6 problems

Let $\mathcal{M}^{p,q}_{s}$ denote a weighted modulation space, with $0<q\leq 1$ allowed, and consider the nonlinear heat equations associated with fractional powers of the general…

Let $u$ be the solution of the Cauchy–Dirichlet problem under the assumptions of the preceding theorem, and let $T$ be its first singular time. Complete blowup conjecture. The blow…

Well-posedness conjecture. The well-posedness result proved in the paper for the stated modulation-space setting should hold as well for initial data in $M^{p,q}_s(\mathbb{R}^d)$ w…

Let $n$ be the dimension, let $p$ satisfy … and let $J$ be the functional on $H^{1}_K(\mathbb R^n_+)\cap L_{K'}^p(\partial\mathbb R^n_+)$ appearing in the paper, where $K$ and $K'$…

Let $u$ solve the classical nonlinear heat equation … For $N\geq 3$, set $p_S=(N+2)/(N-2)$, while $p_S=\infty$ for $N=1,2$. Optimal Liouville conjecture. The Liouville property sho…

Let $\gamma:[0,T]\to\mathbb{R}^{N+1}$ be a $C^1$ curve, write $\gamma(\tau)=(x(\tau),t(\tau))$, and assume $\gamma'(\tau)\neq 0$. For $\tau_0\in(0,T)$, let…