Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions

Wang, Qi; Yang, Jingxian; Yu, Feng

doi:10.3934/dcds.2017216

“…This three coming lemmas will be used in the next logical steps. We start by considering a suitable version of the Gagliardo-Nirenberg interpolation inequality, commonly used to treat nonlinearities appearing in the diffusion and/or chemosensitivity terms (see [20,22,24]), and successively by recalling a particular boundary integral employed to deal with terms defined in non-convex domains.…”

Section: Some Algebraic and Functional Inequalitiesmentioning

confidence: 99%

“…on (0, T max ). (24) Now (recall that p > 2 − 2α by virtue of ( 16)) we can apply Lemma 4.1 with p = 2(p−2+2α)θ p , q = 2 p and, once the following inequality (used in the sequel without mentioning)…”

Section: Estimating 1 P(p−1) D Dtmentioning

confidence: 99%

See 1 more Smart Citation

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

Frassu

¹

,

Viglialoro

²

2021

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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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“…This three coming lemmas will be used in the next logical steps. We start by considering a suitable version of the Gagliardo-Nirenberg interpolation inequality, commonly used to treat nonlinearities appearing in the diffusion and/or chemosensitivity terms (see [18,20,22]), and successively by recalling a particular boundary integral used to deal with terms defined in non-convex domains.…”

Section: Preliminaries: Inequalities and Parametersmentioning

confidence: 99%

“…Subsequently, by collecting ( 23), (24) and adjusting the product between ( 26) and ( 27) by means of the δ-Young inequality, relation (22) becomes…”

Section: 2mentioning

confidence: 99%

Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation

Frassu¹,

Viglialoro²

2020

Preprint

0

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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 problemwhere Ω is a bounded and smooth domain of R n , for n ≥ 2, and f (u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f (u) = u α and g(u) = u l , with proper α, l > 0. After having shown that any sufficiently smooth u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0 emanate a unique classical and nonnegative solution (u, v) to problem (✸), which is defined on Ω × (0, Tmax) with Tmax denoting the maximum time of existence, we establish that for any l ∈ (0, 2 n ) and, Tmax = ∞ and u and v are actually uniformly bounded in time. This paper is in line with the contribution by Horstmann and Winkler ([6]) and, moreover, extends the result by Liu and Tao ([12]). Indeed, in the first work it is proved that for g(u) = u the value α = 2 n represents the critical blow-up exponent to the model, whereas in the second, for f (u) = u, corresponding to α = 1, boundedness of solutions is shown under the assumption 0 < l < 2 n .

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Global in Time and Bounded Solutions to a Parabolic–Elliptic Chemotaxis System with Nonlinear Diffusion and Signal-Dependent Sensitivity

Viglialoro

¹

2019

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Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions

Cited by 3 publications

References 25 publications

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

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

Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation

Global in Time and Bounded Solutions to a Parabolic–Elliptic Chemotaxis System with Nonlinear Diffusion and Signal-Dependent Sensitivity

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