Related papers: Chemotaxis systems with singular sensitivity and l…

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla…

In this work we extend a recent result to chemotaxis fluid systems which include matrix-valued sensitivity functions $S(x,n,c):\Omega\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ in addition to the porous medium type diffusion, which were…

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth…

We study the system \begin{align*}\label{prob:star} \tag{$\star$} \begin{cases} u_t = D_1 \Delta u - \chi_1 \nabla \cdot (u \nabla v) + u(\lambda_1 - \mu_1 u + a_1 v) \\ v_t = D_2 \Delta v + \chi_2 \nabla \cdot (v \nabla u) + v(\lambda_2 -…

Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, $$ \left\{ \begin{array}{l} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad…

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 consider the chemotaxis-consumption system \[ \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) \\ v_t &= \Delta v - uv \end{aligned} \right. \] in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n$, $n \geq…

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1 - u - a_1 v), & x\in\Omega,\ t>0,\\ v_t = d_2…

This paper is concerned with a parabolic-elliptic chemotaxis model with density-suppressed motility and general logistic source in an $n$-dimensional smooth bounded domain with Neumann boundary conditions. Under the minimal conditions for…

We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann…

We study the existence of global boundedness solutions to the fully parabolic chemotaxis systems with logistic sources, $ru- \mu u^2$, under nonlinear Neumann boundary conditions, $\frac{\partial u}{\partial \nu }= |u|^p$ where $p >1 $ in…

We construct global generalized solutions to the chemotaxis system \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \lambda(x) u - \mu(x) u^\kappa,\\ v_t = \Delta v - v + u \end{cases} \end{align*} in smooth,…

We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\…

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 study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when…

In this paper, we study a nonlocal logistic system with nonlinear advection terms \begin{equation*} \left\{ \begin{array}{lcl} -\Delta u+\vec{\alpha}(x)\cdot \nabla (|u|^{p-1}u)&=&\left(a-\int_{\Omega}K_1(x,y)f(u,v)dy \right)u+bv\mbox{ in…

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\…

This work considers the Keller-Segel consumption system \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta (u\phi(v))+au-bu^\gamma,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0 \end{array} \right.…

This paper is concerned with a parabolic-parabolic-parabolic chemotaxis system with indirect signal production, modelling the impact of phenotypic heterogeneity on population aggregation \begin{equation*} \begin{cases} u_t = \Delta u -…

We study the weakly coupled nonlinear Schr\"odinger system \begin{equation*} \begin{cases} -\Delta u_1 = \mu_1 u_1^{p} +\beta u_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } \Omega,\\ -\Delta u_2 = \mu_2 u_2^{p} +\beta…