15 problems

In infinite depth, consider nontrivial periodic water-wave solutions and a branch of such solutions bifurcating from a half-space. Stokes's conjecture. There is a branch of nontriv…

Consider modulated invasion in which the leading edge, the front interface, or the state in the wake may be oscillatory or temporally constant. Oscillatory invasion conjecture. The…

Let $d_1$ be the first critical strip width, and consider solutions with Neumann boundary conditions satisfying the symmetries under discussion. Uniqueness conjecture. This uniquen…

For fixed flux parameter $\alpha$, let the pole move in a direction specified by the angle of the vector $a=|a|(\cos\alpha,\sin\alpha)$. Almost-every-direction bifurcation conjectu…

Let $d_0$ and $d_\infty$ be the local degrees of the critical fixed points at $0$ and $\infty$, respectively, and let $\{F_c\}_{c\in\mathbb{C}^*}$ be the one-parameter family of ra…

Let $\Omega$ be the domain and let $f$ be the nonlinearity in the $k$-Hessian problem … has no nontrivial solution for every $\lambda\in(0,\lambda_{*})$ and at least two nontrivial…

Let $\Omega$ be the domain and let $f$ be the nonlinearity in the $k$-Hessian problem … has at least two convex solutions for every $\lambda>\lambda_{*}$, exactly one convex soluti…

Let $\psi_0$ be the even, single-lobe periodic wave from the even periodic-wave theorem, write … and define $\omega_0=c_0-6a_0^2$. Let $\phi(\cdot,\omega,a)$ be the associated loca…

Consider the bifurcation problem in the parameter $q$ for $p>2$, where the constant solution undergoes bifurcation at $q=p$. Infinite-branch bifurcation conjecture. Infinitely many…

Let $f$ and $g$ be related by … Let $u_0$ be a positive constant and let $\lambda_{k+1}^{\textnormal{rad}}$ denote the $(k+1)$-st radial Neumann eigenvalue, with $k\geq1$. BNW's os…

Let $\Omega$ be a convex domain of second kind. Let the conclusions of Theorem 3 denote the properties established there for the entropy and mean field equation in the paper. Conje…

Let $\sigma_{\nu}=\sigma/\nu$, and define … Let $y$ be distributed according to the corresponding stationary density proportional to $e^{-\mathcal{E}(y)/(2\sigma_{\nu}^2)}$. Second…

Let $u0$ and let $rho$ be a weight whose definition involves a small constant $c0$. Consider the attractor in the deterministic case with no noise, $sigma=0$. Diameter conjecture.…

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…