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We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_\nu u&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady…

We study the Neumann initial-boundary problem for the chemotaxis system \begin{align*} \left\{\begin{array}{c@{\,}l@{\quad}l@{\,}c} u_{t}&=\Delta u-\nabla\!\cdot(u\nabla v),\ &x\in\Omega,& t>0,\\ v_{t}&=\Delta v-v+u+f(x,t),\ &x\in\Omega,&…

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma}…

In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array} \right.…

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 analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing: \begin{equation*} \begin{cases} u_t = \Delta(e^{-v}u)\qquad &\mathrm{in}\ \Omega \times…

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=\Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=\Delta v - v + u^{\theta} \end{cases}…

Given a smooth bounded domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=\lambda a(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\,…

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -\Delta v+v=\lambda e^v, \quad v>0 \quad & \text{in }\Omega,\\ \partial_\nu v=0 &\text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a…

We study the Neumann initial-boundary problem for the chemotaxis system $$ \left\{\begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, 0=\Delta v - \mu(t)+w, & x\in \Omega, \, t>0, \tau w_t + \delta w = u, &…

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{array}…

In this study, the finite volume method is implemented for solving the problem of the semilinear equation: $-d \delta u+ u=u^q (d, q>0) $with a homogeneous Neumann boundary condition. This problem is equivalent to the known stationary…

We consider the heat equation in a smooth bounded convex domain $\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_\nu u = \lambda (u - u^3)$. Stable non-constant stationary solutions do not exist when…

This paper focuses on the following Keller-Segel system with singular sensitivity and logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{ v_t=\Delta v-…

In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a…

In this paper, we study the global stability of classical solutions to a Keller--Segel equations in scaling-invariant spaces. We prove that for any given $0<\mathcal{M}<1+\lambda_1$ with $\lambda_1$ being the first eigenvalue of Neumann…

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa(|x|) u - \mu(|x|) u^p, \\ 0 = \Delta v -…

For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{\beta }u=-(-\Delta )^{\frac{\alpha}{2}}(\rho (v)u),& t>0\\ (-\Delta )^{\frac{\alpha}{2}} v+v=u,& t>0 \end{cases} \end{equation*}…

We consider the quasilinear parabolic-parabolic Keller-Segel system $$ u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \qquad x\in\Omega, \ t>0, v_t=\Delta v -v + u, x\in\Omega, \ t>0, $$ under homogeneous Neumann boundary…

We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (\phi(u)\nabla u) - \nabla \cdot (\psi(u)\nabla v)+f(u), & x \in \Omega, \; t > 0, v_t = \Delta v - v + u, & x \in \Omega, \; t > 0,…