Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Suguro, Takeshi

doi:10.1007/s00028-021-00727-w

Cited by 6 publications

(6 citation statements)

References 30 publications

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“…The assumption of the initial data in Proposition 1.1 is rather stringent than the one appeared in [38]. However, the most importantly, the existence time T = T (u 0 , ψ 0 ) > 0 depends only on the initial data but not on the parameter τ ≥ 1.…”

Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 98%

“…On the other hand, both the systems have the spatially non-local structure and it is interesting to consider the well-posedness of the problems in spatially local function classes. The second author showed that the Cauchy problem (1.3) is time locally wellposed in the uniformly local Lebesgue space in [38]. For 1 ≤ p < ∞, let the uniformly local Lebesgue space is defined as follows:…”

Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 99%

“…Indeed, one can find a well-posedness result more general assumption the initial data (cf. [38]). The added regularity assumption on the data is required for applying maximal regularity.…”

Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 99%

“…The proof of Proposition 1.2 for the parabolicelliptic Keller-Segel system (1.3) is similar to the case for (1.1) and we do not show the case for (1.3) (cf. [38]).…”

Section: Well-posedness Of the Cauchy Problemsmentioning

confidence: 99%

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Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

Ogawa

Suguro

2022

Math. Ann.

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We consider the singular limit problem for the Cauchy problem of the (Patlak–) Keller–Segel system of parabolic-parabolic type. The problem is considered in the uniformly local Lebesgue spaces and the singular limit problem as the relaxation parameter $$\tau $$ τ goes to infinity, the solution to the Keller–Segel equation converges to a solution to the drift-diffusion system in the strong uniformly local topology. For the proof, we follow the former result due to Kurokiba–Ogawa [20–22] and we establish maximal regularity for the heat equation over the uniformly local Lebesgue and Morrey spaces which are non-UMD Banach spaces and apply it for the strong convergence of the singular limit problem in the scaling critical local spaces.

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Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 98%

Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 99%

“…Indeed, one can find a well-posedness result more general assumption the initial data (cf. [38]). The added regularity assumption on the data is required for applying maximal regularity.…”

Section: Keller-segel System and Drift-diffusion Equationmentioning

confidence: 99%

“…The proof of Proposition 1.2 for the parabolicelliptic Keller-Segel system (1.3) is similar to the case for (1.1) and we do not show the case for (1.3) (cf. [38]).…”

Section: Well-posedness Of the Cauchy Problemsmentioning

confidence: 99%

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Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

Ogawa

Suguro

2022

Math. Ann.

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“…Kozono and Sugiyama 18 proved the existence of local strong solutions to PEKS Model () with the initial data

\varphi \in {L}_{\omega}&amp;amp;#x0005E;{\frac{d}{2}}\left({\mathbb{R}}&amp;amp;#x0005E;d\right)

for

d\ge 3

. Suguro 19 proved the well‐posedness of the Keller‐Segel system in uniformly local Lebesgue spaces. Diaz et al 20 proved the existence and uniqueness of the solution with

\varphi \in {L}&amp;amp;#x0005E;1\left({\mathbb{R}}&amp;amp;#x0005E;d\right)\cap {W}&amp;amp;#x0005E;{1,p}\left({\mathbb{R}}&amp;amp;#x0005E;d\right)\left(p&amp;amp;gt;d\right)

.…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problem for Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces

Zhou

Zhang

2023

Math Methods in App Sciences

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In this paper, the local well‐posedness (which means that the initial‐to‐solution map is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces with and was established, and the solution of PEKS (Keller‐Segel system with ) converges to the solution of HEKS (Keller‐Segel system with ) as the diffusion coefficient in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces but ill‐posed (not continuous) in Besov spaces . Moreover, we show a further continuity result that the initial‐to‐solution map is Hölder continuous in Besov spaces equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces was also obtained.

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Stability of constant steady states of a chemotaxis model

et al. 2021

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The Cauchy problem for the parabolic–elliptic Keller–Segel system in the whole n-dimensional space is studied. For this model, every constant $$A \in {\mathbb {R}}$$ A ∈ R is a stationary solution. The main goal of this work is to show that $$A < 1$$ A < 1 is a stable steady state while $$A > 1$$ A > 1 is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

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Well-posedness and unconditional uniqueness of mild solutions to the Keller–Segel system in uniformly local spaces

Cited by 6 publications

References 30 publications

Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

On the Cauchy problem for Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces

Stability of constant steady states of a chemotaxis model

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