Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

Lankeit, Johannes

doi:10.1016/j.jde.2016.12.007

Cited by 99 publications

(61 citation statements)

References 36 publications

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“…To prove the global existence of the smooth solution ( u , v ) to , we have to deal with the lower bound estimate of v at first for characterizing the strength of bacteria concentration involved by the singular sensitive function

\frac{χ}{v^{α}}

. Similarly to Winkler and Lankeit, let

w false(x, t false) : = - \ln \frac{v false(x, t false)}{false‖ v_{0} false(x false) ‖_{L^{\infty} false(normalΩ false)}} .

…”

Section: Global Existencementioning

confidence: 99%

“…Together with Lemma and , we know u , w ≥0 in ( x , t )∈Ω×[0, T ). Moreover, on the basis of the mass conservation , we can get a point‐wise upper bound estimate for w , and then the lower bound estimate for v like that in Lankeit ,. Lemma 3.8 Denote

\overset{ε}{true} = 1 + \frac{1}{ε}

, we have the following lemma.…”

Section: Global Existencementioning

confidence: 99%

“…Remark Recall the known result for the system, where a nonlinear diffusion

D \in C_{δ, m}

is contained with singular sensitive function

χ false(v false) = \frac{1}{v}

and ε =1 in . The global existence condition obtained in Lankeit is

m > 1 + \frac{n}{4}

with n ≥2. We would like to point out that in the case of n =1, the global existence of classical solutions can be ensured by m ≥1, since, firstly, the a priori lower bound estimate of v is due to the mass conservation of u (similar that in Lemma ) independent of m ≥1 and, secondly, the required L p ‐estimate of u with some p >1 can be established via a similar procedure whenever m ≥1.…”

Section: Global Existencementioning

confidence: 99%

“…Moreover, in a radially symmetric setting, a “normalized solution” has been constructed for n ≥3. If

D \in C_{δ, m}

with

m > 1 + \frac{n}{4}

and n ≥2, since the chemotactic sensitivity at large cell densities is weakened by the strong diffusion, there exists a global classical solution of in the nondegenerate case D (0)>0, or a global weak solution under the degenerate case D (0)=0 . Refer to Wang and Liu for the chemotaxis‐fluid model with singular sensitivity.…”

Section: Introductionmentioning

confidence: 99%

“…To prove the global existence of the chemotaxis consumption system , we should pay attention to the lower bound estimate of v , which can be obtained by building the pointwise upper bound of

w : = - \ln \frac{v false(x, t false)}{false‖ v_{0} false(x false) ‖_{L^{\infty} false(normalΩ false)}}

, for controlling the singular chemotactic term

- χ \nabla \cdot false(\frac{u}{v^{α}} \nabla v false)

. In addition, with another transformation

z : = u false/ \exp false(\int_{1}^{v} \frac{χ}{false(1 - ε false) s^{α}} d s false)

inspired by Friedman and Tello, we will get the uniformly lower bound for z in Ω×(0, ∞ ) and hence obtain that v decays to 0 in the L ∞ norm by twice comparison arguments.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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\frac{χ}{v^{α}}

. Similarly to Winkler and Lankeit, let

w false(x, t false) : = - \ln \frac{v false(x, t false)}{false‖ v_{0} false(x false) ‖_{L^{\infty} false(normalΩ false)}} .

…”

Section: Global Existencementioning

confidence: 99%

\overset{ε}{true} = 1 + \frac{1}{ε}

, we have the following lemma.…”

Section: Global Existencementioning

confidence: 99%

“…Remark Recall the known result for the system, where a nonlinear diffusion

D \in C_{δ, m}

is contained with singular sensitive function

χ false(v false) = \frac{1}{v}

and ε =1 in . The global existence condition obtained in Lankeit is

m > 1 + \frac{n}{4}

Section: Global Existencementioning

confidence: 99%

“…Moreover, in a radially symmetric setting, a “normalized solution” has been constructed for n ≥3. If

D \in C_{δ, m}

with

m > 1 + \frac{n}{4}

Section: Introductionmentioning

confidence: 99%

“…To prove the global existence of the chemotaxis consumption system , we should pay attention to the lower bound estimate of v , which can be obtained by building the pointwise upper bound of

w : = - \ln \frac{v false(x, t false)}{false‖ v_{0} false(x false) ‖_{L^{\infty} false(normalΩ false)}}

, for controlling the singular chemotactic term

- χ \nabla \cdot false(\frac{u}{v^{α}} \nabla v false)

. In addition, with another transformation

z : = u false/ \exp false(\int_{1}^{v} \frac{χ}{false(1 - ε false) s^{α}} d s false)

inspired by Friedman and Tello, we will get the uniformly lower bound for z in Ω×(0, ∞ ) and hence obtain that v decays to 0 in the L ∞ norm by twice comparison arguments.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

Zhao

Zheng

2018

Math Methods in App Sciences

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Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity, and logistic source

Marras

Viglialoro

2018

Mathematische Nachrichten

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In this paper we study the zero‐flux chemotaxis‐system trueright126.0pt{ut=∇·false(false(u+1false)m−1∇u−u(u+1)α−1χ(v)∇vfalse)+ku−μu2,x∈Ω,t>0,vt=Δv−vu,x∈Ω,t>0,Ω being a convex smooth and bounded domain of Rn, n≥1, and where m,k∈R, μ>0 and α0. We prove that for nonnegative and sufficiently regular initial data u(x,0) and v(x,0), the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.

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A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients

Winkler

2022

Mathematische Nachrichten

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This manuscript is concerned with the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous issue of key importance in virtually any proof of boundedness features in Keller–Segel type systems. A strategy is proposed which at its core relies on L∞$L^\infty$ bounds for such quantities, conditional in the sense of involving certain Lebesgue norms of solution components that explicitly influence the signal evolution. Applications of this procedure firstly provide apparently novel boundedness results for two particular classes chemotaxis systems, and apart from that are shown to significantly condense proofs for basically well‐known statements on boundedness in two further Keller–Segel type problems.

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Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

Abstract: Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion Johannes Lankeit * We show the existence of locally bounded global solutions to the chemotaxis system

Cited by 99 publications

References 36 publications

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity, and logistic source

A unifying approach toward boundedness in Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients

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