150 problems

Strauss's conjecture. The critical exponent is $\gamma(n-1)$. This conjecture was later confirmed with the complete high-dimensional theory.

Let $u$ solve the wave equation Cauchy problem … with $u(x,0)=f(x)$ and $\partial_tu(x,0)=0$ on $\mathbb{R}^d\times\mathbb{R}$. For $s_p=\frac{d-1}{2}-\frac{d-1}{p}$, Sogge's conje…

Let $u$ solve the radial, defocusing, cubic wave equation … on $\mathbf{R}^{3}$ with initial data $(u_0,u_1)\in \dot{H}^{1/2}(\mathbf{R}^{3})\times\dot{H}^{-1/2}(\mathbf{R}^{3})$,…

Consider the free wave equation with initial data $u_0,u_1\in L^2$ and solution $u(t,\cdot)$. The preceding result gives almost linear growth for data whose Fourier transform satis…

Consider the bilinear restriction estimate for free wave solutions, with data $f$ and $g$ localized at frequency approximately $1$ and with separated Fourier supports. The correspo…

The paper considers Anti-de Sitter spacetime with either reflective or optimally dissipative boundary conditions. Nonlinear instability and dissipative stability conjecture. Anti-d…

Compact-manifold local smoothing conjecture. For all $p\geq p_n^+$ and $\sigma<\frac1p$, one has

Let $u(x,t)$ solve the free wave equation … with initial data $u(x,0)=f(x)$ and $u_t(x,0)=0$. Let $W^{\varepsilon,p}(\mathbb{R}^n)$ be the Sobolev space with norm…

Let $S(f)$ denote the stationary set of the solution of the wave equation in $\mathbb{R}^n$ with compactly supported initial data $f$, and suppose that stationary sets are containe…

The Rayleigh conjecture. The series above converges up to the boundary $S$.

Decay-extension conjecture. The results of Wirth for very slowly oscillating dissipation terms, including decay estimates of the same structure as in that work, should extend to ge…

Uniqueness conjecture. A uniqueness result remains valid even without the assumption that, for almost every $\xi$, the map $t\mapsto U(t,\xi)$ is absolutely continuous and satisfie…

Naive transfer-principle conjecture. If $R\geq 1$, then

Petkov's conjecture. The operator $G$ has no eigenvalues for every strictly convex obstacle $K$.

Localization conjecture. Analogous spatial localization phenomena should also occur beyond the setting covered by the existing theory, including for two-dimensional hyperbolic prob…

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying…

The energy-critical wave equation is an evolutionary dispersive equation whose maximally extended solutions are expected to exhibit soliton resolution. A dynamically scaled soliton…

Asymptotic profile conjecture. As $r\to0^+$, every odd non-radiative solution $N\in\mathcal{N}_0$ satisfies

Let $n\geq2$ and $\frac12<s<\frac n2$. Let $(f_j)_j$ be an orthonormal system in $H^s(\mathbb{R}^n)$, and let $\beta$ be the exponent in the maximal estimate denoted by … holds for…

Let $n\geq2$, $q\geq \frac{2(n+1)}{n-1}$, and let $(f_j)_j$ be an orthonormal system in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^n)$. For a sequence of coefficients…

Generalized Huygens condition. The restricted Huygens condition should hold for arbitrary open and closed Lipschitz curves $\mathcal{C}$ and for all $t>0$: namely,

Wave local smoothing conjecture. The estimate holds for $r\ge 2d/(d-1)$. This is an epsilon-loss local smoothing prediction for the wave propagator; the supplied text gives no reso…

Consider an energy-critical wave system … with $f$ homogeneous of degree $\frac{N+2}{N-2}$, and suppose that $f$ derives from a potential: there is a homogeneous function $F$ of de…

Let $\Phi$ be a global finite-energy solution of an energy-critical wave system, with maximal forward existence time $T_+=\infty$. A decomposition into rescaled stationary solution…

Higher-dimensional cubic-system conjecture. The analogs of the continuous soliton resolution and rigidity theorems should remain valid for a general system of this form with $f$ ho…