32 problems

Critical-exponent conjecture. The critical exponent of the equation is

Consider the three-species food-chain model given by equations (1.3)–(1.1) on a bounded domain with Neumann boundary conditions. Let the parameters satisfy $c<\frac{w_3}{D_3}$, and…

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…

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…

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…

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…

Let $B_1$ be the unit ball, let $\beta$ denote the first Dirichlet eigenvalue parameter, and let $Q_b$ be the least-energy (ground state) solution of the stationary nonlinear Schrö…

Let $\alpha(\{\delta_j\}_{j=1}^n)$ be the exponent associated with the multi-level logarithmically improved criteria, and let $s$ denote the Sobolev regularity exponent. Potential…

Let $u$ be a solution of the cubic wave equation on its maximal interval $[0,t_+)$, with initial data in $\mathcal{H}^{1/2}$. Logarithmic critical-norm criterion. There exists…

Let $\mathcal{H}^s=\dot H^s(\mathbb{R}^3)\times\dot H^{s-1}(\mathbb{R}^3)$ denote the phase space, and consider initial data for the cubic wave equation. The self-similar profile i…

Klein–Saut conjecture. The following assertions hold:

Let $m>3$, let $u_0\in X\cap H^{1/2}(\mathbb R)$ be sufficiently localized, where $X$ is the corresponding local well-posedness space for the given $m$, and let $Q$ be the ground s…

Let $u_0\in H^{1/2}(\mathbb R)$ be sufficiently localized, let $Q$ be the ground state solution of the equation defining $Q$ with $m=3$, and let $u(x,t)$ be the mBO time evolution…

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the focusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Let $Q$…

Well-posedness and blow-up conjecture. If $0<\sigma<\frac{2s}{n}-1$, the solutions are global, whereas if $\sigma\geq\frac{2s}{n}-1$, finite-time blow-up is possible for some initi…

Let $u_0\in H^1(\mathbb R)$ and let $u(t)$ be an evolution of the stochastic nonlinear Schrödinger equation with $\sigma>2$ and the specified space-time white-noise coefficient. Le…

Let $u_0 \in H^1(\mathbb R)$ and let $u(t)$, $t>0$, be an evolution of the stochastic nonlinear Schrödinger equation with $\sigma=2$ and the specified space-time white-noise coeffi…

Let $\boldsymbol{u}_c(t)$ denote the constructed blow-up solution from Part (a) of Theorem, and let $\boldsymbol{u}(t) \in \mathcal{H}_1$ be any other solution that blows up backwa…

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric. Global-existence-versus-blow-up co…

Let $E$ satisfy $|E|\ll 10$. A small-energy blow-up conjecture asserts that the KdV soliton for small $a$ is stable under NV dynamics, while the KdV soliton for large $a$ is unstab…

Let $E\gg 1$. A large-positive-energy behavior conjecture asserts that the KdV soliton for small $a$ is stable under NV dynamics, whereas the KdV soliton for large $a$ is unstable…

Smaller-refuge-with-overcrowding conjecture. There exist a patch $\Omega_2$ and a coefficient $d_4$ such that every single refuge patch of measure at least $|\Omega_2|$, with the s…

Consider rapidly decreasing smooth initial data in $L_{2}(\mathbb{R}^{2})$ with a single maximum, and let $t_c$ be the critical time of the corresponding semiclassical DS II soluti…

Let $u_0\in L_2(\mathbb{R})$ be smooth initial data with a single negative hump. For $-1<\alpha<0$, solutions to the Whitham equation and to the fKdV equations with sufficiently sm…

Let $u0 in L2(\mathbb{R})$ be smooth initial data with a single hump. The fKdV equation is conjectured to have the following behavior. For $\alpha0.5$, solutions remain smooth for al…