37 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…

Let $(\rho_\delta,c_\delta)$ and $(\rho,c)$ be the solutions in Theorem 1.1, and suppose the convergence estimate … holds for every $\lambda>0$ and sufficiently small $\delta>0$. O…

Let $u$ be a radially symmetric solution of the parabolic-elliptic Keller–Segel system in two spatial dimensions that blows up at a finite time. Nanjundiah's conjecture. The densit…

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…

For $d>2$, consider nonnegative, nontrivial subsolutions of the Lane–Emden inequality … and let $M_c^{sub}(d)$ be the corresponding minimal mass. Let $M_c(d)$ denote the Keller–Seg…

Consider the estimate for the difference $Q(u)$ of the second derivatives of the potential $u$ used to derive Li–Yau and Aronson–Bénilan estimates for the Keller–Segel system. Opti…

Let $M_c^{sub}(d)$ denote the minimal mass of nonnegative, nontrivial subsolutions of the Lane–Emden equation … and let $M_c(d)$ be the critical mass of the Keller–Segel system. Cr…

Consider the three-dimensional Keller–Segel equation and the explicit self-similar solution … where … and $Q$ satisfies … Here, $\Lambda$ denotes the scaling operator, and a radial…

The two-dimensional Keller–Segel system is known to admit finite-time blowup solutions when the initial density has total mass greater than $8\pi$ and finite second moment; existin…

For each $n\geq 0$, let $L_n$ be the operator described in the surrounding spectral proposition, with its nonpositive eigenvalues and eigenfunctions as specified there. Zero-eigenv…

The three-dimensional and higher-dimensional Keller–Segel system admits several known localised radial blow-up patterns, including type I self-similar blow-up, type II blow-up conc…

For $d=3,4$, let $\ell=\frac{d}{d-2}$ and consider the linear transport operator … It acts on smooth radial functions. Point-spectrum conjecture. The point spectrum of…

For $d=3,4$ and $s>0$, let … and define … Let $\mathscr{H}[s]:H^2_{\omega[s]}\to L^2_{\omega[s]}$ be the linearized operator acting on radial functions. Spectral conjecture. The op…

The algorithm is tested by varying the particle number $P$, with the fitted convergence rate for the relative error given by … Particle-interaction conjecture. The reduced converge…

Let $(u,v)$ be an entire solution of the Keller–Segel system on $\mathbb{R}^2\times\mathbb{R}$, meaning that $u$ and $\nabla v$ are smooth and satisfy the system, with the second e…

Let $N\geq 3$ and consider the logarithmic Keller–Segel system obtained when $\gamma(s)=s^{-k}$, with parameter $k$ and critical value $k_c=\frac{N}{N-2}$. The threshold conjecture…

Continuity conjecture for blowup. Blowup phenomena should be continuous with respect to the parameter $\tau\to 0$, both for ${\rm (TM)}\to{\rm (NLH)}$ and for…

Let $\varrho$ be a solution of the time-dependent Keller–Segel system, and suppose that for every time $t$ in its domain the second moment … is finite. The mass of static Keller–Se…

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

Let $\Omega=B_R(0)\subset\mathbb{R}^2$ with $R>0$ and $k>0$. A steady state is a solution $(u,v)$ of the stationary problem considered in the paper, with prescribed mass…

Forced-wave uniqueness and stability conjecture. The forced wave solution of the model is unique and stable in certain parameter region.

Heterogeneous forced-wave existence conjecture. If $b>\chi\mu$ and $\lambda_{\infty}(r(\cdot))>0$, there is a forced wave solution connecting $(0,0)$ and $(0,0)$.

Forced-wave existence conjecture. If $b>\chi\mu$ and $c>-2\sqrt{r^*}$, there is a forced wave solution connecting $((r^*/b),(\mu/\nu)(r^*/b))$ and $(0,0)$.

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

Extension conjecture. Theorem holds even without this second restriction, although the constant $C$ may then depend on $T$ as well.