46 problems

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

Scaling-critical regularity conjecture. The equation is well-posed in $B^s$ for $s>s_\text{crit}$ and ill-posed in $B^s$ for $s<s_\text{crit}$.

Consider the De Gregorio model on $S^1$, given by … where $H$ is the Hilbert transform and smooth initial data are prescribed. Global regularity conjecture for the De Gregorio mode…

The Schrödinger equation leads to the eigenvalue problem … For a finite-energy solution, the proposed interpretation of a Bohr transition is the long-time asymptotics … with…

Let $\Omega$ be a ball, and let $u\in H_0(\operatorname{curl};\Omega)\cap L^p(\Omega,\mathbb{R}^3)$ be a ground state solution of … where $2<p<6$. Here, symmetry means the property…

Let $K(m,n)$ denote the generalized Korteweg–de Vries equation admitting compacton and anticompacton solutions. Based on numerical experiments with soliton-antisoliton collisions,…

The paper considers a generalized nonlinear partial differential equation (NPDE), its associated infinite-order linear ordinary differential equation (ODE), and a finite-difference…

For $1<p<N$, consider the critical quasi-linear Hénon equation … where $N\geq2$, $p_\alpha^*=\frac{p(N+\alpha)}{N-p}$, and $\alpha>0$. Gladiali–Grossi–Neves conjecture. Nonradial s…

Let $a\in\mathbb{R}$, and consider the generalized Constantin-Lax-Majda-DeGregorio equation on $\mathbb{S}^1=\mathbb{R}/(2\pi\mathbb{Z})$, … where $\mathcal{H}$ is the Hilbert tran…

Nondegeneracy conjecture. For every $k\in\mathbb{N}$, the only smooth, evanescent solutions of this linearised equation are linear combinations of $\partial_xu_k^\star$ and…

Let $u_k^\star(x,y)=-2\partial_x^2\log \tau_k^\star(x,y)$ be the explicit symmetric lump solutions of the normalised KP equation … where $k\in\mathbb{N}$ and $\tau_k^\star$ is a po…

Let $(M,J, omega)$ be a compact almost-Kähler manifold, where $J$ is an almost-complex structure and $omega$ is its fundamental $2$-form. Let $sigma$ be a smooth volume form on…

The FHD equation is considered in the Black–Scholes framework, and the reconstructions in Figure 3 are reconstructions of local-volatility surfaces from option-price data. Multisol…

Rigidity conjecture. If $u$ is positive, then $(M,g)$ is isometric to Euclidean space and $u$ is an Aubin–Talenti bubble, namely

Let the quadratic case refer to the corresponding equation in which only classical travelling-wave solutions exist, and let $\tau$ denote the parameter in the travelling-wave probl…

Extremal threshold conjecture. For the extremal case $\sigma=1$, there is still a threshold separating dispersion to zero from finite-time ODE-type blowup in the white regions.

One-dimensional dispersive flows with cubic defocusing nonlinearities and small initial data are considered. Global small-data scattering conjecture. Such flows should have global-…

The $u_t+u$ nonlinearity conjecture. Because the decay estimates are the same for the solution and the term $u_t(t,x)+u(t,x)$, this nonlinearity should lead to the same results as…

Complete classification conjecture. Either $u\equiv 0$, or there exist $\lambda:=u(0)^{\frac{p}{N-p}}>0$ and $x_{0}\in\mathbb{R}^{N}$ such that

The Amick-Schonbek system is the dispersive system obtained by setting $\epsilon=1$ in … Its zero-dispersion limit is the corresponding dispersionless Saint-Venant system, and the…

Infinity-harmonic regularity conjecture. Infinity-harmonic functions have $C^{1,1/3}$ regularity and also belong to $W^{2,q}$ for every $q<3/2$.

Thomas–Fermi dynamical neutrality conjecture. For every initial state $(\varphi_0,\rho_0)$ satisfying $0\leqslant\rho_0\in L^1(\mathbb{R}^3)\cap L^{5/3}(\mathbb{R}^3)$ and…

Dynamical Hartree conjecture. For every initial state $u_0\in H^1(\mathbb{R}^3)$ and every $R>0$,

Let $4\Psi(\xi,\eta,t)=Q_{\omega}(\xi,\eta)e^{i\omega t}4$ be a stationary ansatz for the Davey–Stewartson I equation, with $4\omega>04$. The profile $4Q_{\omega}4$ may be chosen t…

Let $u=u(x,t)$ and $\omega=\omega(x,t)$ satisfy the generalized Constantin–Lax–Majda model … Here $a$ is a real parameter, and the initial data are smooth. Finite-time singularity…