8 problems

Consider the damped nonlinear Schrödinger equation … Here the collapsing core is the localized part of the solution near blow-up, and dissipation removes mass from this core. Influ…

Takens' conjecture. The number of determining functionals should be very small; in a generic situation, one has

Inertial-manifold obstruction conjecture. The absence of an inertial manifold is the only natural obstruction to the limit dynamics being finite-dimensional; equivalently, whenever…

An inertial manifold (IM) is a finite-dimensional, invariant, exponentially attracting manifold for the dynamics generated by a dissipative partial differential equation. Zelik's c…

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…

Let $H$ be the phase space, let $H^2$ denote the more regular space, let $F$ be the non-linearity satisfying the reasonable assumptions considered in the paper, and let…

Consider exponential dissipation with Fourier symbol $\mu \mathrm{e}^{|k|/k_{\rm d}}$, and write $\tilde k=k/k_{\rm d}$. Let $\hat u(k)$ be a Fourier coefficient and let $C_\star$…

Consider the expNSE in space dimension $d\ge 2$, with exponential dissipation having Fourier symbol $\mu \mathrm{e}^{|k|/k_{\rm d}}$, and write $\tilde k=k/k_{\rm d}$. Let…