Abstract
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.
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1 Drift-diffusion system in higher dimensions
We consider the finite time blow-up for the solution to the drift-diffusion system in spatially higher dimensions. Let u be a solution to the Cauchy problem of the drift-diffusion system:
where \(u = u(t, x)\) denotes particle density and \(\psi \) denotes a potential of the particle field. The equation (1.1) is relevant to a model of self-interacting particles (see Biler–Nadzieja [7], see also Biler [2]), the semi-conductor device simulations (see [15, 21, 32]) and a model of the aggregation of mold known as the Keller–Segel system (see [22]). In some models, the equation can be derived from a singular limit problem from a fluid mechanical approximation of gravitation gaseous stars (cf. [14, 24]). If the solution has enough regularity and integrability, then the problem exhibits an instability of the solution, namely, under certain conditions on the initial data, the solution blows up in finite time. The blow up results can be found in [2, 6, 13, 18, 27, 28, 33, 34, 37, 43, 44]. On the other hand, the global existence and stability are known for two dimensional case ([20, 25, 26, 35,36,37,38]). This property is naturally inherited to the system (1.1).
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 non-local 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 \(1 \le p \le \infty \),
where \(\langle \cdot \rangle = (1 + |\cdot |^2)^{1/2}\). Noting that \(L^p_s({\mathbb {R}}^n) \subset L^1({\mathbb {R}}^n)\) if \(s > n/p\), we recall the existence and uniqueness of the solution for the n dimensional drift-diffusion equation in a critical space \(L^{\frac{n}{2}}({\mathbb {R}}^n)\).
Definition. Let \(n \ge 3\) and \(1 \le p < \infty \). For \(u_0 \in L^p({\mathbb {R}}^n)\), we call u a mild solution to the system (1.1) if u(t) solves the integral equation
in \(C([0, T); L^p({\mathbb {R}}^n))\), where
with \(\omega _{n - 1} \equiv 2 \pi ^{\frac{n}{2}}/\Gamma \left( \frac{n}{2}\right) \) as the surface volume of a unit sphere and \(\Gamma (\cdot )\) denotes the gamma function.
Proposition 1.1
(Local well-posedness and conservation laws) Let \(n \ge 3\) and \(\frac{n}{2} \le p < n \). For any \(u_0 \in L^p({\mathbb {R}}^n)\), there exists \(T > 0\) and a unique mild solution \((u, \psi )\) to (1.1) with the initial data \(u_0\) such that \(u \in C([0, T); L^p({\mathbb {R}}^n)) \cap L^\theta (0, T; L^q({\mathbb {R}}^n))\) with \(2/\theta + n/q = 2\) and \(q > \frac{n}{2}\). Moreover, the solution has higher regularity \(u\in C([0,T);W^{2, p}({\mathbb {R}}^n))\cap C^1((0, T); L^p({\mathbb {R}}^n))\) and it is a strong solution for (1.1). Besides there exists a maximal existence time \(T = T_* \le \infty \) such that if \(T_* < \infty \), then for any \(\frac{n}{2} < p \le \infty \),
Furthermore, the solution satisfies the following properties:
-
(1)
If the initial data \(u_0 \in L^1_b({\mathbb {R}}^n)\) for \(b > 0\), then the solution satisfies
$$\begin{aligned} u \in C([0, T); L^p({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)). \end{aligned}$$ -
(2)
If \(u_0(x) \ge 0\), then \(u(t, x) \ge 0\) for any \((t, x)\in (0, T) \times {\mathbb {R}}^n\).
-
(3)
If \(u_0 \in L^1({\mathbb {R}}^n)\), then
$$\begin{aligned} \Vert u(t)\Vert _1 = \Vert u_0\Vert _1. \end{aligned}$$(1.2) -
(4)
If in addition \(u_0 \in L^1_b({\mathbb {R}}^n)\), where \(b > 0\) and \(p\ge \frac{n}{2}\), then the solution \((u, \psi )\) satisfies
$$\begin{aligned} H[u(t)] +\int _0^t \int _{{\mathbb {R}}^n}u(t)\big |\nabla (\log {u(\tau )}-\psi (\tau ))\big |^2\, dx\, d\tau = H[u_0], \end{aligned}$$(1.3)where
$$\begin{aligned} H[u(t)] \equiv \int _{{\mathbb {R}}^n} u(t) \log {u(t)}\, dx - \frac{1}{2} \int _{{\mathbb {R}}^n} u(t) \psi (t)\, dx. \end{aligned}$$(1.4) -
(5)
If \(u_0 \in L^1_2({\mathbb {R}}^n)\), then the second moment of the solution satisfies the following identity:
$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^n} |x - x'|^2 u(t)\, dx&= \int _{{\mathbb {R}}^n} |x - x'|^2 u_0(x)\, dx + 2(n-2)\int _0^t H[u(s)]\, ds + 2 n t \Vert u_0\Vert _1 \\&\quad - 2 (n - 2) \int _0^t \int _{{\mathbb {R}}^n} u(s) \log {u(s)}\, dx\, ds, \end{aligned} \end{aligned}$$(1.5)where \(x' \in {\mathbb {R}}^n\) 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 \(\lambda > 0\),
is invariant scaling for the system. The critical space coincide with the invariant scale is
and for two dimensional case it is \(L^\infty ({\mathbb {R}}_{+}; L^1({\mathbb {R}}^2)) \times L^\infty ({\mathbb {R}}_+; L^\infty ({\mathbb {R}}^2))\). The global existence in the scaling critical spaces is a direct consequence of Fujita–Kato’s principle.
Proposition 1.2
(Global existence) Let \(n \ge 3\), \(p \ge \frac{n}{2}\), and \(b > 0\). Assume that the initial data \(u_0\) is non-negative in \(L^p({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\), and for some constant \(B_n > 0\) depending only on n such that \(\Vert u_0\Vert _{\frac{n}{2}} < B_n\).
-
(1)
Then the corresponding solution u obtained by Proposition 1.1 exists globally in time.
-
(2)
The solution decays in time:
$$\begin{aligned} \Vert u(t)\Vert _p \le C (1 + t)^{- \frac{n}{2} \left( 1 - \frac{1}{p}\right) }. \end{aligned}$$
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
for \(t \in [0, T)\). Then \(\Vert u(t)\Vert _{\frac{n}{2}}\) 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 \(u_0 \in L^1({\mathbb {R}}^2) \cap L^2_s({\mathbb {R}}^2)\) satisfying
On the other hand, if
then the solution blows up in a finite time. Namely, there exists \(T_\textrm{m} < \infty \) such that
for all \(1 < p \le \infty \) (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 \(L^{\frac{n}{2}}\) 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] obtained a finite time blow-up result for the case of bounded domain with a boundary condition. Corrias–Perthame–Zaag [13] obtained the finite time blowing up result for the higher dimensional cases who did not use the entropy functional. Namely, there exists a large constant \(M = M(n) > 0\) such that if the initial data satisfies
where \({\bar{x}}\) 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 \(L^1_2({\mathbb {R}}^n)\), 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 \(\Vert u_0\Vert _1 = 1\) 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].
Proposition 1.3
(Blow-up criterion) Let \(n \ge 3\) and \(b > 0\). For \(u_0 \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\), \(u_0 \ge 0\), assume further that for some constant \(C_{n, b} > 0\),
where \({\bar{x}}\) is the b-th mass center of \(u_0\).
-
(1)
When \(b \ge 2\), then
$$\begin{aligned} C_{n, b} \ge \frac{2 \pi e^\frac{n}{n - 2}}{n} \end{aligned}$$(1.7)and the solution u to (1.1) blows up in a finite time. Namely, for any \(\frac{n}{2} < p \le \infty \), there exists some \(T_* < \infty \) such that
$$\begin{aligned} \varlimsup _{t \rightarrow T_*} \Vert u(t)\Vert _p = \infty . \end{aligned}$$(1.8) -
(2)
If \(u_0 \ge 0\) is radially symmetric and \(b > 0\), then
$$\begin{aligned} C_{n, b} \ge \frac{b c_{n, b} e^{1 + \frac{b (1 + \delta /n)}{n - 2}}}{n} \end{aligned}$$(1.9)and the solution u to (1.1) blows up in a finite time in the sense of (1.8), where \(c_{n, b}\) is the constant defined in Proposition 2.1.
-
(3)
When \(0< b < 2\), then the solution does not remain uniformly bounded in \(L^{\frac{n}{2}}({\mathbb {R}}^n)\).
Our main result is the concentration phenomena for the blowing up solution:
Theorem 1.4
Let \(n \ge 3\) and \(b \ge 2\). For \(u_0 \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\), \(u_0 \ge 0\), assume that (1.6) holds with \(C_{n, b} = 2 \pi e^\frac{n}{n - 2}/n\).
-
(1)
Then the blowing up solution u to (1.1) concentrates the following sense: Let \(T_* > 0\) be the blow up time and \(\{t_k\}_{k \in {\mathbb {N}}}\) satisfy \(t_k \rightarrow T_*\) and \(\Vert u(t_k)\Vert _\frac{n}{2} \rightarrow \infty \) as \(k \rightarrow \infty \). Then for any \(\varepsilon > 0\), there exist a subsequence of \(\{t_k\}_{k \in {\mathbb {N}}}\) and \(x_* \in {\mathbb {R}}^n\) such that
$$\begin{aligned} \left( \frac{2 n}{(n - 2) C_\textrm{HLS}}\right) ^\frac{n}{2} \le \varliminf _{k \rightarrow \infty } \int _{B_\varepsilon (x_*)} u(t_k, x)^\frac{n}{2}\, dx, \end{aligned}$$(1.10)where \(C_\textrm{HLS}\) is the best possible constant of the Hardy–Littlewood–Sobolev inequality, that is,
$$\begin{aligned} \int _{{\mathbb {R}}^n}f(-\Delta )^{-1}f\, dx \le C_\textrm{HLS}\Vert f\Vert _1\Vert f\Vert _{\frac{n}{2}}. \end{aligned}$$(1.11) -
(2)
Furthermore, if \(T_* < \infty \), then for any \(\varepsilon > 0\), there exist a subsequence of \(\{t_k\}_{k \in {\mathbb {N}}}\) and \(\{x_k\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}^n\) such that
$$\begin{aligned} \left( \frac{2 n}{(n - 2) C_\textrm{HLS}}\right) ^\frac{n}{2} \le \varliminf _{k \rightarrow \infty } \int _{B_{\varepsilon \sqrt{T_* - t_k}}(x_k)} u(t_k, x)^\frac{n}{2}\, dx, \end{aligned}$$(1.12)where \(C_\textrm{HLS}\) is defined by the above.
By the assumption (1.6) with \(C_{n, b} = 2 \pi e^\frac{n}{n - 2}/n\), 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 \(u_0 \ge 0\) satisfies
The constant of the right hand side coincides with one appearing in (1.10). We also see that
for \(n \ge 3\) (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 \(b \ge 2\), the constant appearing in the concentration phenomena changes as follows:
Theorem 1.5
Let \(n \ge 3\) and \(b > 0\). For a radially symmetric function \(u_0 \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\), \(u_0 \ge 0\).
-
(1)
If \(b \ge 2\) and \(u_0\) satisfies (1.6) with \(C_{n, b} = 2 \pi e^\frac{n}{n - 2}/n\), then the blowing up solution u to (1.1) concentrates the following sense: Let \(0< T_* < \infty \) be the blow up time and \(\{t_k\}_{k \in {\mathbb {N}}}\) satisfy \(t_k \rightarrow T_*\) and \(\Vert u(t_k)\Vert _\frac{n}{2} \rightarrow \infty \) as \(k \rightarrow \infty \). Then for any \(\varepsilon > 0\), there exists a subsequence of \(\{t_k\}_{k \in {\mathbb {N}}}\) such that
$$\begin{aligned} \left( \frac{2 n}{(n - 2) C_\textrm{HLS}}\right) ^\frac{n}{2} \le \varliminf _{k \rightarrow \infty } \int _{B_{\varepsilon \sqrt{T_* - t_k}}(0)} u(t_k, x)^\frac{n}{2}\, dx. \end{aligned}$$(1.14) -
(2)
If \(b > 0\) and \(u_0\) satisfies (1.6) with (1.9) for any \(\delta > 0\), then the blowing up solution u to (1.1) concentrates the following sense: Let \(0< T_* < \infty \) be the blow up time and \(\{t_k\}_{k \in {\mathbb {N}}}\) satisfy \(t_k \rightarrow T_*\) and \(\Vert u(t_k)\Vert _\frac{n}{2} \rightarrow \infty \) as \(k \rightarrow \infty \). Then for any \(\varepsilon > 0\), there exist a subsequence of \(\{t_k\}_{k \in {\mathbb {N}}}\) such that
$$\begin{aligned} \left( \frac{2 (n + \delta )}{(n - 2) C_\textrm{HLS}}\right) ^\frac{n}{2} \le \varliminf _{k \rightarrow \infty } \int _{B_{\varepsilon \sqrt{T_* - t_k}}(0)} u(t_k, x)^\frac{n}{2}\, dx. \end{aligned}$$(1.15)
Theorem 1.5 gives the \(L^\frac{n}{2}\)-concentration rate of the radially symmetric blow-up solution and is the analogous result of a nonlinear Schrödinger equation (see Merle–Tsutsumi [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 \(C_{n, b} = 2 \pi e^\frac{n}{n - 2}/n\), then we see that
While we relax the weight condition from \(b \ge 2\) 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 \(\alpha > 1\) denotes the adiabatic constant originated from the pressure term \(P(\rho ) = c \rho ^\alpha \) in the barotropic damped compressible Navier–Stokes–Poisson system (cf. [14, 24]). It is known that there are two critical exponents \(\alpha _* = 2 - 4/(n + 2)\) and \(\alpha ^* = 2 - 2/n\) (see [8, 12, 40, 42, 45, 46, 49]). In the case \(\alpha _* \le \alpha \le \alpha ^*\) (see [23]), the threshold for the global existence of the weak solution to (1.16) is identified as
where
and V is the optimizer for the Hardy–Littlewood–Sobolev inequality
with
In particular case of \(\beta = 0\) when \(\alpha = \alpha _*\), 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 \(L^1({\mathbb {R}}^n)\). We take \(\{t_k\}_{k \in {\mathbb {N}}}\) such that \(t_k \rightarrow T\) as \(k \rightarrow \infty \) and introduce the rescaled solution sequence
for a blowing up solution u to (1.1). Then \(\Vert u_k\Vert _1 = \Vert u_0\Vert _1\) and \(\Vert u_k\Vert _\frac{n}{2} = 1\). In Bedrossian–Kim [1], they showed the profile decomposition in \(L^1({\mathbb {R}}^n)\), but their technique allows that the profile depends on k. In this paper, we improve the extraction of the profile independent of k from \(\{u_k\}_{k \in {\mathbb {N}}}\). 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]).
2 Preliminaries
The following inequality originally due to Shannon is useful to estimate the logarithmic functional (see [29, 41]).
Proposition 2.1
(Generalized Shannon’s inequality) Let \(n \ge 2\) and \(b > 0\). There exists a constant \(C_{n, b} > 0\) such that for any non-negative function \(f \in L^1_b({\mathbb {R}}^n)\),
where \({\bar{x}}\) is the b-th mass center of f and
is the best possible. In particular, if \(b = 2\), then
Lemma 2.2
Let \(n \ge 2\) and \(b > 0\). There exists a constant \(C > 0\) such that for all \(f \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\),
From this lemma, we see that
and there is a limitation of the blow-up speed of the \(L^{\frac{n}{2}}\)-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 \(n \ge 3\). Then there exists a constant \(C_\textrm{HLS} > 0\) which is depending only on n such that for any function \(f \in L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\),
Moreover, there exists the extremal function \(V \in L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\) such that V is radially symmetric and decreasing function satisfying the Euler–Lagrange equation
for some \(0< R < \infty \).
Proof of Proposition 2.3
The Hardy–Littlewood–Sobolev inequality implies that it follows that
where \(1< \alpha , r < \infty \) satisfy
By Hölder’s inequality, we have
where \(0 \le \theta \le 1\) satisfies
In particular, let \(\alpha = \frac{n}{2}\), then \(\theta = 0\), so that the inequality
holds for any function \(f \in L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\). By the definition of the Poisson kernel, we can rewrite as
where \(C_\textrm{HLS}\) denotes the best possible constant of the Hardy-Littlewood-Sobolev inequality and \(C_\textrm{HLS}=C_n c_n\). 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]).\(\square \)
Proposition 2.4
(Comparison of constants) The best constant \(C_\textrm{HLS}\) is estimated from the above by
where \(S_n\) is the best constant of Sobolev’s inequality
given by
for all \(f \in {\dot{H}}^1({\mathbb {R}}^n)\). In particular, for \(n \ge 3\),
Remark
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 \(C_\textrm{HLS}\) 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 \(S_n^2\) also gives the best constant for the inequality
for any \(f \in L^{2n/(n + 2)}({\mathbb {R}}^n)\) by the duality argument (see for instance [40]). Hence, by Hölder’s inequality, one can observe that
holds for any \(f \in L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\). This shows \(C_\textrm{HLS} \le S_n^2\). To see that \(C_\textrm{HLS} < S_n^2\), we assume on the contrary that \(C_\textrm{HLS} = S_n^2\) for \(n \ge 3\). Then we may show that there exists an extremal function \(V \in L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\) 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,
On the other hand, the extremal function of the Hardy–Littlewood–Sobolev inequality (2.2) must satisfy the Euler–Lagrange equation (2.3). Clearly, the Talenti function (2.6) does not satisfy (2.3). This is a contradiction, and hence, we obtain (2.4). Moreover, it follows from (2.4) that
for any \(n \ge 3\). 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 \(\Vert V\Vert _1\Vert V\Vert _\frac{n}{2}\) and \(\Vert V\Vert _{2n/(n+2)}\) explicitly. Since it holds that
where \(B(\cdot , \cdot )\) 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 computationFootnote 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
\(\square \)
3 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]).
Lemma 3.1
(Profile decomposition in \(L^1\)) Let \(\{f_k\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n) \cap L \log {L}({\mathbb {R}}^n)\) satisfy
for some constant \(M, L > 0\). Then for all \(\varepsilon > 0\), there exists a subsequence of \(\{f_k\}\) (not relabeled), \(J \in {\mathbb {N}}\cup \{0\}\), \(\{x_k^{(j)}\}_{k \in {\mathbb {N}}, j = 1, \dots , J} \subset {\mathbb {R}}^n\), \(\{R^{(j)}\}_{j = 1}^J \subset {\mathbb {R}}_+\), and function sequences \(\{F^{(j)}\}_{j = 1}^J, \{w_k\}_{k \in {\mathbb {N}}}, \{e_k\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n)\) which satisfy
with the following properties:
-
(1)
The profile \(\{F^{(j)}\}\) is a nonnegative function sequence satisfying that for each \(k \in {\mathbb {N}}\), \(\textrm{supp}\,{F^{(j)}} \subset B_{R_j}(0)\) and for \(j \ne j'\), \(|x_k^{(j)} - x_k^{(j')}| \rightarrow \infty \) as \(k \rightarrow \infty \). Moreover, for each \(k \in {\mathbb {N}}\), \(B_{R^{(j)}}(x_k^{(j)})\) and \(\textrm{supp}\,{w_k}\) are all disjoint for any \(j = 1, \dots , J\).
-
(2)
\(\{w_k\}\) is the vanishing part, that is, for all \(R > 0\),
$$\begin{aligned} \lim _{k \rightarrow \infty } \sup _{x \in {\mathbb {R}}^n} \int _{B_R(x)} w_k(x)\, dx = 0, \end{aligned}$$and \(0 \le w_k \le f_k\) almost everywhere \(x \in {\mathbb {R}}^n\).
-
(3)
\(\{e_k\}\) is the error term, that is,
$$\begin{aligned} \varlimsup _{k \rightarrow \infty } \Vert e_k\Vert _1 < \varepsilon \end{aligned}$$and \(e_k \le f_k\) almost everywhere \(x \in {\mathbb {R}}^n\).
In particular, passing \(k \rightarrow \infty \), we have the almost orthogonality
Proof of Lemma 3.1
By the argument of the concentration compactness lemma in [30], there exists a subsequence \(\{f_k\}_{k \in {\mathbb {N}}}\) (not relabeled) such that one of the following situation occurs for \(\{f_k\}\): (1) the compactness, (2) the vanishing, or (3) the dichotomy. We fix \(\varepsilon > 0\) arbitrarily.
Case 1. Compactness: If the compactness occurs, then there exists a sequence \(\{x_k^{(1)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}^n\) such that
for some radius \(R^{(1)} > 0\) independent of k. We set
for \(x \in {\mathbb {R}}^n\). By the assumption (3.1), we see that
On the other hand, we have
Thus, it holds that
which implies that \(\{{\tilde{f}}_k\}\) is uniformly integrable. Then the Dunford–Pettis theorem implies that there exist a subsequence \(\{{\tilde{f}}_{k_l}\}_{l \in {\mathbb {N}}} \subset \{{\tilde{f}}_k\}\) and a nonnegative function \({\tilde{F}}^{(1)} \in L^1(B_{R^{(1)}}(0))\) such that \({\tilde{f}}_{k_l}\) converges weakly to \({\tilde{F}}^{(1)}\) in \(L^1(B_{R^{(1)}}(0))\). Thus, we set a profile \(F^{(1)}\) as the zero expansion of \({\tilde{F}}^{(1)}\), 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 \(w_k \equiv 0\). Then the claim in Lemma 3.1 holds in \(J = 1\).
Case 2. Vanishing: If the vanishing occurs, it follows that
for arbitrary \(R > 0\). Define \(w_k \equiv f_k\) and \(e_k \equiv 0\), then we have the claim of Lemma 3.1 with \(J = 0\).
Case 3. Dichotomy: If the dichotomy occurs, there exists \(\mu \in (0, M)\) such that there exist the non-negative function sequences \(\{f_{k, 1}\}_{k \in {\mathbb {N}}}, \{f_{k, 2}\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n)\) which satisfy
where we define \(h_{k} \equiv f_k - (f_{k, 1} + f_{k, 2})\). More precisely, there exist \(\{x_k^{(1)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}^n\), \(R^{(1)} > 0\), and \(\{R_k^{(1)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}_+\) such that
with the following estimates:
and
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 of \({\tilde{f}}_{k, 1}\) 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 \(\mu \in (0, M)\) such that
then we define \(e_k \equiv f_k - F^{(1)}(x - x_k^{(1)})\), and hence, the claim in Lemma 3.1 holds with \(J = 1\). In the case of
we reset
Then \(\Vert {\bar{f}}_{k, 2}\Vert _1 = 1\). We apply the concentration compactness lemma to \(\{{\bar{f}}_{k, 2}\}_{k \in {\mathbb {N}}}\). Then there exists a subsequence \(\{{\bar{f}}_{k, 2}\}\) (not relabeled) such that one of the following situation occur for \(\{{\bar{f}}_{k, 2}\}\): (1) the compactness, (2) the vanishing, or (3) the dichotomy.
Case 3.1. Compactness: If the compactness occurs, then there exists \(\{x_k^{(2)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}^n\) such that
for some radius \(R^{(2)} > 0\). We note that passing \(k \rightarrow \infty \), then \(|x_k^{(1)} - x_k^{(2)}| \rightarrow \infty \) by (3.2). Then we set
Similarly to Case 1, we get a profile \(F^{(2)}\) as a weak convergence limit of \({\tilde{f}}_{k, 2}\) 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 \(\nu \in (0, 1)\) such that there exists the non-negative function sequences \(\{{\bar{f}}_{k, 21}\}, \{{\bar{f}}_{k, 22}\} \subset L^1({\mathbb {R}}^n)\) which satisfy
where we define \({\bar{h}}_k \equiv {\bar{f}}_{k, 2} - ({\bar{f}}_{k, 2, 1} + {\bar{f}}_{k, 2, 2})\). For each sequences, there exist \(\{x_k^{(2)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}^n\), \(R^{(2)} > 0\), and \(\{R_k^{(2)}\}_{k \in {\mathbb {N}}} \subset {\mathbb {R}}_+\) such that
with the following estimates:
By the definition of \({\bar{f}}_{k, 2}\), we can define \(h_{k, 2} \equiv \Vert f_{k, 2}\Vert _1 {\bar{h}}_k\) so that we have
If we can take \(\nu \in (0, 1)\) such that
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
for any \(l \in {\mathbb {N}}\). Take the sum with respect to l, then the total mass \(\Vert f_k\Vert _1\) diverges, that is,
This contradicts the assumption of Lemma 3.1.\(\square \)
Remark
In [1], they also showed the profile decomposition on \(L^1({\mathbb {R}}^n)\). We emphasize that Lemma 3.1 assures the profile independent of k while the profile in [1] depends on k. The independence of the profile on k is valid in the proof of Theorem 1.4.
4 Concentration
Lemma 4.1
Let the initial data \(u_0\) satisfy the assumption (1.6) with (1.7), then there exists the corresponding solution u(t) to (1.1) blows up in a finite time \(T_*\). Then it holds that for any \(0< t < T_*\),
Proof of Lemma 4.1
For simplicity, we consider the case of \(b = 2\). For the solution u(t) to the equation (1.1), by the entropy dissipation (1.3) and definition of the entropy (1.4), we have
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
This implies that
Thus, it follows from (4.2) and (4.5) that
From the assumption (4.4), we obtain
for any \(0< t < T_*\).\(\square \)
In what follows, we show Theorem 1.4 (1). We assume that the initial data \(u_0 \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\) 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 \(\{t_k\}_{k \in {\mathbb {N}}}\) be a sequence that gives the supremum of \(\Vert u(t)\Vert _{\frac{n}{2}}\). We introduce the scaling transform by \(\lambda > 0\) that
This is the \(L^1\)-invariant scaling \(\Vert S_\lambda u(t)\Vert _1 = \Vert u(t)\Vert _1 = \Vert u_0\Vert _1\) and it also holds that
For a blow-up solution u(t) and blow-up time sequence \(\{t_k\}\), we set
Then we find that the rescale solution
is bounded in \(L^1({\mathbb {R}}^n) \cap L^\frac{n}{2}({\mathbb {R}}^n)\). Note that the scaling is \(L^1\)-invariant and hence the \(L^1\)-norm 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 \(\Vert v(t)\Vert _1 = \Vert u(t)\Vert _1 = \Vert u_0\Vert _1\) for any time \(t \ge 0\) and \(\Vert v(t_k)\Vert _\frac{n}{2} = 1\) at \(t = t_k\).
We set the rescaled solution sequence
satisfies
In this case, one can apply Proposition 3.1 with \(\{u_k\}_{k \in {\mathbb {N}}}\):
Proposition 4.2
(Profile decomposition in \(L^p\)) Let \(\{u_k\}_{k \in {\mathbb {N}}}\) be a non-negative sequence in \(L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\) with
defined by the above. Then for all \(\varepsilon > 0\), there exists a subsequence of \(\{u_k\}\) (not relabeled), \(J \in {\mathbb {N}}\), \(\{x_k^{(j)}\}_{k \in {\mathbb {N}}, j = 1, \dots , J} \subset {\mathbb {R}}^n\), \(\{R^{(j)}\}_{j = 1}^J \subset {\mathbb {R}}_+\), and function sequences \(\{U^{(j)}\}_{j = 1}^J, \{w_k\}_{k \in {\mathbb {N}}}, \{e_k\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n) \cap L^\frac{n}{2}({\mathbb {R}}^n)\) which satisfy the following properties:
-
(1)
\(u_k\) is decomposed as
$$\begin{aligned} u_k(x) = \sum _{j = 1}^J U^{(j)}\Big (x - x_k^{(j)}\Big ) + w_k(x) + e_k(x)\quad \text {a.a.}\ x \in {\mathbb {R}}^n, \end{aligned}$$(4.10)where \(\{U^{(j)}\}_{j = 1}^J\) is a nonnegative function sequence with \(\textrm{supp}\,{U^{(j)}} \subset B_{R^{(j)}}(0)\). Moreover, for each \(k \in {\mathbb {N}}\), \(B_{R^{(j)}}(x_k^{(j)})\) and \(\textrm{supp}\,{w_k}\) are all disjoint for any \(j = 1, 2, \dots , J\). For any \(R > 0\), it holds that
$$\begin{aligned} \lim _{k \rightarrow \infty } \sup _{x \in {\mathbb {R}}^n} \int _{B_R(x)} w_k(x)\, dx = 0 \end{aligned}$$(4.11)and the error term satisfies
$$\begin{aligned} \varlimsup _{k \rightarrow \infty } \Vert e_k\Vert _1 < \varepsilon \end{aligned}$$(4.12) -
(2)
Almost orthogonality:
$$\begin{aligned} \Vert u_k\Vert _1 = \sum _{j = 1}^J \Vert U^{(j)}\Vert _1 + \Vert w_k\Vert _1 + \varepsilon , \end{aligned}$$(4.13)and for any \(j \in \{1, 2, \dots , J\}\), it follows that
$$\begin{aligned} \Vert U^{(j)}\Vert _\frac{n}{2} \le (1 + \varepsilon ) \Vert u_k\Vert _{L^\frac{n}{2}(B_{R^{(j)}}(x_k^{(j)}))}. \end{aligned}$$(4.14) -
(3)
Drift term estimate:
$$\begin{aligned} \varlimsup _{k \rightarrow \infty } \left\| |\nabla |^{- 1} w_k\right\| _2^2 = \varlimsup _{k \rightarrow \infty } \left\| |\nabla |^{- 1} e_k\right\| _2^2 = 0. \end{aligned}$$(4.15)
We note that the vanishing in the sense of Lions’ concentration compactness lemma implies \(J = 0\). We emphasize that the sequence \(\{u_k\}_{k \in {\mathbb {N}}}\) is not vanishing and one can extract at least the profile of \(\{u_k\}\).
Proof of Proposition 4.2
We apply Lemma 3.1 with \(\{u_k\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n)\). We note that \(L^\frac{n}{2}\)-boundedness implies that \(\{u_k\}_{k \in {\mathbb {N}}}\) is uniformly integrable. The decomposition (4.10), (4.11), (4.12), and (4.13) are shown directly in Lemma 3.1. By the construction of the profile, we see that
Since \(\{{\tilde{u}}_k\}\) is uniformly bounded in \(L^\frac{n}{2}(B_{R^{(1)}}(0))\), \(\{{\tilde{u}}_k\}_{k \in {\mathbb {N}}}\) also converges to \(U^{(1)}\) weakly in \(L^\frac{n}{2}(B_{R^{(1)}}(0))\). This argument can be applied for any \(j = 1, 2, \dots , J\). Thus, we obtain (4.14). Moreover, we have
which implies that \(\{e_k\}_{k \in {\mathbb {N}}}\) is uniformly bounded in \(L^\frac{n}{2}\).
In what follows, we show (4.15). By the construction of the error term \(e_k\), we have
We note that \(\{e_k\}_{k \in {\mathbb {N}}}\) is uniformly bounded in \(L^\frac{n}{2}({\mathbb {R}}^n)\). Hence, the Hardy–Littlewood–Sobolev inequality (2.2) shows that
Next, we shall estimate the vanishing term \(w_k\). We fix \(\varepsilon > 0\) arbitrarily. For \(R > 0\), we separate the non-local term into three parts as follows:
where \(c_n = 1/((n - 2) \omega _{n - 1})\). For the integral \(I_1\), by the Hausdorff–Young inequality, we have
where \(\chi _A(\cdot )\) denotes the characteristic function of a set A, \(1 \le p \le \frac{n}{2}\) and \(1 \le q < \frac{n}{n - 2}\) satisfy
Then the integral of the kernel can be estimated as
By the uniform boundedness of \(L^1\) and \(L^{\frac{n}{2}}\) 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 set \({\tilde{R}} \equiv \max \{R_0, R_1\}\). Lastly, the integral \(I_2\) can be estimated as
Since \(w_k\) is the vanishing term, we obtain
Thus, we conclude
In order to extract at least one profile, we shall show that \(\{v(t_k)\}_{k \in {\mathbb {N}} \cup \{0\}}\) is not a vanishing sequence by using the bump function method (see [2, 4, 5]). We set the bump function as
For \(\varepsilon \in (0, 1/\sqrt{3})\), the Hessian of \(\eta \) satisfies
where \(c_\varepsilon \equiv 4 (1 - 3 \varepsilon ^2)\). For \(R > 0\) and \(a \in {\mathbb {R}}^n\), we define
which satisfies
in particular, \(\eta _R\) is concave on \(|x - a| < R \sqrt{n/(n + 2)}\) as (4.16). Multiplying the equation (4.7) by \(\eta _R\) and integrating over \(x \in {\mathbb {R}}^n\), 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 \(\varepsilon \in (0, \sqrt{n/(n + 2)})\). By the concavity of \(\eta _R\) as (4.16), we have
Since we know that
it follows that
If we set a constant as
then we have
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
by the symmetry property of \(J_2\) and the fact that \(\nabla \eta _R(x)\) vanishes on the set \(\{x \in {\mathbb {R}}^n;\ |x - a| > R\}\). For \(|x - a| < R\) and \(|y - a| \ge \varepsilon R\), we have
Thus, we have
By the Hardy–Littlewood–Sobolev inequality and \(\Vert v(t_k)\Vert _\frac{n}{2} = 1\), we have
at \(t = t_k\). Hence, we have
There exists \(k_0 \in {\mathbb {N}}\) and \(\varepsilon _0 > 0\) such that for any \(k \ge k_0\),
for some constant \(C_0 > 0\). Thus, there exists \(R_0 = R(\varepsilon _0) > 0\) such that for any \(k \ge k_0\),
since \(\Vert v(t_k)\Vert _1 = 1\). Therefore, it follows that for any \(k \ge k_0\),
for some constant \(C > 0\), which implies that \(\{v(t_k)\}_{k \in {\mathbb {N}}}\) is not vanishing sequence.\(\square \)
The error estimate (4.15) gives the following decomposition:
Proposition 4.3
(Profile decomposition) There exist an integer \(J \in {\mathbb {N}}\) and a non-negative function sequence \(\{U^{(j)}\}_{j = 1}^J \subset L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\) with \(\textrm{supp}\,{U^{(j)}} \subset B_{R^{(j)}}(x_k^{(j)})\) for some sequences \(\{x_k^{(j)}\}_{k \in {\mathbb {N}}, j = 1, 2, \dots , J} \subset {\mathbb {R}}^n\) and \(\{R^{(j)}\}_{j = 1}^J\subset {\mathbb {R}}_+\) such that for any \(\varepsilon > 0\), there exists \(k_* \ge 1\) such that for all \(k \ge k_*\),
where \(\textrm{supp}\,{U^{(j)}}\) are disjoint and \(|x_k^{(j)} - x_k^{(j')}| \rightarrow \infty \) as \(k \rightarrow \infty \) if \(j \ne j'\).
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 \(\lambda _k\), then the right hand side is rewritten by
By the auxiliary inequality (4.1), it holds that for all \(k = 1, 2, \dots \),
By the above inequality (4.19) and the profile decomposition (4.18), for any \(\varepsilon > 0\), there exists \(k_* \in {\mathbb {N}}\) such that for any \(k \ge k_*\), we see that
By the Hardy–Littlewood–Sobolev inequality (2.2) and (4.13),
Thus, there exists \(j_0 \in \{1, 2, \dots , J\}\) such that
Moreover, by (4.14), there exist \(x_k^{(j_0)} \in {\mathbb {R}}^n\) and \(R_0 > 0\) such that
Substituting this to the inequality (4.20), then by scaling, we conclude that
By passing a subsequence if necessary, there exists a point \(x_*\in {\mathbb {R}}^n\) such that \(y_k \equiv \lambda _k^{-1} x_{k}^{(j_0)} \rightarrow x_*\). Indeed, if we assume that \(\{\lambda _k^{-1}x_{k}^{(j_0)}\}_{k \in {\mathbb {N}}}\) is an unbounded sequence, then \(\lambda _k^{-1}| x_{k}^{(j_0)}|\rightarrow \infty \) \((k\rightarrow \infty )\). For \(k \in {\mathbb {N}}\) sufficiently large, we see that
Then, we have
for any \(k \ge k_0\). While from the assumption of the initial condition, there exists a constant \(C_0 > 0\) such that for any \(k \in {\mathbb {N}}\),
Since \(\{\lambda _k^{- 1} x_k^{(j_0)}\}_{k \in {\mathbb {N}}}\) is not bounded, there exists \(k_1 \in {\mathbb {N}}\) such that for any \(k \ge k_1\),
which contradicts (4.21). Therefore, there exists a point \(x_*\in {\mathbb {R}}^n\) such that \(\lambda _k^{-1} x_{k}^{(j_0)} \rightarrow x_*\), and hence, we obtain (1.10).\(\square \)
5 Proof of Theorem 1.4 (2)
In what follows, we show Theorem 1.4 (2). We assume that the initial data \(u_0 \in L^{\frac{n}{2}}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\) satisfies the blow-up condition (1.6) and the solution u(t) to (1.1) blows up in a finite time. Namely, for some \(T_* < \infty \),
Let \(\{t_k\}_{k \in {\mathbb {N}}}\) be a sequence that gives the supremum of \(\Vert u(t)\Vert _{\frac{n}{2}}\). We introduce the backward self-similar transform
Then \({\tilde{u}}\) satisfies
We note that
We set \(\{s_k\}_{k \in {\mathbb {N}}}\) as
We consider the scaling transform by \(\lambda > 0\) that
This is the \(L^1\)-invariant scaling \(\Vert S_\lambda u(t)\Vert _1 = \Vert u(t)\Vert _1 = \Vert u_0\Vert _1\) and it also holds that
For a blow-up solution u(t) and blow-up time sequence \(\{t_k\}\), we set
Then we find that the rescale solution
is bounded in \(L^\frac{n}{2}({\mathbb {R}}^n)\) and \(\{e^{- \frac{n - 2}{2} s_k} v(s_k)\}_{k \in {\mathbb {N}}}\) is bounded in \(L^1({\mathbb {R}}^n)\). 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 \(\Vert v(s)\Vert _1 = \Vert {\tilde{u}}(s)\Vert _1\) for any time \(s \ge 0\) and \(\Vert v(s_k)\Vert _\frac{n}{2} = 1\).
We set the rescaled solution sequence
satisfies
In this case, one can apply Proposition 3.1 with \(\{v_k\}_{k \in {\mathbb {N}}}\):
Proposition 5.1
(Profile decomposition in \(L^p\)) Let \(\{v_k\}_{k \in {\mathbb {N}}}\) be a non-negative sequence in \(L^1({\mathbb {R}}^n) \cap L^{\frac{n}{2}}({\mathbb {R}}^n)\) with (5.4) defined by the above. Then for all \(\varepsilon > 0\), there exists a subsequence of \(\{v_k\}\) (not relabeled), \(J \in {\mathbb {N}}\), \(\{y_k^{(j)}\}_{k \in {\mathbb {N}}, j = 1, \dots , J} \subset {\mathbb {R}}^n\), \(\{R^{(j)}\}_{j = 1}^J \subset {\mathbb {R}}_+\), and nonnegative function sequences \(\{V^{(j)}_k\}_{k \in {\mathbb {N}}}^{j = 1, \dots , J}, \{w_k\}_{k \in {\mathbb {N}}}, \{e_k\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n) \cap L^\frac{n}{2}({\mathbb {R}}^n)\) which satisfy the following properties:
-
(1)
\(v_k\) is decomposed as
$$\begin{aligned} v_k(x) = \sum _{j = 1}^J V^{(j)}_k(x - y_k^{(j)}) + w_k(x) + e_k(x)\quad \text {a.a.}\ x \in {\mathbb {R}}^n, \end{aligned}$$(5.5)where \(\{V^{(j)}_k\}_{k \in {\mathbb {N}}}^{j = 1, \dots , J}\) is a nonnegative function sequence with \(\textrm{supp}\,{V^{(j)}_k} \subset B_{R^{(j)}}(0)\). Moreover, for each \(k \in {\mathbb {N}}\), \(B_{R^{(j)}}(y_k^{(j)})\) and \(\textrm{supp}\,{w_k}\) are all disjoint for any \(j = 1, 2, \dots , J\). For any \(R > 0\), it holds that
$$\begin{aligned} \lim _{k \rightarrow \infty } \sup _{x \in {\mathbb {R}}^n} \int _{B_R(x)} w_k(x)^p\, dx = 0 \quad \text {for any}\ 1 \le p \le \frac{n}{2} \end{aligned}$$(5.6)and the error term satisfies
$$\begin{aligned} \varlimsup _{k \rightarrow \infty } \Vert e_k\Vert _\frac{n}{2} < \varepsilon \end{aligned}$$(5.7) -
(2)
Almost orthogonality:
$$\begin{aligned} \Vert v_k\Vert _1 = \sum _{j = 1}^J \Vert V^{(j)}_k\Vert _1 + \Vert w_k\Vert _1 + \Vert e_k\Vert _1, \end{aligned}$$(5.8)and for any \(j \in \{1, 2, \dots , J\}\), it holds that
$$\begin{aligned} \Vert V^{(j)}_k\Vert _\frac{n}{2} = \Vert v_k\Vert _{L^\frac{n}{2}(B_{R^{(j)}}(y_k^{(j)}))}. \end{aligned}$$(5.9) -
(3)
Drift term estimate:
$$\begin{aligned} \varlimsup _{k \rightarrow \infty } e^{- \frac{n - 2}{2} s_k} \left\| |\nabla |^{- 1} w_k\right\| _2^2 = \varlimsup _{k \rightarrow \infty } e^{- \frac{n - 2}{2} s_k} \left\| |\nabla |^{- 1} e_k\right\| _2^2 = 0. \end{aligned}$$(5.10)
In addition, if \(\{v_k\}_{k \in {\mathbb {N}}}\) is radially symmetric, then \(J = 1\) and \(\{y_k^{(1)}\}_{k \in {\mathbb {N}}}\) is bounded.
For the proof of Proposition 5.1, see Appendix.
We set
and
Then we see that
The estimate (5.8) implies that
By (5.10) and changing variables, we have
in particular, there exists \(k_* \in {\mathbb {N}}\) such that for all \(k \ge k_*\),
Proof of Theorem 1.4 (2)
By the similar argument of the proof of Theorem 1.4 (1), for any \(\varepsilon > 0\), there exists \(k_* \in {\mathbb {N}}\) such that for any \(k \ge k_*\), we see that
Thus, there exists \(j_0 \in \{1, 2, \dots , J\}\) such that
Moreover, by (5.9), there exist \(x_k^{(j_0)} \in {\mathbb {R}}^n\) and \(R_0 > 0\) such that
Substituting this to the inequality (5.11), then by scaling, we conclude that
Therefore, if we set \(x_k \equiv \lambda _k^{- 1} \sqrt{T - t_k} y_k^{(j_0)}\), then we obtain (1.12).\(\square \)
6 Radially symmetric case
In this section, we consider the case that \(u_0 \in L^\frac{n}{2}({\mathbb {R}}^n) \cap L^1_b({\mathbb {R}}^n)\) is radially symmetric and nonnegative for \(b > 0\). We assume that the initial data \(u_0\) satisfies the assumption (1.6) with (1.9) for \(\delta > 0\). Let u(t, x) be a blowing up solution to (1.1). We set the scaling parameter (5.1) and consider the rescaled solution (5.3).
Since the assumption on the data is different from the case of \(b \ge 2\), the lower estimate has to be arranged.
Lemma 6.1
Let the initial data \(u_0\) be radially symmetric and satisfy (1.6) with (1.9) for \(\delta > 0\), then there exists the corresponding radially symmetric solution u(t) to (1.1) blows up in a finite time T. Then it holds that for any \(0< t < T\),
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 \(\{y_k^{(1)}\}_{k \in {\mathbb {N}}}\) 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 \(\varepsilon > 0\),
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 \(\phi (x)\) be a radially symmetric smooth function such that
and set \(r=|x|\). We define
for any \(R \ge 1\).
Proposition 6.2
(Modified virial law) For \(\frac{n}{2}\le p<n\), \(b>0\) let \(u\in C([0,T);L^1_b({\mathbb {R}}^n)\cap L^{p}({\mathbb {R}}^n))\) be a solution to (1.1) with initial data \(u_{0}\in L^1_b({\mathbb {R}}^n)\) with positive initial data \(u_{0}(x)\ge 0\). Then if \(n\ge 4\) or \(n=3\) and \(b<1\), then it holds for any \(t\in (0,T)\) that
where
is supported in \(B_R^c(0)\).
Lemma 6.3
Let \(\Psi _R(x)\in L^1({\mathbb {R}}^n)\cap L^{\infty }({\mathbb {R}}^n)\) be a radially symmetric function supported in \(B_R^c(0)\) and u(t) is radially symmetric.
-
(1)
Then there exists a constant \(C=C(\phi )>0\) depending on \(\Phi _R\) and \(\eta \) such that
$$\begin{aligned} \int _{{\mathbb {R}}^n} \Psi _R(r)\partial _r^2(-\Delta )^{-1}u(t)\, (-\Delta )^{-1}u(t)\, dx \le CR^{-(n-2)} \Vert u_0\Vert _1^2. \end{aligned}$$ -
(2)
For any \(R>0\), it holds that
$$\begin{aligned} \frac{1}{4}\int _{{\mathbb {R}}^n} \Delta ^2 \Phi _R |\psi (t)|^2\, dx \le C(\phi )R^{-(n-2)} \Vert u_0\Vert _1^2. \end{aligned}$$
Proof for Proposition 6.2 and Lemma 6.3, see [41].
Proposition 6.4
Let the initial data be radially symmetric in \(L^\frac{n}{2}(\mathbb {R}^n) \cap L^1_b(\mathbb {R}^n)\) and satisfy the condition (1.6) with (1.9), then
Proof of Proposition 6.4
By Lemma 6.3, choosing \(R > 0\) sufficiently large such that
By the modified virial law (6.2), (6.4), and Shannon’s inequality (2.1), we have for \(\displaystyle M\equiv \Vert u_0\Vert _1\),
Hence under the condition (1.6), the right hand side of (6.5) is negative if \(t=0\). Then
is decreasing function of t in \([0,\eta )\). Since it holds that
the b-th moment of u does not increase if the initial data satisfies
where \(c_{n, b}\) is the constant appearing in Proposition 2.1. Thus, we obtain the inequality (6.3).\(\square \)
Proof of Lemma 6.1
From (1.3) we have
By the Shannon inequality (Proposition 2.1), it follows that
We obtain from (6.3) and (6.6) that
From the assumption (1.6) with (1.9), we have
and hence, we obtain the inequality (6.1).\(\square \)
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
The authors checked this for n = 4 rigorously and up to \(n = 300\) numerically.
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Acknowledgements
The first author would like to thank Professor Taub Hmidi for his sending his result on the profile decomposition. He is also grateful to Professor Kazuhiro Kuwae for information on the b-th moment center. The work of T. Ogawa is partially supported by JSPS Grant-in-aid for Scientific Research S #19H05597. The work of T. Suguro is partially supported by JSPS Grant-in-aid for JSPS Fellows #19J20763 and Research Activity Start-up #22K20336. The work of H. Wakui is supported by JSPS Grant-in-aid for JSPS Fellows #20J00940.
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Appendix A. Proof of Proposition 5.1
Appendix A. Proof of Proposition 5.1
Proof of Proposition 5.1
We apply the similar argument in the proof of Lemma 3.1 with \(\{v_k^\frac{n}{2}\}_{k \in {\mathbb {N}}} \subset L^1({\mathbb {R}}^n)\). We notice that the profile given by the convergence of \(\{v_k\}\) in \(L^\frac{n}{2}\) is not always non-zero function. From this reason, we define the profile, which depends on k, as
The decomposition (5.5), (5.8), and (5.9) are shown directly in Lemma 3.1. In what follows, we show that the error estimates (5.10).
We note that \(\{e^{- \frac{n - 2}{2} s_k} e_k\}_{k \in {\mathbb {N}}}\) is uniformly bounded in \(L^1({\mathbb {R}}^n)\). Hence, the Hardy–Littlewood–Sobolev inequality (2.2) and (5.7) show that
Next, we shall estimate the vanishing term \(w_k\). We fix \(\varepsilon > 0\) arbitrarily. For \(R > 0\), we separate the non-local term into three parts as follows:
where \(c_n = 1/((n - 2) \omega _{n - 1})\). For the integral \(I_1\), by the Hausdorff–Young inequality, we have
where \(\chi _A(\cdot )\) denotes the characteristic function of a set A, and q satisfies \(q = \frac{n}{2 (n - 2)}\). By the uniform boundedness of \(L^{\frac{n}{2}}\) 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 \({\mathbb {R}}^n\) into countable cubes whose centers are lattice points, that is, \({\mathbb {R}}^n = \bigcup _{j \in {\mathbb {Z}}^n} Q\left( j\right) \), where \(Q(j) \equiv \{y \in {\mathbb {R}}^n;\ \max _{1 \le i \le n} |y_i - j_i| \le 1/2\}\). It follows from the cubic decomposition and the Hölder and weak Hausdorff–Young inequalities that
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 \(k \in {\mathbb {Z}}^n\) satisfying \(\max |k_i| \ge R\), we see that
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
Thus, we see that
Therefore, by the uniform boundedness of \(L^1\) for \(\{e^{- \frac{n - 2}{2} s_k} v_k\}\) and Hölder’s inequality, there exists \(R_1 > 0\) independent of k such that for any \(R > R_1\),
We set \({\tilde{R}} \equiv \max \{R_0, R_1\}\). Lastly, the integral \(I_2\) can be estimated as
Since \(w_k\) is the vanishing term, (5.6) gives
Thus, we conclude
By the similar argument in the proof of Proposition 4.2, we see that \(\{v_k\}_{k \in {\mathbb {N}}}\) is also not a vanishing sequence, and hence, we have \(J \ge 1\).
In the case that \(\{v_k\}_{k \in {\mathbb {N}}}\) is radially symmetric, we see that \(\{y_k^{(j)}\}_{k \in {\mathbb {N}}}\) is bounded for any \(j = 1, 2, \dots , J\). Indeed, if not, it follows from the radially symmetrically and weak \(L^\frac{n}{2}\)-convergence of \(\{v_k\}\) that
for \(k \in {\mathbb {N}}\) sufficiently large. The right hand side diverges to infinity as \(k \rightarrow \infty \), which contradicts with \(\Vert v_k\Vert _\frac{n}{2} = 1\). Moreover, if we assume that \(J \ge 2\), then for \(j \ne j'\), we see that
For the same reason, this is a contradiction.\(\square \)
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Ogawa, T., Suguro, T. & Wakui, H. Finite time blow up and concentration phenomena for a solution to drift-diffusion equations in higher dimensions. Calc. Var. 62, 47 (2023). https://doi.org/10.1007/s00526-022-02345-x
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DOI: https://doi.org/10.1007/s00526-022-02345-x