32 problems

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, and consider the minimal Keller–Segel system, meaning the chemotaxis system with constant sensitivity $\chi$, in spatial dimens…

Sharp four-dimensional PES threshold. The free energy $\mathcal F_{\mathrm{PES}}$ should be bounded below on $\mathcal A_M(\Omega)$ for $M<M_*$, should lose compactness at $M=M_*$,…

Let $N=4$ and let $\Omega$ be $\mathbb{R}^{4}$, $\mathbb{T}^{4}$, or a bounded $C^{2}$ Neumann domain. Let $\mathcal{A}_M(\Omega)$ denote the admissible class of densities of mass…

Let $m,\alpha,\gamma\geq 1$, let $\chi$ be the chemotaxis sensitivity, and consider the traveling-wave solutions $(u(t,x),v(t,x))=(U(x-ct),V(x-ct))$ of the model equation. Strong-p…

Let $m,\alpha,\gamma\geq 1$ be the parameters of the repulsion/attraction chemotaxis model, and let $c^*_{\chi,m,\gamma}$ denote the lower bound for traveling-wave speeds. Speed-up…

Consider the classical Keller–Segel model with no reaction term, so that $f=0$, in spatial dimension $n\ge 3$. A solution has sufficiently large total mass, meaning that its initia…

Let $\chi_c$ denote the largest possible constant such that global solutions exist for all $\chi<\chi_c$, whereas blow-up may occur for $\chi>\chi_c$. Here $n$ is the spatial dimen…

Let $u$, $v$, and $w$ denote respectively the lymphocyte density, the chemical-signal concentration, and the tumor-cell density, and let $\boldsymbol{\mu_1}$ and…

Gagliardo–Nirenberg improvement conjecture. The constant $C_S^2$ in the resulting estimate could be improved to $C_{GN}^4/4$.

Consider the chemotaxis system and its constant solution $(1,1)$, with $\chi^*$ denoting the critical sensitivity parameter at which the constant solution changes local stability a…

Subcritical velocity-bound conjecture. The velocity $\mathbf{u}$ possesses a bound that is uniform in time.

Critical-mass global-existence conjecture. The solution $(n,c,\mathbf{u})$ also exists globally in time in the critical mass case.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, and consider the parabolic problem (CC8) with positive parameters $\beta$ and constant $\gamma$. For a tra…

Let the evolution of bacteria in a sessile droplet be understood through the numerical solutions of the coupled chemotaxis-fluid system, with the droplet's total bacterial amount a…

Minimal-speed selection conjecture. When the Cauchy problem is initiated with localized initial data, the minimal speed given by $$ is selected among the speeds giving rise to admi…

Let $\Omega\subset\mathbb{R}^n$ be the spatial domain, let $n\geq 3$, and consider the system $\eqn{eqn:kinemod}$ with $\tau=1$ and $\gamma(v)=v^{-k}$, where $k\geq\frac{n}{n-2}$.…

Consider the logarithmic Keller–Segel system with sensitivity parameter $k>0$ in a domain $\Omega\subset\mathbb{R}^n$ for $n\geq 3$, where the chemotactic sensitivity is…

Global-existence analogy conjecture. The system $$ with nonconstant $d_1$ and $\chi_1$ should possess properties similar to those of its parabolic–elliptic simplifications, for whi…

Fuhrmann–Senba threshold conjecture. The threshold number separating boundedness and blowup of classical solutions is

The chemotaxis system under consideration is a parabolic–elliptic system with a nonlocal reaction term involving the spatial integral of a power of the population density. Earlier…

Lin–et al. blow-up conjecture. Blow-up should occur for

Let $m_1,m_2$ be the masses, $\tau_1,\tau_2$ the time-scale parameters, and $\tau_1,\tau_2>0$. Let $\tau_1,\tau_2>0$ denote the parabolic case of the system…

Maximal-spreading-speed conjecture. In general, the presence of the chemo-attractant does not increase the maximal spreading speed.

Finite-time blow-up conjecture. There exists finite-time blow-up when

Consider the macroscopic chemotaxis system … A travelling wave is a solution of the form $(\rho(z-ct),S(z-ct),N(z-ct))$ satisfying … Two such waves are identified up to multiplicat…