13 problems

Consider equation … corresponding to sufficiently large initial data should blow up in finite time, in the sense that $w_{rr}(t,0)$ diverges as $t\nearrow T$ for some $T>0$, and th…

Let $g$ satisfy the self-similar boundary value problem referred to as, with the boundary conditions described in the preceding discussion, including $g'(0)<0$. Nonexistence conjec…

Self-similar solution-count conjecture. For fixed $\kappa>0$ and $n>4$, this boundary-value problem has $\nu(n)-2$ non-trivial solutions.

Perelman's conjecture. Every smooth rotated backward self-similar profile with $\alpha\neq 0$ and the stated decay is trivial.

Consider the focusing nonlinear wave equation $P[\phi]=0$ with $p=3$ in spatial dimension $n\geq 5$. Let $(\phi_{0,\omega},\phi_{1,\omega})$, $\omega\in[0,1]$, be a one-parameter f…

Nonexistence conjecture. The equation does not have solutions.

Consider weakly localized solutions and their stability analysis by the method of modulation equations. Let the halo part denote the component outside the core or localized part of…

Branch-counting conjecture. The supercritical CGL equation has a countable number of branches of nontrivial radial self-similar singular solutions. These branches can be labeled by…

Let $u$ solve the cubic wave equation on $[0,\infty)$ with initial data $(u_0,u_1)\in\mathcal{H}^{1/2}\cap\mathcal{H}^1$, where $\boldsymbol{u}(t)=(u(t),\partial_tu(t))$ and…

Global non-scattering solutions conjecture. Let $u$ be a solution of the cubic wave equation with the prescribed initial-data problem, defined for $t\in[0,\infty)$ and not scatteri…

Stability and quenching-profile conjecture. The following statements are conjectured:

Consider the compressible Navier–Stokes equations referred to as, and backward self-similar solutions with radial symmetry. Nonexistence conjecture. There are no backward self-simi…

Let $\psi$ be a peak-type singular solution of the supercritical BNLS, let $L(t)$ be its collapsing length scale, let $T_{\rm c}$ be the collapse time, and set $\rho=r/L(t)$. Defin…