On stability of a class of second alpha-order fractal differential equations

1.   Introduction

The main purpose of this note is to provide an alternative construction of nonnegative and nonradially symmetric initial data for some Keller–Segel-type models which will enforce finite or infinite blowup. Consider the following functional:

where $\Omega \subset \mathbb{R}^2$ is a bounded domain with $C^2$ boundary $\partial \Omega$ and a pair of nonnegative smooth functions $(u, v)$. The main result of this note is stated as follows.

Theorem 1.1. For any $M > 0$ and $\Lambda \in (4\pi, \infty)$ there exists a pair of nonnegative functions $(u_0, v_0) \in (C^\infty(\overline{\Omega}))^2$ satisfying

The above functional $\mathcal{F}(u, v)$ appears in the study of the minimal Keller–Segel system:

and also one of the following chemotaxis model featuring a signal-dependent motility function of the negative exponential type:

Classical positive solutions of (1.1) satisfy the following energy-dissipation identity (^[4,9]):

while for the classical solutions to (1.2), there holds (^[2]):

In both cases, the above energy identities will immediately give rise to the a priori upper bound for $\mathcal{F}(u, v)(t)$. On the other hand, for any given initial data of small total mass such that $\|u_0\|_{L^1(\Omega)} < 4\pi$, one could derive a lower bound for the energy functional and then the classical solutions of both systems (1.1) and (1.2) exist globally in time and remain bounded uniformly in the two-dimensional setting (see ^[2,4,7,9]). For large data, unbounded solutions of the above problems could be constructed based on observations of the variational structure of the stationary problem and by taking an advantage of the subtle connection between its associated functional with the energy $\mathcal{F}$. In ^[5] the authors introduced a transformation problem of the original system (1.1) with the unknowns being the cell density and the relative signal concentration. Then they constructed unbounded solutions for the transformed problem, which in turn implied blowup of the original one.

In this note we would rather to construct an unbounded solution to the original system (1.1) or (1.2) in a more direct way. To this aim, let us sketch the main idea of the construction of an unbounded solution following ^[11] (see also ^[5]). First, the corresponding stationary solutions $(u_s, v_s)$ to (1.1) or (1.2) satisfy the following problem:

for some $\Lambda > 0$. Denote

for $\Lambda > 0$. By [5,Lemma 3.5] and [10,Theorem 1], for $\Lambda \not\in 4\pi \mathbb{N}$ there exists some $C > 0$ such that

and

On the other hand, let $(u, v)$ be the classical positive solution to (1.1) or (1.2) in $\Omega \times (0, \infty)$. If the solution is uniform-in-time bounded, by the compactness method (cf. [13,Lemma 3.1]), there exist a sequence of time $\{t_k\} \subset (0, \infty)$ and a solution $(u_s, v_s)$ to (1.3) with $\Lambda = \|u_0\|_{L^1(\Omega)}$ such that $\lim\limits_{k \rightarrow \infty} t_k = \infty$ and that

as well as

Thus taking account of the above discussion, for a pair of nonnegative functions $(u_0, v_0)$ satisfying

the corresponding solution must be unbounded or blow up in finite time.

Recently in ^[2], we constructed nonnegative initial data satisfying (1.4) when $\Lambda \in (8\pi, \infty)$ in the radially symmetric case, which differs from those given in ^[5]. However, it was left open whether our idea for a construction of adequate initial data can be extended to the nonradial symmetric case if $\Lambda \in (4\pi, 8 \pi)$. Theorem 1.1 of the present work gives an affirmative answer to this question and as a consequence, we have an alternative proof of the following corollaries (^[5]).

Corollary 1.2. For any $\Lambda\in(4\pi, \infty)\backslash4\pi\mathbb{N}$ there exists a nonnegative initial datum $(u_0, v_0)$ satisfying (1.4) such that the corresponding classical solution of $(1.1)$ satisfies either:

● exists globally in time and $\limsup\limits_{t\rightarrow \infty}(\|u(t)\|_{L^\infty(\Omega)} + \|v(t)\|_{L^\infty(\Omega)}) = \infty$;

● blows up in finite time.

Remark 1.3. Finite time blowup solutions of the corresponding parabolic-elliptic system are constructed if $\Lambda > 4\pi$ in ^[8].

As to the system (1.2), global existence of classical solutions with any nonnegative initial data was guaranteed in ^[2], which excluded the possibility of finite-time blowup. Hence, we arrive at the following:

Corollary 1.4. For any $\Lambda\in(4\pi, \infty)\backslash4\pi\mathbb{N}$ there exists a nonnegative initial datum $(u_0, v_0)$ satisfying (1.4) such that the corresponding global classical solution of $(1.2)$ blows up at time infinity.

In previous works ^[3,6,12,13], nonnegative initial data with large negative energy were constructed in several modified situations, e.g., the higher dimensional setting, the nonlinear diffusion case, the nonlinear sensitivity case and the indirect signal case. In those works, the initial datum has a concentration at an interior point of $\Omega$. Similarly, in our precedent work ^[2], we constructed an initial datum which concentrates at the origin based on certain perturbation of the rescaled explicit solutions to the elliptic system

provided that the total mass $\Lambda > 8\pi$. However, without the radially symmetric requirement and when $4\pi < \Lambda < 8\pi$, we need to construct an initial datum that concentrates at a boundary point. To this aim, some cut-off and folding-up techniques are introduced. Besides, a lemma of analysis (Lemma 2.2) plays a crucial role in estimating the value of each individual integral in the energy functional and in order to get vanishing estimations of the error terms, the radius of the cut-off function used in our case needs to depend on the rescaled parameter as well, which in contrast was fixed in the radially symmetric case in ^[2].

2.   Proof of Theorem 1.1

A straightforward calculation leads us to the following lemma.

Lemma 2.1. For any $\lambda \geq 1$ and $r \in (0, 1)$, the functions

satisfy

Since $\partial \Omega$ is $C^2$ class, for any boundary point $P\in \partial \Omega$ there exist some $R^\prime = R^\prime_{P}\in (0, 1)$ and some $C^2$ function $\gamma_{P}: \mathbb{R} \to \mathbb{R}$ such that

(cf. [1,Appendix C.1]). Moreover since $\Omega$ is a bounded domain, we can find some point $P_0 = (P_1, P_2) \in \partial \Omega$ satisfying that there exists $R \in (0, R^\prime)$ such that

By translation, we may assume $P_0 = (0, 0)$. Hereafter we fix the above $R \in (0, 1)$ and $\gamma = \gamma_{P_0}$. In this setting, we have the following lemma:

Lemma 2.2. Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a radially symmetric, nonnegative and continuous function. For any $r\in(0, R)$ it follows that

where

Proof. We first note that for any $r\in (0, R)$,

Since $\gamma(0) = 0$ and the assumption (2.1), it follows by Taylor's theorem that for all $x_1 \in (-R, R)$ we have

where $K(R) : = \max_{|\xi| \leq R}\gamma^{\prime\prime}(\xi) > 0$. Thus we can deduce that

where

By denoting

we confirm that

Since the radial symmetry of $f$ implies

we have

Since

we have that

which concludes the proof.

For any $0 < \eta_1 < \eta_2$ we can construct a radially symmetric function $\phi_{\eta_2, \eta_1} \in C^\infty (\mathbb{R}^2)$ satisfying

For any $\lambda > \max\{1, (\frac{4}{R})^{\frac{6}{5}}\}$, we fix

and then $0 < r_1 < r < \min\{1, \frac{R}{4}\}$. Noting that

and by the increasing property of $f$, we can find $\lambda_\ast > \max\{1, (\frac{4}{R})^{\frac{6}{5}}\}$ such that

where $K(R)$ is defined in (2.2). Here we confirm that for any $\lambda > \lambda_\ast$,

Now we define the pair

with some $a > 0$. Here we remark that $u_0$ and $v_0$ are nonnegative functions belonging to $C^\infty(\overline{\Omega})$.

Lemma 2.3. Let $\Lambda \in (4\pi, \infty)$. For $\lambda > \lambda_\ast$ there exists

such that

Proof. Firstly by changing variables, we see that

and that

Here in light of the radial symmetry of $u_\lambda \phi_{r, r_1}$, we can invoke Lemma 2.2 to have

where we used

By the choice of $r > 0$, we have

Therefore for any $\lambda > \lambda_\ast$ we find some $a = a(\lambda)$ satisfying

and (2.4). We conclude the proof.

Lemma 2.4. There exists $C > 0$ such that for all $\lambda > \lambda_\ast$,

where $a = a(\lambda)$ is defined in Lemma 2.3.

Proof. Since $s\log s \leq t\log t +\frac{1}{e}$ for $s\leq t$ and $u_0 \leq a u_\lambda \chi_{B_r(0)\cap\Omega}$, it follows

Since $\log u_\lambda \leq \log (8\lambda^2) = 2 \log \lambda +\log 8$ and $\int_\Omega u_\lambda \leq 8\pi$, we have

By Lemma 2.2 we obtain

Therefore

where we remark that the constant $C$ is independent of $a$ and $\lambda$ in view of (2.3). We conclude the proof.

Lemma 2.5. There exists $C > 0$ such that for all $\lambda > \lambda_\ast$,

where $a = a(\lambda)$ is defined in Lemma 2.3.

Proof. Using $v_{\lambda } > 0$ in $B(0, r)$, $u_0 = 0$ on $B(0, r)^c$ and $r_1 < \frac{R}{4}$, we see that

Since $u_\lambda v_{\lambda}$ is radially symmetric and

we apply Lemma 2.2 and recall $r_1 = 2^{-1}\lambda^{-\frac{5}{6}}$ to deduce that

Since

we have that

By (2.5), it follows

On the other hand, by (2.3) and direct calculations we see

Combining above estimates, we obtain that

for $\lambda > \lambda_\ast$ with some positive constant $C$, which is independent of $a$ and $\lambda$ due to (2.3).

Lemma 2.6. For any $\varepsilon_1 > 0$ there exists $C(\varepsilon_1) > 0$ such that for all $\lambda > \lambda_\ast$,

where $a = a(\lambda)$ is defined in Lemma 2.3.

Proof. Since

we see that for $\lambda > 1$

Hence it follows from straightforward calculations that there is a positive constant $C$ satisfying

where the constant $C$ is independent of $a$ and $\lambda$ due to (2.3).

Moreover by Young's inequality, for any $\varepsilon_1 > 0$ there exists $C^\prime (\varepsilon_1) > 0$ such that

Since by (2.9) we have some $C > 0$ such that

and by the direct calculations, we have

and then we infer that

with some $C^{\prime\prime}(\varepsilon_1) > 0$. Since $\dfrac{ \lambda^4 |x|^2}{(1+\lambda^2 |x|^2)^2}$ is radially symmetric, we can invoke Lemma 2.2 to see

thus

On the other hand,

Since $\lambda > 1$, it follows

Hence

Therefore we conclude

where the constant $C(\varepsilon_1)$ is independent of $a$ and $\lambda$ due to (2.3).

Proof of Theorem 1.1. For any $\Lambda \in (4\pi, \infty)$, we have $\Lambda/{4\pi} > 1$. In view of (2.3), we can fix $\varepsilon_1 > 0$ independently of $\lambda$ such that $(1- \varepsilon_1)a-1 > (1- \varepsilon_1)\frac{\Lambda}{4\pi}-1 > 0$, where $a = a(\lambda)$ is defined in Lemma 2.3. Then it follows that

Collecting (2.6), (2.7) and (2.8), we infer that there exists some $C > 0$ such that

where

Here (2.10) implies $I_1 < 0$ for all $\lambda > \lambda_{\ast}$. On the other hand, we note

Based on the above discussion, for $\Lambda \in (4\pi, \infty)$ and $M > 0$, we can choose some $\lambda > \lambda_{\ast}$ such that

We conclude the proof.

Acknowledgments

The authors thank the anonymous referee's careful reading and useful suggestions. K. Fujie is supported by Japan Society for the Promotion of Science (Grant-in-Aid for Early-Career Scientists; No. 19K14576). J. Jiang is supported by Hubei Provincial Natural Science Foundation under the grant No. 2020CFB602.

Conflict of interest

The authors declare no conflict of interest.