10 problems

Let $h$ solve the thin-film equation with no-slip mobility in a domain whose contact point is $x=s(t)$, with $h=0$ at the contact line. The no-slip condition is the condition impos…

Existence and selection conjecture. For every $U>0$, there exists a one-parameter family of such fronts; moreover, a proposed criterion should select a unique linear-log front.

Let $u$ be a suitably defined weak solution to the thin-film equation with exponent $n\geq 3$, and let $\operatorname{supp}u(\cdot,t)$ denote its spatial support at time $t$. Const…

Finite dissipation implies fixed support. Then $s(t)=s_0$ is constant on $(0,T)$. Under the stated regularity and finite-dissipation assumptions, the conjecture gives a precise mat…

Consider the thin-film contact-line problem with prescribed contact angle, where the mobility exponent is $n=3$. Prescribed-angle ill-posedness conjecture. The contact line problem…

For the stochastic thin-film equation with mobility exponent $n$ in the range $n\in [8/3,4)$, the existing existence result can be extended to the range $n\in (2,8/3)$ using estima…

Consider source-type self-similar solutions of the thin-film equation with mobility exponent $m$, for which such solutions exist for $m\in(0,3)$ but not for $m=3$. Droplet-spreadin…

Consider the maximal profiles associated with the self-similar eigenvalue problem for the fourth-order thin film equation, parametrized by the exponent $n$ and the similarity param…

Consider the radial self-similar profiles $f_k$ and corresponding eigenvalues $\alpha_k$ for the fourth-order thin film equation, together with their linear limiting profiles at…

Consider the thin film equation with diffusion parameter $\nu=1$, in the strong-slippage regime $0<n<2$ and the weak-slippage regime $2<n<3$, where the analysis requires $m>n/2$. S…