17 problems

Fu0dik spectral conjecture. The infimum of the admissible wave speed range can be described by the Fu0dik spectrum of this simple periodic problem; specifically, for every $q>1$,…

Let $E(t)$ denote the beam's energy, let $\alpha$ be the fractional derivative order, and let $\eta$ be the associated parameter. Conjecture. For $\alpha=0.5$ (or similar values) a…

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…

Let $\alpha>0$, $\beta>0$, and $\xi\in(0,1)$, and consider the characteristic equation associated with the beam model containing pointwise damping and stiffness at $\xi$. Suppose t…

Let $u$ solve the beam equation with positive constants $a$, $b$ and $\alpha$, and with the pointwise damping term $\alpha \frac{\partial u}{\partial t}(\xi,t)\delta_{\xi}$ at…

Let $e_\lambda$ and $e_\nu$ be eigenfunctions associated with consecutive eigenvalues $\lambda$ and $\nu$ of the nonlinear non-homogeneous multiply hinged beam problem. The stabili…

Square-root damping conjecture. The choice $\theta=1/2$, corresponding to square-root damping, is sufficient to obtain energy estimates resulting in well-posedness for the inextens…

Consider the initial amplitudes $\alpha_n$ of the residual components, with $n\ne j$, and the energy threshold of nonlinear instability for the corresponding prevailing mode. The i…

Let $\mathbb{E}_{12}:[0,1]\to[0,+\infty]$ denote the energy threshold of instability before time $T$ for the truncated beam model, defined by … Here $a$ is the pier-location parame…

Consider two hinged Euler–Bernoulli beams connected by a point mass, with variable coefficients, and let the associated eigenvalues and eigenfunctions be those of the corresponding…

Consider the autonomous Galerkin systems associated with the positive cubic beam problem, with Fourier modes $varphi_j$ and the notion of full instability defined by the two condit…

The cubic beam equation exhibits energy-exchange behavior beyond the parity-based closure observed in the half-linear case. Cubic energy-exchange conjecture. The general rule for t…

Endpoint behavior conjecture. There exists $E_k>0$ such that if

Let $m,n$ be the mode indices, let $P$ be the parameter, and define … Let $\Theta_m$ and $\Theta_n$ be the corresponding modes with energy $E_{\Theta_m}$. Assume … Let $E_1$ and…

Let $m,n$ be the mode indices, let $P$ be the parameter, and let $\Theta_m$ and $\Theta_n$ denote the corresponding modes with energy $E_{\Theta_m}$. Assume … Let $E_2$ be the thre…

All-energy stability conjecture. Under these assumptions, $\Theta_m$ is linearly stable with respect to $\Theta_n$ for every $E_{\Theta_m}>0$.

Middle-rectangle conjecture. Although the existence of an optimal rectangle and its dependence on $t\in(0,T)$ are posed as problems, the natural conjecture is that, at least as…