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In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v -m(t)+ u , \quad \int_{\Omega}v \,dx=0, & \\[2mm] u(x,0)= u_0(x), & \end{cases} \end{equation*} in $\Omega \times (0,\infty)$, with $\Omega$ a ball in $\mathbb{R}^N$, $N\geq 3$, under homogeneous Neumann boundary conditions, where $g(u)= \lambda u - \mu u^k$ , $\lambda >0, \ \mu >0$, and $ k >1$, $f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-\alpha}$, $\alpha>0$, which describes gradient-dependent limitation of cross diffusion fluxes. The function $m(t)$ is the time dependent spatial mean of $u(x,t)$ i.e. $m(t) := \frac 1 {|\Omega|} \int_{\Omega} u(x,t) \,dx$. Under smallness conditions on $\alpha$ and $k$, we prove that the solution $u(x,t)$ blows up in $L^{\infty}$-norm at finite time $T_{max}$ and for some $p>1$ it blows up also in $L^p$-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on $\alpha$ or $k$, we prove that the solution is global and bounded in time., Comment: 32 pages. arXiv admin note: text overlap with arXiv:2201.08716

This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u (1- u^{\kappa_1-1} - a_1 v^{\lambda_1-1}), &\quad x \in \Omega,\ t>0,\\ % v_t = d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v (1- a_2 u^{\lambda_2-1} - v^{\kappa_2-1}), &\quad x \in \Omega,\ t>0,\\ % 0 = d_3 \Delta w + \alpha u + \beta v - h(u,v,w), &\quad x \in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega \subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=\gamma w$ or $h=\frac{1}{|\Omega|}\int_\Omega(\alpha u+ \beta v)\,dx$. In the case that $\kappa_1=\lambda_1=\kappa_2=\lambda_2=2$ and $h=\gamma w$, it is known that smallness conditions for the chemotacic effects lead to boundedness of solutions (Math.\ Methods Appl.\ Sci.; 2018; 41; 234--249). However, the case that the chemotactic effects are large seems not to have been studied yet; therefore it remains to consider the question whether the solution is bounded also in the case that the chemotactic effects are large. The purpose of this paper is to give a negative answer to this question.

The diffusive Lotka-Volterra predator-prey model \begin{eqnarray*} \left\{ \begin{array}{rcll} u_t &=& \nabla\cdot \left[ d_1\nabla u + \chi v^2 \nabla \Big(\dfrac{u}{v}\Big)\right] +u(m_1-u+av), \qquad & x\in\Omega, \ t>0, \\ v_t &=& d_2\Delta v+v(m_2-bu-v), \qquad & x\in\Omega, \ t>0, \end{array} \right. \end{eqnarray*} is considered in a bounded domain $\Omega\subset\mathbb{R}^n$, $n \in\{2,3\}$, under Neumann boundary condition, where $d_1, d_2, m_1, \chi, a, b$ are positive constants and $m_2$ is a real constant. The purpose of this paper is to establish global existence and boundedness of classical solutions in the case $n=2$ and global existence of weak solutions in the case $n=3$ as well as show long-time stabilization. More precisely, we prove that the solutions $(u(\cdot,t), v(\cdot,t))$ converge to the constant steady state $(u_*, v_*)$ as $t \to \infty$, where $u_*, v_*$ solves $u_*(m_1-u_*+av_*)=v_*(m_2-bu_*-v_*)=0$ with $u_* > 0$ (covering both coexistence as well as prey-extinction cases).

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 u-\beta v, \\[1.05mm] 0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0$ and $f$ is a function of logistic type such as $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved under the condition $p0$. This paper classifies boundedness and blow-up into the cases $pq$ without any condition for the sign of $\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or $\chi\alpha-\xi\gamma>0$.

This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=\Delta u-\nabla \cdot (u\chi(v)\nabla v) +\nabla \cdot (u\xi(w)\nabla w), &x \in \Omega,\ t>0,\\[1.05mm] v_t=\Delta v-v+u, &x \in \Omega,\ t>0,\\[1.05mm] w_t=\Delta w-w+u, &x \in \Omega,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $\chi, \xi$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the system with logistic source have already been obtained by taking advantage of the effect of logistic dampening (J. Math. Anal. Appl.; 2020;489;124153). This paper establishes existence of global bounded classical solutions despite the loss of logistic dampening.

This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u \nabla v) + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k, \quad &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta v + \alpha u - \beta v, \quad &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta w + \gamma u - \delta w, \quad &x \in \Omega,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $\Omega \subset \mathbb{R}^n$ ($n \ge 3$), with constant parameters $\lambda \in \mathbb{R}$, $k>1$, $\mu, \chi, \xi, \alpha, \beta, \gamma, \delta>0$. Blow-up phenomena in the system have been well investigated in the case $\lambda=\mu=0$, whereas the attraction-repulsion chemotaxis system with logistic degradation has been not studied. Under the condition that $k>1$ is close to $1$, this paper ensures a solution which blows up in $L^\infty$-norm and $L^\sigma$-norm with some $\sigma>1$ for some nonnegative initial data. Moreover, a lower bound of blow-up time is derived.

This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\Delta u-\chi\nabla \cdot (u\nabla v)+\xi \nabla\cdot(u \nabla w), \quad v_t=\Delta v-v+u, \quad w_t=\Delta w-w+u, \quad x \in \Omega,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $\chi, \xi>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the case $n=3$. This paper provides an answer to this open problem by using a transformation which leads to a system presenting structural advantages respect to the original.

This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)\chi(v)\nabla v) +\nabla\cdot(H(u)\xi(w)\nabla w), \quad v_t=d_1\Delta v+\alpha u-\beta v, \quad w_t=d_2\Delta w+\gamma u-\delta w, \quad x \in \Omega,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, \alpha, \beta, \gamma, \delta>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \in \mathbb{R}$; $0 \le G(s) \le b_0(s+1)^{q-1}$ with $b_0>0$ and $q<\min\{2,\ m+1\}$; $0 \le H(s) \le c_0(s+1)^{r-1}$ with $c_0>0$ and $r<\min\{2,\ m+1\}$, and the signal-dependent sensitivities $\chi, \xi$ satisfy $0<\chi(s)\le \frac{\chi_0}{s^{k_1}}$ with $\chi_0>0$ and $k_1>1$; $0<\xi(s)\le \frac{\xi_0}{s^{k_2}}$ with $\xi_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (J. Math. Anal. Appl.; 2018;461;1260-1270) and Jia-Yang (J. Math. Anal. Appl.; 2019;475;139-153). However, there is no work on the above fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system by introducing a new test function.

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -\chi \nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in \Omega,\ t>0, \\[1mm] 0 = \Delta v - \mu + u, &x\in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega := B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $\chi >0$, $p,q\geq1$, $\mu := \frac 1{|\Omega|} \int_\Omega u_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;2017;4;31--67) established existence of an initial data such that the corresponding solution blows up in finite time when $p=q=1$. This paper gives existence of blow-up solutions under some condition for $\chi$ and $u_0$ when $1\leq p\leq q$., Comment: 30 pages

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -\chi \nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = \Delta v - \mu + u \end{cases}\end{equation*} under no-flux boundary conditions in balls $\Omega\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overline{\Omega})$, where $\chi>0$ and $\mu:=\frac{1}{|\Omega|}\int_{\Omega}u_0$. Bellomo--Winkler (Comm.\ Partial Differential Equations;2017;42;436--473) proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when $p=q=1$ under some conditions for $\chi$ and $\int_\Omega u_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\geq 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$., Comment: 44 pages

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begin{equation*} \begin{cases} u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & {\rm in} \; \Omega \times (0,T), \\[1mm] v_t=\Delta v-v+u & {\rm in} \; \Omega \times (0,T) \end{cases} \end{equation*} under Neumann boundary conditions and initial conditions, where $\Omega$ is a general bounded domain in $\mathbb{R}^n$ with smooth boundary, $\alpha>0$, $\chi>0$, $m_1, m_2 \in \mathbb{R}$ and $T>0$. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that $m_1=1$ and $\Omega$ is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that $m_1 \neq 1$ and $\Omega$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $\Omega$.

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.

This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier--Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization., Comment: 12 pages

This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion type equations. The other is the Cahn--Hilliard type system. The present paper applies Br\'ezis theory directly to both equations and gives existence results for these two equations even if the domain is unbounded. Moreover, an error estimate is also proved via apriori estimates obtained directly., Comment: 22 pages

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{equation*} u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v), \quad v_t=\Delta v + u - v, \quad x\in\Omega,\ t>0, \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $\chi$ is a function satisfying $\chi(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted global existence of bounded solutions for arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for $\chi$ and to build a mathematical bridge between the cases $k=1$ and $k>1$., Comment: 15pages