Chemotaxis Consumption Research Articles

Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source

Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Refined existence theorems for doubly degenerate chemotaxis–consumption systems with large initial data

Refined existence theorems for doubly degenerate chemotaxis–consumption systems with large initial data

Roles of density-related diffusion and signal-dependent motilities in a chemotaxis–consumption system

Roles of density-related diffusion and signal-dependent motilities in a chemotaxis–consumption system

Boundedness of classical solutions to a chemotaxis consumption model with signal-dependent motility

Boundedness of classical solutions to a chemotaxis consumption model with signal-dependent motility

Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

Global solutions in a chemotaxis consumption model with singular sensitivity

Global solutions in a chemotaxis consumption model with singular sensitivity

On the steady state problem of the chemotaxis-consumption model with logistic growth and Dirichlet boundary condition for signal

This paper concerns the steady state problem for chemotaxis consumption system with logistic growth and constant concentration of chemoattractant on the boundary of the domain. We establish the existence of a non-constant positive solution to this problem. The uniqueness of this solution is obtained under the smallness assumption on the boundary data. Some qualitative properties of the solutions and numerical results are presented.

Global dynamics for a chemotaxis consumption system with signal-dependent motility and logistic source

This paper deals with the chemotaxis consumption model{ut=Δ(uϕ(v))+au−buγ,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1). Here the parameters a>0, b>0 and γ>1, and the motility function ϕ satisfies ϕ∈C3([0,∞)) with ϕ(s)>0 for all s≥0. For all suitably regular initial data, if one of the following cases holds: (i) n≤2, γ>1; (ii) n≥3, γ>2; (iii) n≥3, γ=2 and b is large, then this system possesses global bounded classical solutions, while if n≥3 and γ∈(1,2], then this system admits at least one global weak solution, which becomes smooth after some waiting time and which will converge to the constant equilibrium ((ab)1γ−1,0).

Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility

This work considers the chemotaxis model with density-dependent motility ut=Δ(ϕ(v)u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2, with the nonnegative initial data (u0,v0)∈[W1,∞(Ω)]2. The motility function ϕ(v) satisfies ϕ(v)∈C3([0,∞)) with ϕ(v)>0 and ϕ′(v)<0 for all v≥0. For all suitably regular initial data, we prove that this system has a unique global bounded classical solution. The purpose of this work is to remove the smallness assumption on ‖v0‖L∞(Ω) in Li and Zhao (2021).

Boundedness to a quasilinear chemotaxis–consumption system with singular sensitivity in dimension one

Boundedness to a quasilinear chemotaxis–consumption system with singular sensitivity in dimension one

Boundedness and stabilization in the chemotaxis consumption model with signal-dependent motility

Boundedness and stabilization in the chemotaxis consumption model with signal-dependent motility

Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

This paper deals with the following chemotaxis–consumption system with signal-dependent motility $$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$ under no-flux boundary conditions in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^{n}$$ . $$\gamma (s)$$ is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.

Large time behavior to a chemotaxis–consumption model with singular sensitivity and logistic source

This paper studies the chemotaxis–consumption model with singular sensitivity and logistic source under homogenous Neumann boundary condition in a smooth bounded domain , with positive parameters χ, ϵ, μ, and f(u) ≃ uα, g(u) ≃ uβ for and β &gt; 0. It is shown that the system possesses a global classical solution when α &lt; 1 and . Furthermore, the asymptotic behavior of solutions is determined for 0 &lt; α &lt; 1 and 0 &lt; β &lt; α in a two‐dimensional setting, provided a sufficiently large μ.

Long-term behaviour in a parabolic–elliptic chemotaxis–consumption model

Global existence and boundedness of classical solutions of the chemotaxis–consumption systemnt=Δn−∇⋅(n∇c),0=Δc−nc, under no-flux boundary conditions for n and Robin-type boundary conditions∂νc=(γ−c)g for c (with γ>0 and C1+β(∂Ω)∋g>0 for some β∈(0,1)) are established in bounded domains Ω⊂RN, N≥1. Under a smallness condition on γ, moreover, we show convergence to the stationary solution.

Eventual smoothness and stabilization of renormalized radial solutions in a chemotaxis consumption system with bounded chemotactic sensitivity

In this paper, a chemotaxis model with bounded chemotactic sensitivity and signal absorption is considered under homogeneous Neumann boundary conditions in the ball $$\Omega =B_R(0)\subset {\mathbb {R}}^n$$, where $$R>0$$ and $$n\ge 2$$. Here, S is a scalar function with $$S(s,t)\in C^2([0,\infty )\times [0,\infty ))$$. Moreover, for some positive constant K, $$|S(s,t)|\le K$$ for all $$s,t\in [0,\infty )$$. For all appropriately regular and radially symmetric initial data ($$u_0,v_0$$) fulfilling $$u_0\ge 0$$ and $$v_0>0$$, the present paper shows that there is a globally defined pair (u, v) of radially symmetric functions which are continuous in $$({{\overline{\Omega }}} \backslash \{0\}) \times [0, \infty )$$ and smooth in $$({{\overline{\Omega }}} \backslash \{0\}) \times (0, \infty )$$, and which solve the corresponding initial-boundary value problem for ($$\star $$) with $$(u(\cdot , 0), v(\cdot , 0))=\left( u_{0}, v_{0}\right) $$ in an appropriate generalized sense. Moreover, in the two-dimensional setting, it is shown that these solutions are global mass-preserving in the flavor of the identity $$\begin{aligned} \int \limits _\Omega u(x,t)=\int \limits _\Omega u_0(x)\quad \text {for all }t>0 \end{aligned}$$and any nontrivial of these globally defined solutions eventually becomes smooth and satisfies $$\begin{aligned} u(\cdot , t) \rightarrow \frac{1}{|\Omega |}\int \limits _\Omega u_{0},\quad \text { and } \quad v(\cdot , t) \rightarrow 0 \quad \text { as } t \rightarrow \infty \end{aligned}$$uniformly with respect to $$x\in \Omega $$.

Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

We consider a chemotaxis consumption system with singular sensitivity , vt=εΔv−uv in a bounded domain with χ,α,ε&gt;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 .

Global existence and asymptotic behavior to a chemotaxis–consumption system with singular sensitivity and logistic source

We consider a chemotaxis–consumption system with singular sensitivity and logistic source: ut=Δu−∇⋅(uϕ(v)∇v)+ru−μuk, vt=Δv−uv in a smooth bounded domain Ω⊂Rn(n≥1), where r,μ>0, k>1, and ϕ(s)∈C1(0,∞) satisfying ϕ(s)→∞ as s→0. It is proved that there exists a global classical solution if k>1 for n=1 or k>1+n2 for n≥2. The asymptotic behavior of solutions is determined as well for ϕ(v)=1v, n=2 that if k>2, there exists μ∗>0 such that (u,v,|∇v|v)→((rμ)1k−1,0,0) as t→∞ provided μ>μ∗.

Global classical solution to a chemotaxis consumption model with singular sensitivity

In this paper, we are concerned with the chemotaxis consumption model with singular sensitivity ut=Δu−∇⋅(f(u)v∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥1), for 0<f(u)≤K(u+1)α with some K>0 and α<2. It is shown that for any sufficiently smooth initial data, the above system admits a global classical solution when either n=1andα<2, or n≥2andα<1−n4.