9 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…

Iterative BHM-BE accuracy conjecture. For large numbers of iterations, specifically $J\geq 4$, the error curve could approach the error produced by the fully implicit nonlinear bac…

Existence and uniqueness conjecture. There exists a unique solution to equation with the contact-line conditions and.

The thin-film velocity is described by an ODE whose solutions are classified into types $\mathrm{I}$–$\mathrm{IV}$ in the phase portrait. More complicated rupture scenarios may con…

The paper studies contact-line singularities in thin-film flows, including regularity near free boundaries for a fourth-order, non-uniformly parabolic equation. Extension conjectur…

Let $u0$, let $h0$ be general smooth initial data, and let $h=h(t,x)$ be the corresponding solution to the regularized thin film epitaxy equation … Here $abla h$ denotes the spat…

Algorithmic preference conjecture. The preference of the numerical algorithm for one solution when multiple steady state solutions exist is related to the proximity of the non-exis…

Operator-theoretic approach conjecture. To ascertain the global bifurcation analysis for these similarity fourth-order thin-film equations, a new and different operator theory appr…

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…