Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity

Fujie, Kentarou; Senba, Takasi

doi:10.1088/0951-7715/29/8/2417

Cited by 68 publications

(52 citation statements)

References 30 publications

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“…Moreover, Fujie–Senba established global existence and boundedness in the two‐dimensional parabolic‐elliptic system with more general sensitivity function. On the other hand, also in the parabolic‐parabolic case, it was shown that some smallness condition for χ implies global existence and boundedness; in the case that

χ false(v false) = \frac{χ_{0}}{v}

(

χ_{0} > 0

) Winkler obtained global existence of classical solutions under the condition that

χ_{0} < \sqrt{\frac{2}{n}}

and Fujie established boundedness of these solutions; in the case that

χ false(v false) \leq \frac{χ_{0}}{{false(a + v false)}^{k}}

(

χ_{0} > 0

a \geq 0

k > 1

) some smallness condition for χ 0 leads to global existence and boundedness (); recently, Fujie–Senba showed global existence and boundedness of radially symmetric solutions to the parabolic‐parabolic system with more general sensitivity function and small λ in a two‐dimensional ball.…”

Section: Introductionmentioning

confidence: 99%

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

Mizukami

2018

Mathematische Nachrichten

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Abstract. This paper gives a first insight into making a mathematical bridge between the parabolic-parabolic signal-dependent chemotaxis system and its parabolic-elliptic version.To be more precise, this paper deals with convergence of a solution for the parabolicparabolic chemotaxis system with strong signal sensitivityto that for the parabolic-elliptic chemotaxis systemwhere Ω is a bounded domain in R n (n ∈ N) with smooth boundary, λ > 0 is a constant and χ is a function generalizingIn chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, the relation between parabolicelliptic systems and parabolic-parabolic systems has not been studied. Namely, it still remains to analyze on the following question: Does a solution of the parabolic-parabolic system converge to that of the parabolic-elliptic system as λ ց 0? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.2010Mathematics Subject Classification. Primary: 35A09; Secondary: 35K51, 35J15, 92C17.

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χ false(v false) = \frac{χ_{0}}{v}

(

χ_{0} > 0

) Winkler obtained global existence of classical solutions under the condition that

χ_{0} < \sqrt{\frac{2}{n}}

and Fujie established boundedness of these solutions; in the case that

χ false(v false) \leq \frac{χ_{0}}{{false(a + v false)}^{k}}

(

χ_{0} > 0

a \geq 0

k > 1

Section: Introductionmentioning

confidence: 99%

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

Mizukami

2018

Mathematische Nachrichten

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“…Remark For n =1, it is easy to find from the arguments for proving Theorem with Lemmas , and that Theorem would be still true if replacing the sensitivity function

\frac{χ}{v^{α}}

in the model by the general χ ( s )∈ C 1+ θ (0, ∞ ) with θ ∈(0,1) and χ ( s )→0 as s → ∞ , similarly to those in Fujie and Senba …”

Section: Global Existencementioning

confidence: 85%

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

Zhao

Zheng

2018

Math Methods in App Sciences

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We consider a chemotaxis consumption system with singular sensitivity ut=normalΔu−χ∇·false(uvα∇vfalse), vt=εΔv−uv in a bounded domain normalΩ⊂Rn with χ,α,ε>0. The global existence of classical solutions is obtained with n=1. Moreover, for any global classical solution (u,v) to the case of n,α≥1, it is shown that v converges to 0 in the L∞‐norm as t→∞ with the decay rate established whenever ε∈(ε0,1) with ε0=maxfalse{0,1−χαfalse‖v0‖L∞false(normalΩfalse)α−1false}.

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“…We next recall the well‐known result about local existence of solutions to (see [, Theorem 2.3], [, Proposition 2.2] and [, Lemma 2.1]). Lemma Assume that χ satisfies with some

λ > 0

k \geq 1

a \geq 0

K > 0

and the initial data

u_{0}, v_{0}

fulfil for some

q > n

.…”

Section: Preliminariesmentioning

confidence: 99%

“…To overcome the difficulty in a singular sensitivity, Fujie established the uniform‐in‐time lower estimate for v , and showed global existence of classical bounded solutions to when

χ false(v false) = \frac{K}{v}

with

K > 0

satisfying

\begin{matrix} true \\ right K < \sqrt{\frac{2}{n}} . \end{matrix}

Moreover, Stinner–Winkler proved global existence of weak power‐λ solutions to for all

K > 0

in the radial setting. As to the problem with

χ false(v false) = \frac{K}{v}

(

K > 0

) in the 2‐dimensional setting, Lankeit obtained global existence of classical bounded solutions when

K < χ_{0} false(χ_{0} > 1 false) .

Futhermore Fujie–Senba dealt with the 2‐dimensional problem which was replaced

v_{t}

with

τ v_{t}

in and showed global existence and boundedness of radially symmetric solutions under the condition that

χ (s) \to 0

(s \to \infty)

and

τ > 0

is sufficiently small. On the other hand, in the case that

χ false(v false) \leq \frac{K}{{false(1 + α v false)}^{k}} false(k > 1, α > 0, K > 0 false),

Winkler established global existence and boundedness in…”

Section: Introductionmentioning

confidence: 99%

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A unified method for boundedness in fully parabolic chemotaxis systems with signal‐dependent sensitivity

Mizukami

Yokota

2017

Mathematische Nachrichten

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This paper deals with the Keller–Segel system {ut=Δu−∇·false(uχ(v)∇vfalse),x∈Ω,t>0,vt=Δv+u−v,x∈Ω,t>0,where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n≥2; χ is a nonnegative function satisfying χfalse(sfalse)≤K(a+s)−k for some k≥1 and a≥0. In the case that k=1 and a=0, Fujie established global existence of bounded solutions under the condition 0false01, Winkler asserted global existence of bounded solutions for arbitrary K>0. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary K>0. Moreover, the condition for K when k>1 cannot connect with the condition when k=1. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases k=1 and k>1.

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Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity

Cited by 68 publications

References 30 publications

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

A unified method for boundedness in fully parabolic chemotaxis systems with signal‐dependent sensitivity

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