14 problems

Consider the generalized thin-film equation … with $a_0>0$, $a_1>0$, and nonnegative solutions subject to boundary conditions preserving the mass $\int h(x,t)\,dx$. The supercritic…

There exist a time $T>0$ and smooth functions … with smooth initial data … for smooth $f_1,f_2,f_3$ and a smooth even angular profile $\phi(\theta)$ adapted to…

The universal amplitude equation derived for weakly nonlinear disturbances of vorticity interfaces describes the wave evolution for a straight interface, a circular patch in the pl…

Consider solutions $(\rho,u)$ of the pressureless Euler–Poisson system, where $\rho$ is the density and $u$ is the velocity. A solution undergoes $C^1$ blow-up when its $C^1$ regul…

Critical-curve conjecture. Based on the work of Kubo–Kubota–Sunagawa, the critical curve for system (2) is

Finite-time predator blow-up conjecture. For the delayed system, the predator population explodes to infinity in finite time.

Consider the Cauchy problem studied in the paper, with parameter $L$ defined by the equation referenced as $(\mathrm{const})$. A solution is non-trivial if it is not identically ze…

Let the incompressible Navier–Stokes equations be denoted by INSE, and consider finite-time blow-up of enstrophy in the inviscid Euler limit. Buaria et al.'s conjecture. Such finit…

Let $w$ solve the equation denoted by … lambda=1,quad 0<muleq 2,qquadtext{or}qquad lambda1,quad mu0. … will blow up in finite time for a family of smooth initial data…

Velocity blow-up rate conjecture. The velocity field satisfies

Let $\beta(t)=h_{xx}|_{x=0}$ and let $V(t)$ denote the variables arising in the reduced dynamical model for the contact line. The conjectured finite-time blow-up occurs as…

Blow-up–condensation time conjecture. The physical-literature conjecture is

Behr–Nečas–Wu blow-up conjecture. The exact solution $u(t)$ blows up in $\mathbb{H}^{3}_{\Sigma 0}$ as $t\rightarrow T_{\star}$, for some $T_{\star}\in(0.32,0.35)$.

Consider the generalized thin-film equation … where $a_0>0$, $a_1>0$, and $h$ is real valued. Hocherman–Rosenau's finite-time blow-up conjecture. If $n>m$, solutions might blow up…