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

“…By putting together the two inequalities and choosing 1 = 2 , the first part of the lemma is concluded. All the details of the second inequality can be found in [15,Lemma 3.3].…”

Section: Lemma 42 Let Be a Bounded And Smooth Domain Of R N With N ≥ 1 And Q ∈ [1 ∞)mentioning

confidence: 99%

Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

Frassu

Viglialoro

2021

Acta Appl Math

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This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem $$ \textstyle\begin{cases} u_{t}= \Delta u - \nabla \cdot (f(u) \nabla v) & \text{ in } \Omega \times (0,T_{max}), \\ v_{t}=\Delta v-v+g(u) & \text{ in } \Omega \times (0,T_{max}), \end{cases} $$ { u t = Δ u − ∇ ⋅ ( f ( u ) ∇ v ) in Ω × ( 0 , T m a x ) , v t = Δ v − v + g ( u ) in Ω × ( 0 , T m a x ) , where $\Omega $ Ω is a bounded and smooth domain of $\mathbb{R}^{n}$ R n , for $n\geq 2$ n ≥ 2 , and $f(u)$ f ( u ) and $g(u)$ g ( u ) are reasonably regular functions generalizing, respectively, the prototypes $f(u)=u^{\alpha }$ f ( u ) = u α and $g(u)=u^{l}$ g ( u ) = u l , with proper $\alpha , l>0$ α , l > 0 . After having shown that any sufficiently smooth $u(x,0)=u_{0}(x)\geq 0$ u ( x , 0 ) = u 0 ( x ) ≥ 0 and $v(x,0)=v_{0}(x)\geq 0$ v ( x , 0 ) = v 0 ( x ) ≥ 0 produce a unique classical and nonnegative solution $(u,v)$ ( u , v ) to problem (◊), which is defined on $\Omega \times (0,T_{max})$ Ω × ( 0 , T m a x ) with $T_{max}$ T m a x denoting the maximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ l ∈ ( 0 , 2 n ) and $\frac{2}{n}\leq \alpha <1+\frac{1}{n}-\frac{l}{2}$ 2 n ≤ α < 1 + 1 n − l 2 , $T_{max}=\infty $ T m a x = ∞ and $u$ u and $v$ v are actually uniformly bounded in time.The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for $g(u)=u$ g ( u ) = u the value $\alpha =\frac{2}{n}$ α = 2 n represents the critical blow-up exponent to the model, whereas in the second, for $f(u)=u$ f ( u ) = u , corresponding to $\alpha =1$ α = 1 , boundedness of solutions is shown under the assumption $0< l<\frac{2}{n}$ 0 < l < 2 n .

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“…Proof. Regard the proof of inequalities ( 8) and ( 9), we refer the reader to [16,Lemma 3.1]. As to (10)…”

Section: Some Preparatory Toolsmentioning

confidence: 99%

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

Frassu¹,

Viglialoro²

2020

Preprint

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We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production g for the chemorepellent and sublinear rate f for the chemoattractant:in Ω × (0, Tmax).In this problem, Ω is a bounded and smooth domain of R n , for n ≥ 1, χ, ξ, δ > 0, f (u) and g(u) reasonably regular functions generalizing the prototypes f (u) = Ku α and g(u) = γu l , with K, γ > 0 and proper α, l > 0. Once it is indicated that any sufficiently smooth u(x, 0) = u 0 (x) ≥ 0 and v(x, 0) = v 0 (x) ≥ 0 produce a unique classical and nonnegative solution (u, v, w) to (✸), which is defined in Ω × (0, Tmax), we establish that for any such (u 0 , v 0 ), the life span Tmax = ∞ and u, v and w are uniformly bounded in Ω × (0, ∞), (i) for l = 1, n ∈ {1, 2}, α ∈ (0, 1 2 + 1 n ) ∩ (0, 1) and any ξ > 0, (ii) for l = 1, n ≥ 3, α ∈ (0, 1 2 + 1 n ) and ξ larger than a quantity depending on χ v 0 L ∞ (Ω) , (iii) for l > 1 any ξ > 0, and in any dimensional settings. Finally, an indicative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in [11].

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

Cited by 15 publications

References 46 publications

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

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