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

Mizukami, Masaaki; Yokota, Tomomi

doi:10.1002/mana.201600399

Cited by 40 publications

(23 citation statements)

References 14 publications

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“…Proof For a function

f : Ω \times [0, T) \to R

putting

\tilde{f} false(x, t false) : = f false(x, λ t false)

for

false(x, t false) \in Ω \times (0, \frac{T}{λ})

, we see that

\overset{v_{λ}}{true}

satisfies

\begin{matrix} true \\ right \{\begin{matrix} left {false(\overset{v_{λ}}{true} false)}_{t} = normalΔ \overset{v_{λ}}{true} - \overset{v_{λ}}{true} + \overset{u}{true}, & left x \in normalΩ, t \in (() 0, \frac{T}{λ}), \\ left \nabla \overset{v_{λ}}{true} \cdot ν = 0, & left x \in \partial normalΩ, t \in (() 0, \frac{T}{λ}), \\ left \overset{v_{λ}}{true} (x, 0) = v_{init} (x), & left x \in normalΩ . \end{matrix} \end{matrix}

Thus since the mass conservation yields that

0true \int_{Ω} u (\cdot, λ t) = m

for all

t \in (0, \frac{T}{λ})

and any

λ > 0

, we infer from (see also [, Lemma 2.1]) that

\overset{η}{true}

defined as satisfies

\begin{matrix} true \\ righ... \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…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%

“…In order to obtain an answer to this question we first deal with the chemotaxis system with strong signal sensitivity, because we have already provided all tools to establish the

L^{\infty} false(normalΩ \times [0, \infty) false)

‐estimate for solutions via a priori estimate in (This is likely to enable us to see a uniform‐in‐λ estimate). The purpose of this work is to obtain some positive answer to this question in the chemotaxis system with strong signal sensitivity.…”

Section: Introductionmentioning

confidence: 99%

“…In order to use a compactness method some uniform‐in‐λ estimate for the solution is required. The strategy of seeing an estimate independent of λ is to modify the methods in . One of keys for this strategy is to derive the differential inequality

0true \frac{d}{d t} \int_{Ω} u_{λ}^{p} φ (v_{λ}) \leq - c_{1} {(\int_{normalΩ} u_{λ}^{p} φ false(v_{λ} false))}^{b} + c_{2} \int_{Ω} u_{λ}^{p} φ (v_{λ}) + c_{3},

where φ is some function, and

b > 1

and

c_{1}, c_{2}, c_{3} > 0

are some constants.…”

Section: Introductionmentioning

confidence: 99%

“…in the case that ( ) ≤ 0 ( + ) ( 0 > 0, ≥ 0, > 1) some smallness condition for 0 leads to global existence and boundedness ( [22]); recently, Fujie-Senba [10] showed global existence and boundedness of radially symmetric solutions to the parabolicparabolic system with more general sensitivity function and small in a two-dimensional ball.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Proof For a function

f : Ω \times [0, T) \to R

putting

\tilde{f} false(x, t false) : = f false(x, λ t false)

for

false(x, t false) \in Ω \times (0, \frac{T}{λ})

, we see that

\overset{v_{λ}}{true}

satisfies

\begin{matrix} true \\ right \{\begin{matrix} left {false(\overset{v_{λ}}{true} false)}_{t} = normalΔ \overset{v_{λ}}{true} - \overset{v_{λ}}{true} + \overset{u}{true}, & left x \in normalΩ, t \in (() 0, \frac{T}{λ}), \\ left \nabla \overset{v_{λ}}{true} \cdot ν = 0, & left x \in \partial normalΩ, t \in (() 0, \frac{T}{λ}), \\ left \overset{v_{λ}}{true} (x, 0) = v_{init} (x), & left x \in normalΩ . \end{matrix} \end{matrix}

Thus since the mass conservation yields that

0true \int_{Ω} u (\cdot, λ t) = m

for all

t \in (0, \frac{T}{λ})

and any

λ > 0

, we infer from (see also [, Lemma 2.1]) that

\overset{η}{true}

defined as satisfies

\begin{matrix} true \\ righ... \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

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%

“…In order to obtain an answer to this question we first deal with the chemotaxis system with strong signal sensitivity, because we have already provided all tools to establish the

L^{\infty} false(normalΩ \times [0, \infty) false)

Section: Introductionmentioning

confidence: 99%

0true \frac{d}{d t} \int_{Ω} u_{λ}^{p} φ (v_{λ}) \leq - c_{1} {(\int_{normalΩ} u_{λ}^{p} φ false(v_{λ} false))}^{b} + c_{2} \int_{Ω} u_{λ}^{p} φ (v_{λ}) + c_{3},

where φ is some function, and

b > 1

and

c_{1}, c_{2}, c_{3} > 0

are some constants.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Mizukami

2018

Mathematische Nachrichten

Self Cite

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Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source

2021

Mathematische Nachrichten

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In this paper, we study the following chemotaxis system with singular sensitivity and logistic source trueright85.0pt{ut=Δu−χ∇·0trueuv∇v+ru−μuk,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,in a smooth bounded convex domain Ω⊂double-struckRn (n≥2) with the non‐flux boundary, where χ,r,μ>0,k>2. The boundedness of solutions has been proved in the case that n≥2, k>3(n+2)n+4 and r, χ>0 satisfying χ2

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Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

Lin

et al. 2017

Math Methods in App Sciences

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This paper deals with the attraction-repulsion chemotaxis system with nonlinear diffusion u t = ∇ · (D(u)∇u) − ∇ · (u (v)∇v) + ∇ · (u (w)∇w), 1 v t = Δv − 1 v + 1 u, 2 w t = Δw − 2 w + 2 u, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain Ω ⊂ R n (n ⩾ 2), where the coefficients i , i , and i ∈ {0, 1}(i = 1, 2) are positive. The function D fulfills D(u) ⩾ C D u m−1 for all u > 0 with certain C D > 0 and m > 1. For the parabolic-elliptic-elliptic case in the sense that 1 = 2 = 0 and = 1, we obtain that for any m > 2 − 2 n and all sufficiently smooth initial data u 0 , the model possesses at least one global weak solution under suitable conditions on the functions and . Under the assumption m > − 2 n , it is also proved that for the parabolic-parabolic-elliptic case in the sense that 1 = 1, 2 = 0, and ⩾ 2, the system possesses at least one global weak solution under different assumptions on the functions and .KEYWORDS attraction-repulsion, boundedness, chemotaxis, logistic source, weak solutionwhich also models the quorum sensing effect in the chemotactic movement (see Painter et al 2 ). Ω ⊂ R n (n ⩾ 2) is a bounded domain with smooth boundary Ω, denotes the differentiation with respect to the outward normal derivative 7368 ;40 7368-7395. : 2 n − for all u ⩾ 0 with some > 0 and c > 0, the corresponding solutions are global and bounded provided that D has algebraic upper and lower growth estimates as u → ∞. Wang et al 47 removed the condition of upper growth estimate of D in Tao et al. 44 More results about Cauchy problem are obtained (see previous studies 8,9,18,25,26,34,[48][49][50] ).Compared to the classical Keller-Segel chemotaxis (1.3), the coupled attraction-repulsion system (1.1) is much less understood because of the lack of Lyapunov functional. As for the system (1.1) with D ≡ 1, ≡ 0 and ≡ 0 , where the coefficients 0 , 0 are positive constants. When 1 , 2 = 0, the authors Hillen et al 42 and Jin et al 51 investigated the 2 −p 1 |Ω| ) C 7 .

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

Cited by 40 publications

References 14 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

Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source

Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

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