Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

By Xinru Cao. 2014

266 citations recorded by the source corpus.

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array} \right. \qquad \qquad (\star) \end{equation} is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and approaches the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_{\Omega} u_0$. Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/a0e3f54608ac245cfe18f397d319840368a6b67d

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Citations in the corpus

Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components (2018, background): “  n ∈ L∞([0,∞);L1(Ω)), c ∈ L∞(Ω× (0,∞)) ∩ L(2)loc([0,∞);W 1,2(Ω)) and u ∈ L(2)loc(Ω̄× [0,∞)) ∩ ⋂ p∈[1,2) L p loc([0,∞);W 1,p 0 (Ω)), such that (n, c, u) is a global mass-preserving generalized solution of (1.”
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity (2017, background): “existence and characteristics of the large time behavior of solutions can be found in [15, 3].”
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up (2016, background): “ u(·, 0) is small, or if n = 1, then for various types of initial-boundary value problems, global bounded solutions are known to exist ([13, 14, 27, 28, 32]).”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, background): “12), are known to enforce global boundedness also in the minimal Keller-Segel system when n ≥ 3 ([4], [33]), whereas the total cell mass apparently loses its relevance in this respect, yet present in the case n = 2 ([22], [19]), in such high-dimensional settings in the sense that for arbitrary m > 0”
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties (2016, background): “Similar perturbation arguments have been used in detecting global existence and essentially diffusive behavior of small-data solutions in various related chemotaxis systems ([38], [6], [43]), but as evident from complementary results on the occurrence of explosions e.”