6 problems

For problem, consider positive, radially symmetric solutions with the setting described above. Fila–King's conjecture. Infinite-time blow-up occurs only in dimensions $3$ and $4$.…

Let $A(t)$ be a flow of connections on a vector bundle $E$ over a manifold $M$, and let $U(t)=U^{A(t)}$ denote the corresponding flow of parallel transports, regarded as sections o…

Consider the reaction-diffusion equation … with periodic boundary conditions and periodized Gaussian initial data. Finite-time blow-up conjecture. For periodic boundary conditions,…

Unique optimal-domain conjecture. There exists a unique best control domain, namely a unique minimizer of $C_T(\omega)$ over $\mathcal{U}_L$.

Sharp half-order convergence conjecture. If $\omega\subsetneq\Omega$, the estimate is not sharp, and the order of convergence $1/2$ should hold as well.

Precise $L^p$ spectrum conjecture. For every $p\in(1,\infty)$,