Global generalized solutions to a Keller–Segel system with nonlinear diffusion and singular sensitivity

Yan, Jianlu; Li, Yuxiang

doi:10.1016/j.na.2018.06.016

Cited by 20 publications

(8 citation statements)

References 28 publications

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“…Moreover, he took into account asymptotic behavior of solutions to the system (1.5), and proved that v(•, t) * 0 in L ∞ (Ω) and v(•, t) → 0 in L p (Ω) as t → ∞ provided Ω u 0 ≤ m, − Ω ln( v0 v0 L ∞ (Ω) ) ≤ M , where m, M are positive constants. When ∆u is replaced by ∆u m (m ≥ 1), Yan et al [52] showed that if m > 1 + N −2 2N (N ≥ 2), the corresponding Neumann initial-boundary value problem admits a global generalized solution. When v does not stand for a nutrient to be consumed but a signalling substance produced by the bacteria themselves, i.e.…”

Section: Guoqiang Ren and Bin Liumentioning

confidence: 99%

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

Ren¹,

Liu²

2020

Communications on Pure &Amp; Applied Analysis

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In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.

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Section: Guoqiang Ren and Bin Liumentioning

confidence: 99%

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

Ren¹,

Liu²

2020

Communications on Pure &Amp; Applied Analysis

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“…No matter biological relevance or mathematical meaning, we find it is worth addressing the basic solvability theory of the model (3). Inspired by the arguments in previous studies [8,13,14,26,27], we mainly investigate the global classical solution in a chemotactic movement with singular sensitivity by two stimuli. eorem 1 partially generalizes and improves previously known ones.…”

Section: Introductionmentioning

confidence: 99%

Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

2021

Discrete Dynamics in Nature and Society

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“…Yan and Li 27 studied the global generalized solutions to () by maximum principle and parabolic regularity theory. The global weak solution to the following chemotaxis system with additional cross‐diffusion

({, \begin{matrix} left left left \\ array u_{t} & array = \nabla \cdot (\nabla u^{m} - u \nabla v), & array in Ω, t > 0, \\ array τ v_{t} & array = Δ v + δ Δ (u^{n}) + u - v, & array in Ω, t > 0, \\ array u (x, 0) & array = u_{0} (x), v (x, 0) = v_{0} (x), & array in Ω, \end{matrix})

with no‐flux boundary condition

(\nabla u^{m} - u \nabla v) \cdot ν = \nabla (v + δ u^{n}) = 0, on \partial Ω, t > 0

is proved by Carrillo et al 28 with the help of Alikakos iteration method.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative solutions to time fractional Keller–Segel system

Akilandeeswari

Tyagi

2020

Math Methods in App Sciences

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We establish the existence of nonnegative weak solutions to time fractional Keller-Segel system with Dirichlet boundary condition in a bounded domain with smooth boundary. Since the considered system has a cross-diffusion term and the corresponding diffusion matrix is not positive definite, we first regularize the system. Then under suitable assumptions on the initial conditions, we establish the existence of solutions to the system by using the Galerkin approximation method. The convergence of solutions is proved by means of compactness criteria for fractional partial differential equations. The nonnegativity of solutions is proved by the standard arguments. Furthermore, the existence of the weak solution to the system with Neumann boundary condition is discussed.

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Global generalized solutions to a Keller–Segel system with nonlinear diffusion and singular sensitivity

Cited by 20 publications

References 28 publications

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

Nonnegative solutions to time fractional Keller–Segel system

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