Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space

Abstract

A chemotaxis system with Newtonian attraction and fractional dissipation of order $\alpha \in (0,2)$ is considered in ${ \mathbb{R} }^{N}$. For initial data belonging to $L^{1}\cap H^{4}$ but small in $L^{\frac{N}{ \alpha }}$, $N=2,3$, the temporal decay and the asymptotic behavior of a global classical solution are established. In particular, we derive a precise decay estimate for higher Sobolev norms.

Appendix: Proof of Lemma 1

Regarding the proof of Lemma 1 in the two-dimensional case, we refer to Theorem 1.1 and Theorem 1.4–Theorem 1.5 provided by Li–Rodrigo–Zhang [29] (see also [23, Theorem 2.1]). The proofs are based on the a priori estimates and the classical contraction arguments. More precisely, for a given non-negative initial data $u_{0}\in (L^{1}\cap H^{s})({ \mathbb{R} }^{2})$, $s>3$, they constructed a unique and non-negative local-in-time solution of (2) in the class $\mathcal{C}([0,T]; (L^{q} \cap H^{s})( { \mathbb{R} }^{2}))$, $1< q<2$, satisfying (10)–(11). In our cases, $u\in \mathcal{C}([0,T]; (L^{q} \cap H^{4})({ \mathbb{R} }^{2}))$, $1< q<2$ satisfying (2) and (10)–(11) can be constructed as [29] and moreover, it is easy to verify that $u\in L^{2}([0,T]; H^{4+\frac{ \alpha }{2}}({ \mathbb{R} }^{2}))$ and (9). Indeed, a uniform (in $n\ge 1$) bound of $u_{n}$ in $L^{2}([0,T]; H^{4+\frac{ \alpha }{2}}({ \mathbb{R} }^{2}))$ was obtained in the approximation scheme [29, (3.30)] using (34), but Li–Rodrigo–Zhang did not state $u\in L^{2}([0,T]; H^{4+\frac{\alpha }{2}}({ \mathbb{R} }^{2}))$ in their theorem as they also considered (2) with no dissipation term $\varLambda ^{\alpha }u$ (i.e. inviscid case).

Because the extension of the two-dimensional proofs in [29] to a three-dimensional case is rather straightforward, we shall summarize the main modifications in the following instead of repeating the procedures.

1.1 A Priori Estimates, Mass Conservation, Blow-up Criteria

We first note that the a priori estimates on $\mathcal{C}([0,T]; (L ^{q} \cap H^{4})({ \mathbb{R} }^{3}))$, $1< q<2$, the blow-up criteria (11), and the total mass conservation property (10) can be obtained as [29] without any serious difficulties. We refer interested readers to [29, Sect. 3.1] for the a priori estimates, to [29, Theorem 1.4] for the blow-up criteria (11), and to [29, Theorem 1.5] for the total mass conservation property (10). Note that the regularity of the solution, (9), can be obtained as in the two-dimensional case above.

1.2 Contraction Arguments

We note a simple modification in the contraction arguments because the other parts of the proofs in [29] can be reused in the three-dimensional case. As in [29], we consider a sequence of functions $\{u_{n}\}_{n=1}^{\infty }$ defined by $u_{1}(x,t)=u_{0}(x)$,

$$ \left \{ \textstyle\begin{array}{l} \partial _{t}u_{n+1}+(-\Delta )^{\frac{\alpha }{2}}u_{n+1}= -\nabla \cdot (u_{n+1}\nabla (-\Delta )^{-1}u_{n}) \\ u_{n+1}(x,0)=u_{0}(x) \end{array}\displaystyle \right . \quad \mbox{for} \,\,\, n\ge 1. $$

(34)

For a given $u_{n+1}\in \mathcal{C}([0,T];H^{4}({ \mathbb{R} }^{3}))$, a bound of $u_{n+1}$ in $L^{\infty }([0,T];L^{q}({ \mathbb{R} }^{3}))$, $1< q<2$ can be obtained via

$$\begin{aligned} & \bigl\Vert u_{n+1}(t) \bigr\Vert _{L^{q}} \\ &\quad \lesssim \bigl\Vert G_{\alpha }(t)u_{0} \bigr\Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert G_{\alpha }(t-\tau )\nabla \cdot \bigl(u_{n+1}\nabla (- \Delta )^{-1}u_{n}\bigr) (\tau ) \bigr\Vert _{L^{q}}d\tau \\ &\quad \lesssim \Vert u_{0} \Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert \nabla u _{n+1}(\tau ) \bigr\Vert _{L^{3}} \bigl\Vert \nabla (- \Delta )^{-1}u _{n} (\tau ) \bigr\Vert _{L^{\frac{3q}{3-q}}}d\tau \\ & \qquad {}+ \int _{0}^{t} \bigl\Vert u_{n+1}(\tau ) \bigr\Vert _{L^{\infty }} \bigl\Vert u_{n}(\tau ) \bigr\Vert _{L^{q}}d\tau \\ &\quad \lesssim \Vert u_{0} \Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert u _{n+1}(\tau ) \bigr\Vert _{H^{2}} \bigl\Vert u_{n}(\tau ) \bigr\Vert _{L^{q}} d\tau . \end{aligned} $$

Here, the main modification is the use of the three-dimensional Hardy–Littlewood–Sobolev inequality instead of its two-dimensional version. We refer interested readers to [29, p. 313].

1.3 Non-negativity

We provide an elementary proof for $N=2,3$. For $u_{-}:=-\min \{u,0\}$, let us note that the weak derivative of $u_{-}$ is $-\nabla u$ if $u<0$ and vanishes otherwise. Taking the $L^{2}$ scalar product of (2) with $u_{-}$ yields

$$\frac{1}{2}\frac{d}{dt} \Vert u_{-} \Vert _{L^{2}}^{2}- \int _{{ \mathbb{R} }^{N}}u_{-}\varLambda ^{\alpha }u = \frac{1}{2} \int _{{ \mathbb{R} }^{N}} \nabla (-\Delta )^{-1}u\cdot \nabla (u_{-})^{2}- \int _{{ \mathbb{R} }^{N}}u^{2}u_{-}. $$

Employing a point-wise expression of the fractional Laplacian,

$$\int _{{ \mathbb{R} }^{N}}u_{-}\varLambda ^{\alpha }u =C \iint _{{ \mathbb{R} }^{N}\times { \mathbb{R} }^{N}}\frac{ (u(x)-u(y))(u _{-}(x)-u_{-}(y) ) }{ \vert x-y \vert ^{N+\alpha } }dxdy\le 0 $$

for some positive number $C$ (see for instance [19, p. 1247]). We also note from the integration by parts that

$$\int _{{ \mathbb{R} }^{N}} \nabla (-\Delta )^{-1}u\cdot \nabla (u_{-})^{2}=- \int _{{ \mathbb{R} }^{N}} u(u_{-})^{2}. $$

Thus, combining above, we have

$$\frac{d}{dt} \Vert u_{-} \Vert _{L^{2}}^{2} \le 3 \Vert u \Vert _{L^{\infty }} \Vert u_{-} \Vert _{L^{2}}^{2}. $$

As $u\in L^{\infty }([0,T]; L^{\infty }({ \mathbb{R} }^{N}))$ for $T< T_{\mathrm{max}}$ by (9), the non-negativity of $u$ can be deduced by the Grönwall lemma and $u_{0}\ge 0$. This completes the proof.