Related papers: Two-species chemotaxis-competition system with sin…
The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N \tau v_t=\Delta…
We consider the potentially degenerate haptotaxis system \begin{equation*} \left\{ \begin{aligned} u_t &= \nabla \cdot (\mathbb{D} \nabla u + u \nabla \cdot \mathbb{D}) - \chi \nabla \cdot (u\mathbb{D}\nabla w) + \mu u(1-u^{r- 1}), \\ w_t…
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…
In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$,…
It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(\Omega \subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t =…
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided…
In this article we investigate a parabolic-parabolic-elliptic two-species chemotaxis system with weak competition and show global asymptotic stability of the coexistence steady state under a smallness condition on the chemotactic strengths,…
In a ball $\Omega\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - uv, \end{array} \right. \] is…
Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= \Delta n - \nabla \cdot (n \nabla c), \\ 0 &= \Delta c - nc, \end{align*} under no-flux boundary conditions for $n$ and…
This paper is concerned with a three-component chemotaxis model accounting for indirect signal production,reading as $u_t = \nabla\cdot(\nabla u - u\nabla v)$,$v_t = \Delta v - v + w$ and $0 = \Delta w - w + u$,posed in a ball of $\mathbb…
This paper investigates the global well-posedness of a class of reaction-advection-diffusion models with nonlinear diffusion and Lotka-Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the…
We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = \Delta u - \chi\nabla\cdot(u\nabla v_1) + \xi\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = \Delta v_1 - \beta v_1 + \alpha u \\ \partial_t v_2 =…
We consider the system \begin{align*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \chi \nabla \cdot (\frac{n}{c} \nabla c),
We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+\kappa u-\mu u^2$ $ 0 =\Delta v-v+u$ $\partial_\nu v|_{\partial\Omega} = \partial_\nu u|_{\partial\Omega}=0,$ $…
We consider the coupled chemotaxis Navier-Stokes model with logistic source terms \[ n_t + u\cdot \nabla n = \Delta n - \chi \nabla \cdot (n \nabla c) + \kappa n - \mu n^2\] \[ c_t + u\cdot \nabla c = \Delta c - nc\] \[ u_t + (u\cdot…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha…
In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain $\Omega \subset \mathbb{R}^N$ ($N\in\{2,3\}$), not necessarily being convex. Thanks to the elementary lemma…
This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} \Delta u-\beta u+\lambda\frac{e^u}{\int_{\Omega}e^u}=0~\mathrm{in}~\Omega,…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla…
This paper deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=\epsilon\Delta u-\nabla\cdot(u\nabla…