13 problems

Consider a third-order scalar evolutionary differential equation of the form … A conservation law has a weight as defined by the differential-system and characteristic-cohomology f…

A system of $m$ evolution equations is said to be integrable when it possesses sufficiently many higher-order generalized symmetries. Fokas's conjecture. A system of $m$ evolution…

Finite-time convergence conjecture. Then $u(t,\cdot)$ converges to $u_\phi$ in finite time.

Conti–Ortiz relaxation conjecture. As $\varepsilon\to0$, minimizers $u^\varepsilon$ of $\overline W^\varepsilon$ converge to solutions of

Removable-singularity conjecture. The roots of $\Omega+n\omega_0$ that must be removable singularities of $\widehat U_n(k)$ are precisely the two roots lying along $\partial D^-$,…

Consider real-life dynamical systems modeled by differential equations and subject to intrinsic phenomena such as impulses, delays, memory, nonlocal conditions, and noise. Regard t…

Let a dynamical system be subject to perturbations involving delays, impulses, or memory. Controllability conjecture. Controllability of the system should not change under these pe…

Finite-time extinction conjecture. For every $q\gg1$ large enough and $h_0\in L^q(\partial\Omega)$, the trajectory

Let $\mathcal{E}$ be the infinite prolongation of a $\textup{(1+1)}$-dimensional evolution equation. For $p\in\mathbb{Z}_{\ge 0}$ and $a\in\mathcal{E}$, let…

A scalar evolution equation in one space dimension has the form … where $\nu_k$ denotes the $k$th derivative with respect to the space variable. It is formally integrable when it a…

Consider the potential Burgers equation … Let $D_x$ denote total differentiation with respect to $x$, and let $Q_1=u_x$, $Q_2=u_{xx}+\varepsilon u_x^2$, $Q_7=\varepsilon u_x$, and…

Let $u(t,\theta,k)$ solve … where $|V(t,\theta)|\lesssim (1+t)^{-\gamma}$ with $\gamma\in(0.5,1)$, and $k$ and $V$ are real. Sobolev growth conjecture. Generically in $k$, one shou…

Vu's conjecture. The condition