8 problems

Let $\mathbb{G}=(\mathbb{R}^n,\circ)$ be a stratified Lie group of homogeneous dimension $Q$, let $\Omega\subset\mathbb{G}$ be a bounded open set with smooth boundary, let…

Logarithmic grow-up conjecture. In the critical case $\sigma=-2$, solutions to Eq. (1) exhibit logarithmic grow-up as $t\to\infty$.

Blow-up analysis conjecture. A blow-up analysis can be performed in the quasilinear context to prove the existence of extremal functions for $\Phi$ in ${\mathcal B}_1$. This concer…

Let $s$ denote the Sobolev regularity exponent, and let Theorems and be the local well-posedness results stated in the source for the two-dimensional and $n$-dimensional cubic prob…

Consider quasilinear dispersive flows in two and higher spatial dimensions with cubic nonlinearities and sufficiently small initial data. Two-dimensional and higher-dimensional qua…

Extension conjecture. Theorem $7.4$ remains true if $a(x)$ satisfies $(a_5)$ instead of $(a_6)$; namely, under $(f_2)$, $(a_2)$, and $(6.18)$, problem $(1.1)$ cannot admit regular…

In higher-order equations with sub-homogeneous terms, consider certain homogeneous higher-order operators for which T-accretivity in $L^p$ fails. Higher-order $L^2$ contraction con…

Let $1<p<N-1$, let $q_c$ be the critical exponent defined in the paper, and let $S\subset S^{N-1}_+$ be star-shaped. Consider the boundary-value problem … is that this problem admi…