178 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 initial data for the Novikov–Veselov equation be classified as critical, subcritical, or supercritical according to the behavior of the associated Schrödinger operator at the z…

Klein–Roudenko–Stoilov's conjecture. Blow-up happens in finite time, and blow-up solutions have some resemblance to being self-similar: the blow-up core forms a rightward-moving se…

Let $v$ be a solution of the forced Navier–Stokes equations on $\mathbb{R}^3$, and suppose that $v$ first loses smoothness at time $T^*>0$. For a fixed constant $c>0$, consider the…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, and consider the minimal Keller–Segel system, meaning the chemotaxis system with constant sensitivity $\chi$, in spatial dimens…

Subcritical lifespan conjecture. If $1<p<p_0(n)$ with $n\geq3$, or $2<p<p_0(2)$ with $n=2$, then

Let $Q$ be the unique positive radial solution of … Consider solutions $v$ of the $L^2$-critical nonlinear Schrödinger equation with critical mass $\|v\|_{L^2}=\|Q\|_{L^2}$, where…

Klein–Saut conjecture. The following assertions hold:

Let $u(t)$ solve the two-dimensional generalized Zakharov–Kuznetsov equation. In the $L^2$-critical case, the blow-up core is expected to form a rightward-moving self-similarly res…

Bizoń et al.'s blow-up profile conjecture. The rescaled solution converges to the static profile:

Consider the focusing energy-critical wave equation … and the Talenti–Aubin solution $W(r)=(1+r^2/3)^{-1/2}$. Let $\mathcal M$ be the small codimension-one manifold around $W$ whos…

Energetic criterion for blow-up for $p>2$. If $p>2$, then $u(t)$ should blow up in finite time.

Translated rigidity conjecture. Theorem 5.1 remains true if

Critical-norm blow-up conjecture. Finite-time blow-up implies

Consider the damped cubic Schrödinger equation in dimension two … with $(t,x)\in\mathbb{R}_+\times\mathbb{R}^2$, initial datum $\psi|_{t=0}=u_0$, and damping parameter $\delta>0$.…

Consider the critical focusing nonlinear Schrödinger equation in one spatial dimension and its standing-wave blow-up solutions. The generalized root space of the linearized operato…

Let $B_\delta(0)$ be a neighborhood of the origin in the data space, and let $\Sigma$ be the stable manifold of the stationary solution, so that $B_\delta^+(0)$ and $B_\delta^-(0)$…

Let $f_0$ be the closed-form self-similar solution for co-rotational wave maps from Minkowski space to the three-sphere, whose existence was established for the blow-up profile und…

Consider the mass-critical one-dimensional focusing half-wave equation … where $|D|$ is the Fourier multiplier operator with symbol $|\xi|$. The paper proves an upper bound of log-…

Elgindi's blow-up conjecture. For every $\alpha\in(0,1/3)$, there exists a compactly supported $C_c^{1,\alpha}$ axisymmetric swirl-free local solution to the Euler equations that d…

Let $u$ be a radially symmetric solution of the parabolic-elliptic Keller–Segel system in two spatial dimensions that blows up at a finite time. Nanjundiah's conjecture. The densit…

Critical-power conjecture. For $\alpha=0$, the small-data Cauchy problem admits the critical power

Assume $(B,T)$, $(u_{0},u_{1})$, $d$, and $\Omega$ are given, and let $u$ be the unique maximal solution to the hyperbolic problem. Critical-threshold conjecture. There exist criti…

Consider the classical Keller–Segel model with no reaction term, so that $f=0$, in spatial dimension $n\ge 3$. A solution has sufficiently large total mass, meaning that its initia…

Quadratic-phase conjecture. For the combined nonlinear Schrödinger equation, if $b>0$, the solution with initial condition $v_0$ exists globally and scatters, whereas if $b<0$ and…