18 problems

Let $\mathcal{A}_L$ be the global attractor of the one-dimensional Kuramoto–Sivashinsky equation with periodic boundary conditions of period $L$, and let $d_H(\mathcal{A}_L)$ and…

The attractor-structure conjecture. There is a heteroclinic connection $z\rightsquigarrow y$ if and only if either $z=0$, $y=v_n^{\pm}$, $n\geq1$, or $z=v_n^{\pm}$ and…

Invariant-measure existence conjecture. If the dynamical system generated by the deterministic equation has a global attractor $\mathscr A$ in $X$, then the Markovian transition se…

Let $\lambda$ and $\nu$ be the two eigenvalues of the planar microinertia block, with the microstructure called inertially oblong when $\lambda>\nu$ and inertially oblate when…

Let $\beta_0$ be the critical parameter of the nonlocal thermostat model, and let $\beta\in(\beta_0,\infty)$. Let $\mathcal{A}_\beta$ and $\hat{\mathcal A}_\beta$ be the attractor…

Let $\beta_0$ be the critical parameter of the nonlocal thermostat model, and let $\beta>0$. Denote by $\mathcal{A}_\beta$ and $\hat{\mathcal A}_\beta$ the two attractor sets defin…

Let $H$ be the phase space, and let $\mathscr{A}_1$ and $\mathscr{A}_2$ be the global attractors for the partly diffusive Hindmarsh–Rose equations. Finite-dimensionality conjecture…

Trivial-symmetry global attraction conjecture. The solution satisfies

Global attraction conjecture. For generic such equations, every finite-energy solution admits the asymptotics

Let $G$ be a Lie symmetry group acting by a linear representation $T$ on a suitable Hilbert or Banach phase space $\mathcal E$, and consider a generic $G$-invariant autonomous Hami…

Maxwell–Schrödinger global attraction conjecture. These asymptotics should hold in $H^1$ norms on every bounded region of $\mathbb R^3$ for all finite-energy solutions.

Global Attractor Conjecture. Any vertex balanced polynomial dynamical system has a globally attracting point within any affine invariant set. This extends the global attractor conj…

Let $\alpha>0$ and let $U(\alpha)$ be the forward extension by the semiflow of a local unstable manifold at zero for Wright's equation. Write … Let $\overline{U(\alpha)}$ denote th…

Consider the quintic wave equation with damping parameter $\theta$ in the range $0<\theta<\frac12$, in the setting of the analogous results established in the paper for the borderl…

Uniform asymptotic Strichartz estimate. At least in the autonomous case, the displayed estimate should hold.

Let $q_h$ be the heteroclinic solution of the Kuramoto equation connecting the incoherent equilibrium to the synchronized equilibrium, and let $W^{\frac{1}{2\pi}}$ denote the globa…

Let $g$ satisfy condition (L), and let $\bar J=[\bar\alpha,\bar\beta]$ be the smallest globally attracting interval for the difference equation … Assume that $g$ does not have a un…

Let $(1.1)$ be the competitive Lotka–Volterra system, let $I_n$ denote the set of species indices, let $\pi_i$ be the relevant coordinate hyperplane, and let $\gamma_i$ be the thre…