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.
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JA was supported by NRF, Republic of Korea-2018R1D1A1B07047465. JL was supported by SSTF-BA1701-05 (Samsung Science & Technology Foundation, Republic of Korea).
Appendix: Proof of Lemma 1
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 \(u_{1}(x,t)=u_{0}(x)\),
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
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
Employing a point-wise expression of the fractional Laplacian,
for some positive number \(C\) (see for instance [19, p. 1247]). We also note from the integration by parts that
Thus, combining above, we have
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.
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Ahn, J., Lee, J. Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space. Acta Appl Math 169, 199–215 (2020). https://doi.org/10.1007/s10440-019-00296-8
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DOI: https://doi.org/10.1007/s10440-019-00296-8