Logistic Source Research Articles

Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source

This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, \t\t\t{ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} u_{t}=\\Delta u-\\chi _{1}\\nabla \\cdot (\\frac{u}{w}\\nabla w)+\\mu _{1}u-\\mu _{1}u^{\\alpha }, &x\\in \\varOmega , t>0, \\\\ \\upsilon _{t}=\\Delta v-\\chi _{2}\\nabla \\cdot ( \\frac{\\upsilon }{w}\\nabla w)+\\mu _{2}\\upsilon -\\mu _{2}\\upsilon ^{\\beta } , &x\\in \\varOmega , t>0, \\\\ w_{t}=\\Delta w-(u+\\upsilon )w, &x\\in \\varOmega , t>0, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain varOmega subset mathbb{R}^{n} (ngeq 1) with smooth boundary, where chi _{i}, mu _{i}>0(i=1, 2) and α, beta >1. It is proved that there exists a global classical solution if max {chi _{1}, chi _{2}}<sqrt{frac{2}{n}}, min {mu _{1}, mu _{2}}>frac{n-2}{n}, alpha =beta =2 for ngeq 2 or any chi _{i}>0(i=1,2), mu _{i}>0 (i=1,2), α, beta >1 for n=1.

Global existence, boundedness and asymptotic behavior to a chemotaxis model with singular sensitivity and logistic source

ABSTRACT This article deals with the chemotaxis model with singular sensitivity and logistic source under homogenous Neumann boundary condition in a smooth bounded domain , with and for all s>0 with . For , we show that the system admits a global classical solution that provides , and . Furthermore, for N=2, under the assumptions above, we obtain the boundedness and asymptotic behavior of solutions that provide , , and μ sufficiently large.

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, [Formula: see text]where [Formula: see text], a, b, [Formula: see text], and [Formula: see text] are positive constants. Assume [Formula: see text] . Then if in addition [Formula: see text] holds, it is proved that [Formula: see text] is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function [Formula: see text] with nonempty compact support, that is, [Formula: see text]and [Formula: see text]where [Formula: see text] is the unique global classical solution of (0.1) with [Formula: see text]. It is also proved that, if [Formula: see text] and [Formula: see text] holds, then [Formula: see text] is the minimal speed of the traveling wave solutions of (0.1) connecting (0,0) and [Formula: see text], that is, for any [Formula: see text], (0.1) has a traveling wave solution connecting (0,0) and [Formula: see text] with speed c, and (0.1) has no such traveling wave solutions with speed less than [Formula: see text]. Note that [Formula: see text] is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, [Formula: see text]Hence, if [Formula: see text] and [Formula: see text], or [Formula: see text] and [Formula: see text], then the chemotaxis neither speeds up nor slows down the spatial spreading in (0.1).

Boundedness in a quasilinear attraction\u2013repulsion chemotaxis system with nonlinear sensitivity and logistic source

In this paper, we deal with the following quasilinear attraction–repulsion model: \t\t\t{ut=∇⋅(D(u)∇u)−∇⋅(S(u)χ(v)∇v)+∇⋅(F(u)ξ(w)∇w)+f(u),x∈Ω,t>0,vt=Δv+βu−αv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}=\\nabla \\cdot (D(u)\\nabla u)-\\nabla \\cdot ( S(u)\\chi (v)\\nabla v)+ \\nabla \\cdot ( F(u)\\xi (w)\\nabla w)+f(u), &x\\in \\varOmega , t>0, \\\\ v_{t}=\\Delta v+\\beta u-\\alpha v, &x\\in \\varOmega ,t>0, \\\\ 0=\\Delta w+\\gamma u-\\delta w, &x\\in \\varOmega , t>0, \\\\ u(x,0)=u_{0}(x), \\quad\\quad v(x,0)= v_{0}(x), &x\\in \\varOmega \\end{cases} $$\\end{document} with homogeneous Neumann boundary conditions in a smooth bounded domain varOmega subset R^{n} (ngeq 2). Let the chemotactic sensitivity chi (v) be a positive constant, and let the chemotactic sensitivity xi (w) be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.

Global boundedness of solutions resulting from both the self-diffusion and the logistic-type source

This paper is concerned with the zero-flux logistic chemotaxis system with nonlinear signal production: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u)\), \(v_t=\Delta v-v+g(u)\) in a bounded and smooth domain \(\Omega \subset {\mathbb {R}}^n\) (\(n\ge 1\)), where \(\chi >0\) is a constant, and \(f,g\in C^1({\mathbb {R}})\) generalize the prototypes \(f(s)=s-\mu s^{\alpha }\) and \(g(s)=s(s+1)^{\beta -1}\) with \(\alpha >1\) and \(\beta ,\mu >0\). The existing studies, for the case that the self-diffusion or the logistic-type source prevails over the cross-diffusion singly, have asserted the global boundedness of solutions under the assumption that \(\beta 0\) suitably large. In the present paper, we prove that if \(\alpha -1=\beta =\frac{2}{n}\), then due to the self-diffusion and the logistic kinetics working together, the solutions are globally bounded regardless of the size of \(\mu >0\). Note that \(\alpha -1=\beta =\frac{2}{n}\) reduces to \(\alpha =2\) and \(\beta =1\) if \(n=2\). Hence, our result covers that of the two-dimensional classical chemotaxis system with logistic source: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+u-\mu u^2\), \(v_t=\Delta v-v+u\) (Osaki et al. in Nonlinear Anal 51:119–144, 2002).

Convergence rate of a quasilinear parabolic-elliptic chemotaxis system with logistic source

This paper deals with the quasilinear parabolic-elliptic chemotaxis system{ut=∇⋅(D(u)∇u)−∇⋅(χu∇v)+μu(1−u),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with smooth boundary, where χ>0 and μ>0. D(u) is supposed to satisfyD(0)>0,D(u)≥uαwithα∈(0,1). When n≥2, the convergence rate of the solution is investigated.

A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: ut=∇⋅(D(u)∇u−S(u)∇v)+f(u), vt=Δv−a1v+b1w, wt=Δw−a2w+b2u, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥1), where ai,bi>0 (i=1,2), D,S∈C2([0,∞)) and f:R→R is a smooth function generalizing the logistic source f(s)=b−μsr for all s≥0 with b≥0, μ>0 and r≥1. We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with μ>0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with μ>0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: u˜t=∇⋅(D(u˜)∇u˜)−χ∇⋅(u˜∇z)+ξ∇⋅(u˜∇w˜)+f(u˜), zt=Δz−ρz+ηu˜, w˜t=Δw˜−δw˜+γu˜, where χ,η,ξ,γ,ρ,δ>0.

Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source

In this paper we study the global existence of solutions to the fully parabolic chemotaxis system: ut=Δu−χ∇⋅(uv∇v)+f(u), vt=Δv−v+u in a smooth bounded domain Ω⊂Rn (n≥3) subject to the non-flux boundary conditions, where χ>0 and the logistic function f∈C1[0,∞) satisfies f(s)≤r−μsγ with r≥0 and γ,μ>0. It is shown that the problem possesses a global and classical solution as long as γ>2. Moreover, the global existence of the weak solution is also established provided that f(s)=rs−μs2 with any r,μ>0.

A new result for global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

We consider the following chemotaxis model{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+μ(u−u2),x∈Ω,t>0,vt−Δv=−uv,x∈Ω,t>0,(D(u)∇u−χu⋅∇v)⋅ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω on a bounded domain Ω⊂RN(N≥1), with smooth boundary ∂Ω,χ and μ are positive constants. Here, D is supposed to be smooth positive function satisfying D(u)≥CD(u+1)m−1 for all u≥0 with some CD,m>0. Besides appropriate smoothness assumptions, in this paper it is only required thatm>{1−μ2χ[1+max1≤s≤2⁡λ0(s)‖v0‖L∞(Ω)23]ifN≤2,1ifN≥3, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where λ0 is a positive constant which is corresponding to the maximal Sobolev regularity. The results of this paper extends the results of Jin (J. Differential Equations 263 (9) (2017) 5759–5772), who proved the possibility of boundness of weak solutions, in the case m>1 and N=3. To the best of our knowledge, this is the first result which gives the relationship between m and μχ that yields to the boundedness of the solution.

Global boundedness to a parabolic-parabolic chemotaxis model with nonlinear diffusion and singular sensitivity

This article deals with the parabolic-parabolic chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ φ ( v ) ) + f ( u ) , x ∈ Ω , t > 0 , v t = △ v − v + u , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R n ( n ≥ 1 ) with smooth boundary conditions, D , S ∈ C 2 ( [ 0 , + ∞ ) ) nonnegative, with D ( u ) = a 0 ( u + 1 ) − α for a 0 > 0 and α < 0 , 0 ≤ S ( u ) ≤ b 0 ( u + 1 ) β for b 0 > 0 , β ∈ R , and where the singular sensitivity satisfies 0 < φ ′ ( v ) ≤ χ v k for χ > 0 , k ≥ 1 . In addition, f : R → R is a smooth function satisfying f ( s ) ≡ 0 or generalizing the logistic source f ( s ) = r s − μ s m for all s ≥ 0 with r ∈ R , μ > 0 , and m > 1 . It is shown that for the case without a growth source, if 2 β − α < 2 , the corresponding system possesses a globally bounded classical solution. For the case with a logistic source, if 2 β + α < 2 and n = 1 or n ≥ 2 with m > 2 β + 1 , the corresponding system has a globally classical solution.

Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production

This work is concerned with a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production: u t = ∇ ⋅ ( ( 1 + u ) − α ∇ u ) − ∇ ⋅ ( u ( 1 + u ) β − 1 ∇ v ) + f ( u ) , v t = Δ v − v + u γ , with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain, where α , β ∈ R and γ > 0 . The logistic type source term f ( u ) satisfies that either f ( u ) ≡ 0 or f ( u ) = r u − μ u k with r ∈ R , μ > 0 and k > 1 . The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameters conditions, which improves the known results.

Global asymptotic stability in a chemotaxis-growth model for tumor invasion

This paper presents global existence and asymptotic behavior of solutions to the chemotaxis-growth system$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \qquad x\in \Omega, \ t&gt;0, \\ \ v_t = \Delta v + wz, \qquad x\in \Omega, \ t&gt;0, \\ \ w_t = -wz, \qquad x\in \Omega, \ t&gt;0, \\ \ z_t = \Delta z - z + u, \qquad x\in \Omega, \ t&gt;0, \end{array} \right. $in a smoothly bounded domain $ \Omega \subset \mathbb{R}^n $, $ n \le 3 $, where $ r&gt;0 $, $ \mu&gt;0 $ and $ \alpha&gt;1 $. Without the logistic source $ ru-\mu u^\alpha $, the stabilization of this system has been shown by Fujie, Ito, Winkler and Yokota (2016), whereas especially about asymptotic behavior, the logistic source disturbs applying this method directly. In the present paper, a way out of this difficulty is introduced and the asymptotic behavior of solutions to the system with logistic source is precisely determined.

Decay in chemotaxis systems with a logistic term

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain $Ω$ of $\mathbb{R}^N, $ for N∈{2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established.

Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

The current paper is devoted to the study of traveling wave solutions of the following parabolic-elliptic-elliptic chemotaxis systems, \begin{document}$\begin{equation}\label{main-eq-abstract}\begin{cases}u_{t} = Δ u- \nabla · ({ χ_1 u} \nabla v_1)+ \nabla · ({ χ_2 u} \nabla v_2) + u(a-bu), \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_1-λ_1v_1+μ_1u, \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_2-λ_2v_2+μ_2u, \;\;\;\; x∈\mathbb{R}^N, \end{cases}\;\;\;\;\;\;\;\;(0.1)\end{equation}$ \end{document} where \begin{document}$a>0, b>0, $\end{document} \begin{document}$u(x, t)$\end{document} represents the population density of a mobile species, \begin{document}$v_1(x, t), $\end{document} represents the population density of a chemoattractant, \begin{document}$v_2(x, t)$\end{document} represents the population density of a chemorepulsion, the constants \begin{document}$χ_1≥ 0$\end{document} and \begin{document}$χ_2≥ 0$\end{document} represent the chemotaxis sensitivities, and the positive constants \begin{document}$λ_1, λ_2, μ_1$\end{document} , and \begin{document}$μ_2$\end{document} are related to growth rate of the chemical substances. It was proved in an earlier work by the authors of the current paper that there is a nonnegative constant \begin{document}$K$\end{document} depending on the parameters \begin{document}$χ_1, μ_1, λ_1, χ_2, μ_2$\end{document} , and \begin{document}$λ_2$\end{document} such that if \begin{document}$b+χ_2μ_2>χ_1μ_1+K$\end{document} , then the positive constant steady solution \begin{document}$(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$\end{document} of (0.1) is asymptotically stable with respect to positive perturbations. In the current paper, we prove that if \begin{document}$b+χ_2μ_2>χ_1μ_1+K$\end{document} , then there exists a positive number \begin{document}$c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)≥ 2\sqrt{a}$\end{document} such that for every \begin{document}$ c∈ ( c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, , ∞)$\end{document} and \begin{document}$ξ∈ S^{N-1}$\end{document} , the system has a traveling wave solution \begin{document}$(u(x, t), v_1(x, t), v_2(x, t)) = (U(x·ξ-ct), V_1(x·ξ-ct), V_2(x·ξ-ct))$\end{document} with speed \begin{document}$c$\end{document} connecting the constant solutions \begin{document}$(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$\end{document} and \begin{document}$(0, 0, 0)$\end{document} , and it does not have such traveling wave solutions of speed less than \begin{document}2\sqrt a $\end{document} . Moreover we prove that \begin{document}$\begin{equation*}\lim\limits_{(χ_{1}, χ_2)?(0^+, 0^+)}c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2) = \begin{cases} 2\sqrt{a} \;\;\text{if}\;\; a≤ \min\{λ_1, λ_2\}\\\frac{a+λ_1}{\sqrt{λ_1}} \;\;\text{if}\;\; λ_1≤ \min\{a, λ_2\}\\\frac{a+λ_2}{\sqrt{λ_2}} \;\;\text{if}\;\; λ_2≤ \min\{a, λ_1\}\end{cases}\end{equation*}$ \end{document} for every \begin{document}$ λ_1, λ_2, μ_1, μ_2>0$\end{document} , and \begin{document}$\begin{equation*}\lim\limits_{x?∞}\frac{U(x)}{e^{-\sqrt a μ x}} = 1, \end{equation*}$ \end{document} where \begin{document}$μ$\end{document} is the only solution of the equation \begin{document}$μ+\frac{1}{μ} = \frac{c}{\sqrt{a}}$\end{document} in the interval \begin{document}$(0 , \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$\end{document} .

Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

In this paper, we study traveling wave solutions of the chemotaxis system \begin{document}$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad x\in \mathbb{R} \tau {{\partial }_{t}}{{v}_{1}} = (\Delta -{{\lambda }_{1}}I){{v}_{1}}+{{\mu }_{1}}u,\qquad x\in \mathbb{R}, \tau \partial {{v}_{2}} = (\Delta -{{\lambda }_{2}}I){{v}_{2}}+{{\mu }_{2}}u,\qquad x\in \mathbb{R}, \\\end{array} \right. & (0.1) \\\end{matrix} $\end{document} where \begin{document}$ \tau>0,\chi_{i}> 0,\lambda_i> 0, \mu_i>0 $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) and \begin{document}$ a>0, b> 0 $\end{document} are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) such that for every \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c , (0.1) has a traveling wave solution \begin{document}$ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $\end{document} connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} satisfying \begin{document}$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $\end{document} where \begin{document}$ \mu\in (0,\sqrt a) $\end{document} is such that \begin{document}$ c = c_\mu: = \mu+\frac{a}{\mu} $\end{document} . Moreover, \begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $\end{document} and \begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $\end{document} where \begin{document}$ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $\end{document} . We also show that (1) has no traveling wave solution connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} with speed \begin{document}$ c .

Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source \begin{document}$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), {v_t = \Delta v- v +u}, {w_t = - vw}, \quad {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, x\in \partial\Omega, t>0, {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), x\in \Omega, \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$ \end{document} in a smooth bounded domain \begin{document}$ \mathbb{R}^N(N\geq1) $\end{document} , with parameter \begin{document}$ r>1 $\end{document} . the parameters \begin{document}$ a\in \mathbb{R}, \mu>0, \chi>0 $\end{document} . It is shown that when \begin{document}$ r>2 $\end{document} , or \begin{document}$ \begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \end{array} \end{equation*} $\end{document} the considered problem possesses a global classical solution which is bounded, where \begin{document}$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $\end{document} is a positive constant which is corresponding to the maximal sobolev regularity. Here \begin{document}$ C_{\beta} $\end{document} is a positive constant which depends on \begin{document}$ \xi $\end{document} , \begin{document}$ \|u_0\|_{C(\bar{\Omega})}, \|v_0\|_{W^{1, \infty}(\Omega)} $\end{document} and \begin{document}$ \|w_0\|_{L^\infty(\Omega)} $\end{document} . This result improves or extends previous results of several authors.

Risk factor analysis, antimicrobial resistance and pathotyping of Escherichia coli associated with pre- and post-weaning piglet diarrhoea in organised farms, India.

A cross-sectional study was conducted from 2014 to 2017 in 13 organised pig farms located in eight states of India (Northern, North-Eastern and Southern regions) to identify the risk factors, pathotype and antimicrobial resistance of Escherichia coli associated with pre- and post-weaning piglet diarrhoea. The data collected through questionnaire survey were used to identify the risk factors by univariable analysis, in which weaning status, season, altitude, ventilation in the shed, use of heater/cooler for temperature control in the sheds, feed type, water source, and use of disinfectant, were the potential risk factors. In logistic regression model, weaning and source of water were the significant risk factors. The piglet diarrhoea prevalence was almost similar across the regions. Of the 909 faecal samples collected (North - 310, North-East - 194 and South - 405) for isolation of E. coli, pathotyping and antibiotic screening, 531 E. coli were isolated in MacConkey agar added with cefotaxime, where 345 isolates were extended spectrum β-lactamase (ESBL) producers and were positive for blaCTX-M-1 (n = 147), bla TEM (n = 151), qnrA (n = 98), qnrB (n = 116), qnrS (n = 53), tetA (n = 46), tetB (n = 48) and sul1 (n = 54) genes. Multiple antibiotic resistance (MAR) index revealed that 14 (2.64%) isolates had MAR index of 1. On the virulence screening of E. coli, 174 isolates harboured alone or combination of Stx1, Stx2, eaeA, hlyA genes. The isolates from diarrhoeic and post-weaning samples harboured higher number of virulence genes than non-diarrhoeic and pre-weaning. Alleviating the risk factors might reduce the piglet diarrhoea cases. The presence of multidrug-resistant and ESBL-producing pathogenic E. coli in piglets appears a public health concern.

Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production

In this paper we study the fully parabolic chemotaxis system with logistic-type source and nonlinear production: ut=Δu−χ∇⋅(u∇v)+f(u), vt=Δv−v+g(u), subject to the non-flux boundary conditions with a smooth and bounded domain Ω⊂Rn (n≥1), and the nonnegative initial data u0∈C0(Ω̄) and v0∈W1,∞(Ω), where the sensitivity χ>0, the logistic-type source f(s)≤s−μsα for s≥0 with α>1, μ>0, and the nonlinear production g(s)≤s(s+1)β−1 for s≥0 with β>0. It is obtained that the problem possesses a globally bounded classical solution if the self-restriction mechanism of the logistic-type source dominates the nonlinear production that β<α−1, or β=α−1 with μ>0 sufficiently large. This extends the current global existence result by Nakaguchi and Osaki (2018). The situation with the nonlinear production controlled by the linear diffusion that β∈(0,2n) is considered as well, which is parallel to the model with nonlinear diffusion and subcritical chemotactic sensitivity without logistic source studied by Tao and Winkler (2012).

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization

The Keller–Segel–Navier–Stokes system(⋆){nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ρn−μn2,ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇ϕ+f(x,t),∇⋅u=0, is considered in a bounded convex domain Ω⊂R3 with smooth boundary, where ϕ∈W1,∞(Ω) and f∈C1(Ω¯×[0,∞)), and where χ>0,ρ∈R and μ>0 are given parameters.It is proved that under the assumption that supt>0⁡∫tt+1‖f(⋅,s)‖L65(Ω)ds be finite, for any sufficiently regular initial data (n0,c0,u0) satisfying n0≥0 and c0≥0, the initial-value problem for (⋆) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L1(Ω)×L6(Ω)×L2(Ω;R3).Moreover, under the explicit hypothesis that μ>χρ+4, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying(n(⋅,t),c(⋅,t))→(ρ+μ,ρ+μ)in L1(Ω)×Lp(Ω)for all p∈[1,6)as t→∞. Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that u(⋅,t)→0 in L2(Ω;R3) as t→∞.

Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source

In this paper, we deal with the chemotaxis system with logistic source under homogeneous Neumann boundary conditions in a bounded convex domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by developing some Lp-estimate techniques, we show that the system possesses at least one global and bounded weak solution. Our results generalized and improved previous results, and partially results are new.