Finite time blow up and concentration phenomena for a solution to drift-diffusion equations in higher dimensions

We show the finite time blow up of a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If the initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not exist globally in time and blows up in a finite time for the scaling critical space. Besides there exists a concentration point such that the solution exhibits the concentration in the critical norm. This type of blow up was observed in the scaling critical two dimensions. The proof is based on the profile decomposition and the Shannon inequality in the weighted space.

Since u denotes the density of particles, it is natural to consider a non-negative solution, and in this case, the L 1 norm of the solution u(t) is preserved in time and this corresponds to the mass conservation law. Under this setting the question of whether the solution exists globally in time or blows up in a finite time is a basic problem. If the initial data is large and decays fast at spatially infinity, the solution blows up in a finite time. In this paper, we discuss such instability of solution under weaker assumptions on the initial data that decays slower at space infinity.
The system (1.1) involves the Poisson equation and the solution u is influenced by a nonlocal effect from the Green's function of the Poisson equation. Hence, the large time behavior of the solution is largely depending on the behavior of the solution at spatial infinity. Hence, the weight condition on the data may give a subtle effect on the large time behavior of the solution. Therefore, to eliminate the weight condition or reduce the condition is an interesting problem to (1.1).
The local existence of the solution in both the semi-group approach and the energy method is now well established (cf. [27]). To state results, we define some function spaces: For s > 0 and 1 ≤ p ≤ ∞, where · = (1 + | · | 2 ) 1/2 . Noting that L p s (R n ) ⊂ L 1 (R n ) if s > n/ p, we recall the existence and uniqueness of the solution for the n dimensional drift-diffusion equation in a critical space L n 2 (R n ).
Definition. Let n ≥ 3 and 1 ≤ p < ∞. For u 0 ∈ L p (R n ), we call u a mild solution to the system (1. Proposition 1.1 (Local well-posedness and conservation laws) Let n ≥ 3 and n 2 ≤ p < n. For any u 0 ∈ L p (R n ), there exists T > 0 and a unique mild solution (u, ψ) to (1.1) with the initial data u 0 such that u ∈ C([0, T ); L p (R n )) ∩ L θ (0, T ; L q (R n )) with 2/θ + n/q = 2 and q > n 2 . Moreover, the solution has higher regularity u ∈ C([0, T ); W 2, p (R n )) ∩ C 1 ((0, T ); L p (R n )) and it is a strong solution for (1.1). Besides there exists a maximal existence time T = T * ≤ ∞ such that if T * < ∞, then for any n 2 < p ≤ ∞, lim t→T * u(t) p = ∞.
(1.2) (4) If in addition u 0 ∈ L 1 b (R n ), where b > 0 and p ≥ n 2 , then the solution (u, ψ) satisfies If u 0 ∈ L 1 2 (R n ), then the second moment of the solution satisfies the following identity:

5)
where x ∈ R n is an arbitrary point.
The local well-posedness in L p space is essentially due to the earlier works by Weissler [50] and Giga [15], where they consider the nonlinear heat equations and incompressible Navier-Stokes equations. Since the scaling structure to (1.1) is similar to the Navier-Stokes equation, we may apply those theories and obtain the existence and uniqueness of the mild solution by Kurokiba-Ogawa [28] (see also for the critical case [25] and in the weighted space [27]). If the initial data is non-negative and integrable, then the weak maximum principle and the conservation law of the average assure that the weak solution preserves the total mass (1.2). This is natural consequence from the equation originally appears from the conservation laws (cf. [14,24]).
On the other hand, if we consider the invariant scaling property, namely, the equation has a scaling invariant property that for λ > 0, is invariant scaling for the system. The critical space coincide with the invariant scale is and for two dimensional case it is L ∞ (R + ; L 1 (R 2 )) × L ∞ (R + ; L ∞ (R 2 )). The global existence in the scaling critical spaces is a direct consequence of Fujita-Kato's principle. Proposition 1.2 (Global existence) Let n ≥ 3, p ≥ n 2 , and b > 0. Assume that the initial data u 0 is non-negative in L p (R n ) ∩ L 1 b (R n ), and for some constant B n > 0 depending only on n such that u 0 n 2 < B n . (1) Then the corresponding solution u obtained by Proposition 1.1 exists globally in time.
(2) The solution decays in time: One can find the constant B n can be chosen as (see for instance [13], see also [11] for an improved constant), where S n denotes the best possible constant of Sobolev's inequality (cf. Talenti [47]). Under this assumption we see that for some T > 0, it holds Then u(t) n 2 is uniformly bounded by B n . This implies a uniform bound for the solution.
The main subject of this paper is to show an instability result of the mild solution to (1.1). When n = 2, the solution to (1.1) exists globally in time or non-negative initial data On the other hand, if then the solution blows up in a finite time. Namely, there exists T m < ∞ such that lim t→T m u(t) p = ∞ for all 1 < p ≤ ∞ (see one dimensional modification [9]). In Biler [2] and Nagai [33,34], the finite time blow up of the positive solutions are shown (cf. Kurokiba-Ogawa [27]), under certain conditions for two dimensional case.
While if we consider the higher dimensional case, the invariant space is shifted in L n 2 and the L 1 conservation law does not work very well. In this sense, the problem is a "super critical" case. On the other hand, the usage of the entropy functional yields new difficulty to show the finite time blow-up. Biler [3] wherex is the second mass center u 0 , that is, Then the solution blows up in a finite time. In both results, the initial data and solutions are both assumed to be in L 1 2 (R n ), namely, the second moment of the solution remains finite for all time up to the maximal existence time. This is indeed a natural condition of the system if we regard that the u(t, x) dx as a probability measure assuming u 0 1 = 1 with positivity. The following statement is essentially due to Biler [3] in a bounded domain and refined by Calvez-Corrias-Ebde [11] for the whole space case and Ogawa-Wakui [41].
wherex is the b-th mass center of u 0 .
(1) When b ≥ 2, then and the solution u to (1.1) blows up in a finite time. Namely, for any n 2 < p ≤ ∞, there exists some T * < ∞ such that (1.8) (2) If u 0 ≥ 0 is radially symmetric and b > 0, then Our main result is the concentration phenomena for the blowing up solution: (1) Then the blowing up solution u to (1.1) concentrates the following sense: Let T * > 0 be the blow up time and {t k } k∈N satisfy t k → T * and u(t k ) n 2 → ∞ as k → ∞. Then for any ε > 0, there exist a subsequence of {t k } k∈N and x * ∈ R n such that where C HLS is the best possible constant of the Hardy-Littlewood-Sobolev inequality, that is, (1.11) (2) Furthermore, if T * < ∞, then for any ε > 0, there exist a subsequence of {t k } k∈N and {x k } k∈N ⊂ R n such that

12)
where C HLS is defined by the above.
By the assumption (1.6) with C n,b = 2πe n n−2 /n, we see that It follows from the Hardy-Littlewood-Sobolev inequality (1.11) and Shannon inequality (2.1) (see Proposition 2.1 below) that which implies that the initial data u 0 ≥ 0 satisfies (1.13) The constant of the right hand side coincides with one appearing in (1.10). We also see that for n ≥ 3 (see Proposition 2.4 below). For a radially symmetric solution to (1.1), one can extend the condition of the weighted Lebesgue space. Since the assumption on the initial data is different from the case of b ≥ 2, the constant appearing in the concentration phenomena changes as follows: (1) If b ≥ 2 and u 0 satisfies (1.6) with C n,b = 2πe n n−2 /n, then the blowing up solution u to (1.1) concentrates the following sense: Let 0 < T * < ∞ be the blow up time and {t k } k∈N satisfy t k → T * and u(t k ) n 2 → ∞ as k → ∞. Then for any ε > 0, there exists a subsequence of {t k } k∈N such that (1.14) (2) If b > 0 and u 0 satisfies (1.6) with (1.9) for any δ > 0, then the blowing up solution u to (1.1) concentrates the following sense: Let 0 < T * < ∞ be the blow up time and {t k } k∈N satisfy t k → T * and u(t k ) n 2 → ∞ as k → ∞. Then for any ε > 0, there exist a subsequence of {t k } k∈N such that  [31] and Tsutsumi [48]).

Remark
In the derivation of (1.13), if we assume that u 0 satisfies (1.6) with (1.9) instead of C n,b = 2πe n n−2 /n, then we see that While we relax the weight condition from b ≥ 2 to b > 0 in Theorem 1.5, the assumption of the initial data needs to be tightened when b > 0. The result of Theorem 1.5 (2) follows from the analogous argument of Theorem 1.4 naturally.
It is worth comparing the above result and the case of the degenerate drift-diffusion system: where α > 1 denotes the adiabatic constant originated from the pressure term P(ρ) = cρ α in the barotropic damped compressible Navier-Stokes-Poisson system (cf. [14,24]). It is known that there are two critical exponents α * = 2 − 4/(n + 2) and α * = 2 − 2/n (see [8,12,40,42,45,46,49]). In the case α * ≤ α ≤ α * (see [23]), the threshold for the global existence of the weak solution to (1.16) is identified as and V is the optimizer for the Hardy-Littlewood-Sobolev inequality In particular case of β = 0 when α = α * , the threshold is From this estimate, the above lower bound for the concentration coincides the threshold for the global existence of the weak solution to the degenerate drift-diffusion equation in higher dimensions.
The proof of Theorem 1.4 is based on the profile decomposition in L 1 (R n ). We take {t k } k∈N such that t k → T as k → ∞ and introduce the rescaled solution sequence for a blowing up solution u to (1.1). Then u k 1 = u 0 1 and u k n 2 = 1. In Bedrossian-Kim [1], they showed the profile decomposition in L 1 (R n ), but their technique allows that the profile depends on k. In this paper, we improve the extraction of the profile independent of k from {u k } k∈N . In order to deny the possibility that {u k } is vanishing, we use the bump function method for the rescaled equation (see Biler [2], Biler-Cieślak-Karch-Zienkiewicz [4], Biler-Karch-Zienkiewicz [5]).

Preliminaries
The following inequality originally due to Shannon is useful to estimate the logarithmic functional (see [29,41]).
wherex is the b-th mass center of f and From this lemma, we see that and there is a limitation of the blow-up speed of the L n 2 -norm of the solution u to (1.1) and the speed of the b-th moment.
The following inequality is well-known and will be used in later often:

Proposition 2.3 (Hardy-Littlewood-Sobolev inequality) Let n ≥ 3. Then there exists a constant C HLS > 0 which is depending only on n such that for any function f
Moreover, there exists the extremal function V ∈ L 1 (R n ) ∩ L n 2 (R n ) such that V is radially symmetric and decreasing function satisfying the Euler-Lagrange equation

Proof of Proposition 2.3 The Hardy-Littlewood-Sobolev inequality implies that it follows that
By Hölder's inequality, we have In particular, let α = n 2 , then θ = 0, so that the inequality By the definition of the Poisson kernel, we can rewrite as where C HLS denotes the best possible constant of the Hardy-Littlewood-Sobolev inequality and C HLS = C n c n . This implies the inequality (2.2). The attainability and Euler-Lagrange equation (2.3) of the inequality (2.2) is proved by Kimijima-Nakagawa-Ogawa [23] (see also [10]).

Proposition 2.4 (Comparison of constants)
The best constant C HLS is estimated from the above by

4)
where S n is the best constant of Sobolev's inequality given by 2 n for all f ∈Ḣ 1 (R n ). In particular, for n ≥ 3, The precise constant for S n is known by Talenti [47]. In the view of (1.10), it is interesting to observe that Indeed, the best constant C HLS can be decomposed into the explicit constant and implicit one as where C n is the best constant defined by In particular, the best constant C n becomes 1 when n = 2. In this case, the role of the constant coincides with the threshold in the case of n = 2.

Proof of Proposition 2.4
It is easy to see that the best constant S 2 n also gives the best constant for the inequality for any f ∈ L 2n/(n+2) (R n ) by the duality argument (see for instance [40]). Hence, by Hölder's inequality, one can observe that . This shows C HLS ≤ S 2 n . To see that C HLS < S 2 n , we assume on the contrary that C HLS = S 2 n for n ≥ 3. Then we may show that there exists an extremal function V ∈ L 1 (R n ) ∩ L n 2 (R n ) such that it attains the best constant, that is, Then by the Hölder and Sobolev inequalities, This shows that V is also the extremal function that attains the best possible constant of Sobolev's inequality, that is, Since the extremal function of this inequality is uniquely identified up to translation and dilation, namely, < 2n (n − 2)C HLS for any n ≥ 3. Thus, we conclude that (2.5) holds.
As an alternative proof of (2.4), since the extremal function of this inequality is uniquely identified up to translation and dilation, one can compute V 1 V n 2 and V 2n/(n+2) explicitly. Since it holds that where B(·, ·) is the beta function, the problem can be reduced as the comparison between One can check those values are different from each other by numerical computation 1 . For the higher dimensions, one can show it by applying the Stirling formula The ratio of norms of V is estimated by For example, in the case of n = 200, we compute

Profile decomposition in L 1
The following decomposition is originally due to Gerard [16] (cf. Nawa [39]) and extended by many authors. Here we show a slightly modified version of the result due to Bedrossian-Kim [1] (see also Hmidi-Keraani [19]).
with the following properties: In particular, passing k → ∞, we have the almost orthogonality for some radius R (1) > 0 independent of k. We set for x ∈ R n . By the assumption (3.1), we see that On the other hand, we have Thus, it holds that which implies that {f k } is uniformly integrable. Then the Dunford-Pettis theorem implies that there exist a subsequence {f k l } l∈N ⊂ {f k } and a nonnegative functionF (1) ∈ L 1 (B R (1) (0)) such thatf k l converges weakly toF (1) in L 1 (B R (1) (0)). Thus, we set a profile F (1) as the zero expansion ofF (1) , that is, Moreover, by the weak lower continuity of norm, we have On the other hand, we set then e k l converges weakly to 0 in L 1 . In this case, we define w k ≡ 0. Then the claim in with the following estimates: These properties imply that f k,1 , f k,2 , and h k are the compact, escape, and error term, respectively. Firstly, similarly to Case 1, we set and use the Dunford-Pettis theorem, we can get a weak convergence subsequence off k,1 so that we can set a profile F (1) , which has a compact support and satisfies On the other hand, the escape term f k,2 satisfies Secondary, we divide some cases whether the mass of the escape term is small or not. If we can take μ ∈ (0, M) such that for some radius R (2) > 0. We note that passing k → ∞, then |x Similarly to Case 1, we get a profile F (2) as a weak convergence limit off k,2 and define Then the claim in Lemma 3.1 holds in J = 2. Case 3.2. Vanishing: If the vanishing occurs, similarly to Case 2, we define Then we have the claim of Lemma 3.1 with J = 1. Case 3.3. Dichotomy: If the dichotomy occurs, there exists ν ∈ (0, 1) such that there exists the non-negative function sequences {f k, 21 2,1 +f k,2,2 ). For each sequences, there exist {x with the following estimates: , .
By the definition off k,2 , we can define h k,2 ≡ f k,2 1hk so that we have If we can take ν ∈ (0, 1) such that lim l→∞ f k,2,2 1 < ε 2 , f k,2,2 ≡ f k,2 1fk,2,2 then we take a subsequence {k l } which satisfies the above estimates and define Then the claim in Lemma 3.1 holds with J = 2. In the case of we apply the same argument above many times. The proof of Lemma 3.1 relies upon the induction argument. We omit to prove in detail. Note that the argument must be terminated in a finite step. Indeed, we assume on the contrary, these steps continue infinitely. Then it holds that lim k→∞ f k,l,2 1 ≥ ε 2 for any l ∈ N. Take the sum with respect to l, then the total mass f k 1 diverges, that is, This contradicts the assumption of Lemma 3.1.

Remark
In [1], they also showed the profile decomposition on L 1 (R n ). We emphasize that By Shannon's inequality (2.1), it follows that Since the mass conservation (1.2) holds, then we can rewrite as While by the virial law (1.5) for (1.1), we have and hence, under the assumption From the assumption (4.4), we obtain In what follows, we show Theorem 1.4 (1). We assume that the initial data u 0 ∈ L n 2 (R n )∩ L 1 b (R n ) satisfies the blow-up condition (1.6) with (1.7) and the solution u(t) to (1.1) blows up at T * > 0. Namely, Let {t k } k∈N be a sequence that gives the supremum of u(t) n 2 . We introduce the scaling transform by λ > 0 that This is the L 1 -invariant scaling S λ u(t) 1 = u(t) 1 = u 0 1 and it also holds that  (4.6) Then we find that the rescale solution Note that the scaling is L 1 -invariant and hence the L 1norm of v is invariant. The rescaled solution v(t, x) now solves the Cauchy problem of a semistationary equation where t 0 = 0. The solution preserves its total mass v(t) 1 = u(t) 1 = u 0 1 for any time t ≥ 0 and v(t k ) n 2 = 1 at t = t k . We set the rescaled solution sequence satisfies u k 1 = u 0 1 , u k n 2 = 1.
which satisfy the following properties: (1) u k is decomposed as (2) Almost orthogonality: (4.13) and for any j ∈ {1, 2, . . . , J }, it follows that . (4.14) (3) Drift term estimate: We note that the vanishing in the sense of Lions' concentration compactness lemma implies J = 0. We emphasize that the sequence {u k } k∈N is not vanishing and one can extract at least the profile of {u k }.
We note that {e k } k∈N is uniformly bounded in L n 2 (R n ). Hence, the Hardy-Littlewood-Sobolev inequality (2.2) shows that |∇| −1 e k 2 2 ≤ C HLS e k 1 e k n 2 ≤ C e k 1 → 0 as k → ∞. Next, we shall estimate the vanishing term w k . We fix ε > 0 arbitrarily. For R > 0, we separate the non-local term into three parts as follows: where c n = 1/((n − 2)ω n−1 ). For the integral I 1 , by the Hausdorff-Young inequality, we have where χ A (·) denotes the characteristic function of a set A, 1 ≤ p ≤ n 2 and 1 ≤ q < n n−2 Then the integral of the kernel can be estimated as By the uniform boundedness of L 1 and L n 2 for {u k }, there exists R 0 > 0 independent of k such that for any R > R 0 , For the integral I 3 , it follows that Thus, there exists R 1 > 0 independent of k such that for any R > R 1 , We setR ≡ max{R 0 , R 1 }. Lastly, the integral I 2 can be estimated as Since w k is the vanishing term, we obtain Thus, we conclude lim k→∞ |∇| −1 w k 2 = 0.
In order to extract at least one profile, we shall show that {v(t k )} k∈N∪{0} is not a vanishing sequence by using the bump function method (see [2,4,5]). We set the bump function as For ε ∈ (0, 1/ √ 3), the Hessian of η satisfies H η ≤ −c ε I for |x| ≤ ε, (4.16) where c ε ≡ 4(1 − 3ε 2 ). For R > 0 and a ∈ R n , we define in particular, η R is concave on |x − a| < R √ n/(n + 2) as (4.16). Multiplying the equation (4.7) by η R and integrating over x ∈ R n , then we have By the property (4.17), we see that On the second term in the right hand side, using Poisson representation and symmetry, we decompose where we set For J 1 , the integrand is rewritten by We fix any ε ∈ (0, √ n/(n + 2)). By the concavity of η R as (4.16), we have Since we know that If we set a constant as Thus, we obtain On the other hand, we recall that the integration over other regions is defined as In order to estimate J 2 , it suffices to consider the integral over {(x, y) ∈ R n × R n ; |x − a| < R, |y − a| ≥ ε R} by the symmetry property of J 2 and the fact that ∇η R (x) vanishes on the set {x ∈ R n ; |x − a| > R}. For |x − a| < R and |y − a| ≥ ε R, we have Thus, we have By the Hardy-Littlewood-Sobolev inequality and v(t k ) n 2 = 1, we have There exists k 0 ∈ N and ε 0 > 0 such that for any k ≥ k 0 , for some constant C 0 > 0. Thus, there exists R 0 = R(ε 0 ) > 0 such that for any k ≥ k 0 , since v(t k ) 1 = 1. Therefore, it follows that for any k ≥ k 0 , for some constant C > 0, which implies that {v(t k )} k∈N is not vanishing sequence.

Proposition 4.3 (Profile decomposition) There exist an integer J ∈ N and a non-negative function sequence {U
where supp U ( j) are disjoint and |x Proof of Theorem 1.4 (1) On the right hand side of the inequality (4.1), putting t = t k and scaling with respect to λ k , then the right hand side is rewritten by By the auxiliary inequality (4.1), it holds that for all k = 1, 2, . . . , By the above inequality (4.19) and the profile decomposition (4.18), for any ε > 0, there exists k * ∈ N such that for any k ≥ k * , we see that By the Hardy-Littlewood-Sobolev inequality (2.2) and (4.13), Thus, there exists j 0 ∈ {1, 2, . . . , J } such that 2n (n − 2)C HLS Moreover, by (4.14), there exist x Substituting this to the inequality (4.20), then by scaling, we conclude that By passing a subsequence if necessary, there exists a point x * ∈ R n such that Then, we have for any k ≥ k 0 . While from the assumption of the initial condition, there exists a constant C 0 > 0 such that for any k ∈ N, k } k∈N is not bounded, there exists k 1 ∈ N such that for any k ≥ k 1 , which contradicts (4.21). Therefore, there exists a point x * ∈ R n such that λ −1 k x ( j 0 ) k → x * , and hence, we obtain (1.10).

Proof of Theorem 1.4 (2)
In what follows, we show Theorem 1.4 (2). We assume that the initial data u 0 ∈ L n 2 (R n ) ∩ L 1 b (R n ) satisfies the blow-up condition (1.6) and the solution u(t) to (1.1) blows up in a finite time. Namely, for some T * < ∞, Let {t k } k∈N be a sequence that gives the supremum of u(t) n 2 . We introduce the backward self-similar transform We note that We set {s k } k∈N as We consider the scaling transform by λ > 0 that This is the L 1 -invariant scaling S λ u(t) 1 = u(t) 1 = u 0 1 and it also holds that For a blow-up solution u(t) and blow-up time sequence {t k }, we set (5.1) Then we find that the rescale solution The rescaled solution v(t, x) now solves the Cauchy problem of a semi-stationary equation where s 0 = 0. The solution preserves its total mass v(s) 1 = ũ(s) 1 for any time s ≥ 0 and v(s k ) n 2 = 1. We set the rescaled solution sequence  (2) Almost orthogonality:
Then we see that The estimate (5.8) implies that Proof of Theorem 1.5 By applying the similar argument in the proof of Theorem 1.4 (2), we obtain (1.14). By Proposition 5.1, we note that the center {y (1) k } k∈N of the profile V (1) is bounded. Thus, we have After admitting Lemma 6.1, we conclude the analogous result to (1.12): For any ε > 0, which implies that (1.15).
To see Lemma 6.1, we need to introduce the modified moment and derive its dynamics from the modified virial law. Let φ(x) be a radially symmetric smooth function such that and set r = |x|. We define

Proposition 6.2 (Modified virial law) For n
) be a solution to (1.1) with initial data u 0 ∈ L 1 b (R n ) with positive initial data u 0 (x) ≥ 0. Then if n ≥ 4 or n = 3 and b < 1, then it holds for any t ∈ (0, T ) that (2) For any R > 0, it holds that Proof for Proposition 6.2 and Lemma 6.3, see [41].
Next, we shall estimate the vanishing term w k . We fix ε > 0 arbitrarily. For R > 0, we separate the non-local term into three parts as follows: where c n = 1/((n − 2)ω n−1 ). For the integral I 1 , by the Hausdorff-Young inequality, we have where χ A (·) denotes the characteristic function of a set A, and q satisfies q = n 2(n−2) . By the uniform boundedness of L n 2 for {v k }, there exists R 0 > 0 independent of k such that for any R > R 0 , For the integral I 3 , we decompose R n into countable cubes whose centers are lattice points, that is, R n = j∈Z n Q ( j), where Q( j) ≡ {y ∈ R n ; max 1≤i≤n |y i − j i | ≤ 1/2}. It follows from the cubic decomposition and the Hölder and weak Hausdorff-Young inequalities that j, j ∈Z n , | j− j |≥R (| · | −(n−2) χ Q( j ) ) * (w k χ Q( j) )(y) dy ≤ C w k 2n n+2 | j− j |≥R (| · | −(n−2) χ Q( j ) ) * (w k χ Q( j) ) 2n , where the last estimate is derived by Hölder's inequality and the covering. By the radially decreasing property, |x| −(n−2) attains the maximum in Q( j ) at the closest point y j to the origin. For k ∈ Z n satisfying max |k i | ≥ R, we see that | · | −(n−2) χ Q( j ) We draw a line from the origin to y j and order cubes intersecting with this line. The first cube is Q(0) and the second cube is the cube which the line meets when it goes out of Q(0), and so on. We denote I (Q( j )) the second-to-last cube in this order. Then we have |y j | −2n ≤ I (Q( j , 1 2 )) |x| −2n dx.
Thus, we see that Therefore, by the uniform boundedness of L 1 for {e − n−2 2 s k v k } and Hölder's inequality, there exists R 1 > 0 independent of k such that for any R > R 1 , We setR ≡ max{R 0 , R 1 }. Lastly, the integral I 2 can be estimated as By the similar argument in the proof of Proposition 4.2, we see that {v k } k∈N is also not a vanishing sequence, and hence, we have J ≥ 1.
In the case that {v k } k∈N is radially symmetric, we see that {y for k ∈ N sufficiently large. The right hand side diverges to infinity as k → ∞, which contradicts with v k n 2 = 1. Moreover, if we assume that J ≥ 2, then for j = j , we see that For the same reason, this is a contradiction.