13 problems

GGMT threshold conjecture. There exists a threshold $c^*$ close to $0.85$ such that, for all $\kappa>1$ and $\delta>0$, one has $G_{c,\delta}(\kappa)>1$ for every $c\in(0,c^*)$, wh…

Codimension-one instability conjecture. The solution $u_T$ is co-dimension one unstable for $c$ sufficiently close to zero.

For $d=3,4$, let $\ell=\frac{d}{d-2}$ and consider the linear transport operator … It acts on smooth radial functions. Point-spectrum conjecture. The point spectrum of…

For $d=3,4$ and $s>0$, let … and define … Let $\mathscr{H}[s]:H^2_{\omega[s]}\to L^2_{\omega[s]}$ be the linearized operator acting on radial functions. Spectral conjecture. The op…

Monotonicity and continuity conjecture. For any $a_1\leq a_2\leq 1$,

Existence conjecture for suitable profiles. There exists $b(p)>0$ and a smooth radially symmetric profile $Q_b$ with the properties listed in the conjecture.

Let $B_1\subset\mathbb R^2$ be the unit ball, and consider the Dirichlet problem … with $\lambda>0$ and $\gamma_1\in\mathbb N$. A sequence $u_k$ with $\lambda_k\to0^+$ is a blow-up…

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…

Consider a solution of the semilinear heat equation in the supercritical case, and let its singular set at a fixed time slice be the set of spatial points where the solution is sin…

Let $\phi_1$ denote the first member of the infinite sequence of globally regular shrinker solutions of … Here the index $1$ denotes that $\phi_1$ has one intersection with…

The equations under study are quasilinear wave equations with derivative-quadratic nonlinearities, and their solutions may develop a blowup surface in spacetime. Denote by…

Let $u^k$ be a sequence of blowup solutions of a rank $n$ Toda system, with $n\geq 3$. A component $u_i^k$ is not concentrating if it does not satisfy $u_i^k(x)\to -\infty$ away fr…

Suppose that $u \in C^0([0,T);H^{1/2}(\mathbb{R}^3))$ solves the boson star equation with $m\geqslant 0$ and blows up at a finite time $0<T<+\infty$. Let $u^*\in L^2(\mathbb{R}^3)$…