15 problems

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

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 $\Omega\subset\mathbb{R}^n$ 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…

Extension conjecture. With the exception of Lemma …

Consider locally bounded weak solutions of the Stefan problem … with $\beta$ the enthalpy graph and $p\ge2$. The modulus … for some $c,\sigma>0$ is conjectured to be optimal as a s…

Sire–Wei–Zheng's conjecture. Bubble tower solutions to with $b=0$ should exist.

Let $d\geq 1$, let $r>0$ describe the power-like growth of the local coupling, and consider the mean field games system with a uniformly convex Hamiltonian, including the purely qu…

Let $u$ solve the one-dimensional $p$-parabolic boundary-value problem … on $(-1,1)\times(0,T)$ with boundary values $u(-1,t)=0$, $u(1,t)=1$, and initial value…

Let $U_T$ be the spacetime domain, let $U_{(x,t)}$ be the backwards light cone of $(x,t)\in U_T$, and let $\widetilde{Q}$ be the spatial operator in the relativistic heat equation.…

Let $U_T$ be the spacetime domain, let $U_{(x,t)}$ denote the backwards light cone … and let $\widetilde{Q}$ be the spatial operator in the relativistic heat equation. Strong maxim…

Let $u$ be a solution of the ultraparabolic equation considered in Theorem 1.1, and suppose that the theorem's other hypotheses hold. Boundedness-removal conjecture. Theorem 1.1 st…

Consider the parabolic-parabolic Keller–Segel model with diffusion exponent $m$ and initial mass $M$. Finite-time blow-up conjecture. The cell density should blow up in finite time…

Let $N$ be the spatial dimension and let $\gamma^*=\frac{N+6}{N+2}$ denote the threshold appearing in the preceding global-existence result for the reaction–diffusion system. The a…