6 problems

Let $u(x,t)$ solve the Neumann problem … with … and initial data $u(x,0)=u_0(x)$, where $u_0(x)>0$ for some $x$. Long-time convergence conjecture. … The claim describes the predict…

Half-ground-state threshold conjecture. The energy threshold ensuring energy trapping and a blow-up/scattering dichotomy for the Neumann problem is exactly

Geometric and radiality conjecture. Theorem cannot be generalised to other geometries, and the radiality assumptions in Theorems and are necessary.

Let $f$ satisfy the three hypotheses stated in the source, let $u_0>0$ satisfy $f(u_0)=u_0$, and consider the Neumann problem on the ball $B$...

Let $u$ be a weak solution of the nonlocal elliptic equation on $\Omega$, with the nonlocal operator $\mathcal{L}$ and the associated bilinear form as in the paper. The preceding a…

Let $\Omega$ be a bounded convex domain in the plane symmetric with respect to the $x$-axis, and let $p^N_{\Omega}(t,z,w)$ be its Neumann heat kernel. Let $f:\mathbb{B}\to\Omega$ b…