediate conclusion from [2, Theorem 2.1 and Proposition 2.3]. Thus, we omit the proof.

YŪKI NAITO AND TAKASI SENBA
where W 0 (X) = |X| −2 M 0 (|X|). Here, we write X = (X 1 , X 2 , X 3 , X 4 ), For each b > 0 and each function g satisfying 0 ≤ |X| 2 g(X) < 8π in R 4 we defined a functional H b as which is essentially the same as the functional F b . For b > 0, the function W b is a stationary solution to (17), and is stable in the following sense. (17) and (18) satisfies Since the following lemma is an immediate conclusion from the comparison theorem, we omit the proof. (17) and (18). Proof of Proposition 3. The solution W to (17) and (18) can be represented as where |S 3 | is the area of the unit sphere in R 4 and B(R) is a closed ball with center 0 and radius R in R 4 . We can represent W − W b as Since we infer from the assumption and Lemma 2.1 that and that with a positive constant C, we get Here and henceforth, C represents a positive constant independent of C I = {t, x, X, r, ε, δ}. Then, each C may vary from line to line and may depend on constants a, b and d. For any sufficiently small δ > 0 and T ∈ (0, t − δ), put We deduce from (20) and (21) that for t > T + δ, R > 1 and δ ∈ (0, T ). Take R = δ −3 , T = t/2 and a sufficiently small δ > 0. Thanks to Proposition 2, we see that Since δ is an arbitrary positive constant, we get Thus, we finish the proof.
3. Continuity of solutions with respect to initial data. In this section, we consider the continuity of solutions with respect to initial data u 0 .
For > 0, we define W (·, · : ) as a solution to a Cauchy problem Proposition 4. Assume that W 0 1 and W 0 2 are functions satisfying for X ∈ R 4 and i = 1, 2.
We use the following lemma in the proof of Proposition 4.
with some positive constants A and B. Then, the function f satisfies Proof. Putting F (t) = max 0≤s≤t f (s) for t ≥ 0, we obtain We deduce from this that Putting we see G(t) ≥ F (t)/2 and G(t) ≥ A. These imply So the Gronwall inequality gives Thus, we finish the proof.
Proof of Proposition 4. The comparison theorem yields We can write W 1 (·, · : ) − W 2 (·, · : ) as We see Since we deduce from (23) that and that These imply from which and Lemma 3.1 it follows that Thus, we finish the proof.
4. Special version of Theorem 1.1 and its proof. The following proposition is a special version of Theorem 1.1 and is shown by a similar argument as that in main results. Then, we write the proposition and its proof.
Proof. Let W 0 be a radial, continuous function satisfying for some positive constant L 1 ≥ 1. This entails .

5.
Proofs of Theorems 1.1 and 1.2. The proof of Theorem 1.1 is essentially the same as the one in Section 4. Therefore, we describe a sketch of these proofs.
Proof of Theorem 1.1. There exists a countable set {b j } j≥1 such that and that b i = b j if i = j, where {b j } j≥1 is the closure of the set {b j } j≥1 . For the set {b j } j≥1 we defined a set {β j } j≥1 as −1+j = b j for j = 1, 2, 3, · · · , k + 1 and k ≥ 1. According to the representation in Remark 1, we define W 0 as follows: for X ∈ B(L j+1 ) \ B(L j ) and j ≥ 1.
Taking {L j } j≥1 such that L j+1 L j for j ≥ 1, we can find a sequence {T j } j≥1 such that T j+1 ≥ T j + 2 for each j ≥ 1, W 0 is continuous in R 4 and that W (·, t) − W βj Cω ≤ 1 j for t ∈ [T j − 1, T j + 1] and j ≥ 1 by using a similar argument as that in Lemma 4.1, where W (·, ·) is a solution to (17) with W (·, 0) = W 0 . Furthermore, using a similar argument as that in Proposition 5, we obtain that the corresponding solution (u, v) to (1), (2) and (3) satisfies {u bj } j≥1 ⊂ ω(u) and (10). Combining this with the way to choose the sequence {b j } j≥1 implies that {u b } b∈E ⊂ ω(u). By using sup t≥0 W (·, t) Cω < ∞ and a similar argument as that in Proposition 5, we obtain that Thus, we finish the proof.
We can obtain also Theorem 1.2, by using a similar argument as that in Proposition 5. However, we must note the way to choose the constant , when we use Proposition 4. Because, the sequence {W bj } j≥1 satisfies lim j→∞ W bj (0) = ∞.
Proof of Theorem 1.2. For the sequence {b j } j≥1 , take the same sequence {β j } j≥1 as the one in the proof of Theorem 1.1. For k ≥ 1, put A(k) = max 1≤j≤k b j and B = min j≥1 b j . We assume that b j+1 = b j for j ≥ 1 without loss of generality. Let W 0 be a radial continuous function satisfying for some positive constant L 1 ≥ 1. This entails .