53 problems

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

Consider the radial semilinear wave equation … where $f(u)=-3u(1-u^2)$, and let … Here $U_0$ is the self-similar solution written in terms of the similarity variable $\rho=r/(T-t)$…

Let $u(t)$ solve the two-dimensional generalized Zakharov–Kuznetsov equation. In the $L^2$-critical case, the blow-up core is expected to form a rightward-moving self-similarly res…

Quasi-fractal intersection conjecture. The Zariski closure of $Z(\bar K)\cap Q$ is a union of finitely many points and finitely many components $B_i$ such that, for each $i$, the i…

Let $f$ be a complex rational function, and let $J(f)$ denote its Julia set. A compact metrizable space is finitely self-similar if it arises from a finite self-similarity system,…

A finite simplicial complex is the topological space obtained by gluing finitely many simplices along faces. A compact metrizable space is finitely self-similar if it arises from a…

Physical relevance conjecture. The solutions constructed by the numerical scheme are physically relevant.

Massless-field gravitational collapse can exhibit black-hole formation, critical solutions, self-similarity and mass scaling. A supercritical evolution PDE is an evolution partial…

Let $u(t)$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation with parameter $a<\sqrt b$, and let $Q^{(0)}$ be the scale-invariant ground-state…

Let $u(t)$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation in the supercritical regime $\alpha>8$, with parameter $a$ and ground state…

Let $\Lambda<0$ be a negative cosmological constant, and let the gluing solution consist of the $q=4$ Garfinkle continuously self-similar interior joined to a Vaidya exterior. A so…

The one-dimensional wave equation under consideration admits generalized blow-up solutions $u_{\alpha,\beta,\kappa,T,x_0}$, where $\alpha$, $\beta$, $\kappa$, $T$, and $x_0$ are th…

Assume the critical parameter conditions considered in the paper, in particular $\gamma+2\lambda=1$ and $\gamma\leq -1$, and consider the associated standard self-similar solutions…

Consider the critical parameter regime … and let $\Phi$ be the profile of a standard self-similar solution of the coagulation equation with growth. Singular self-similar solutions…

Let $f$ solve the coagulation equation under the complementary parameter conditions … Let equation (B6) denote the corresponding small-cluster-size equation with constant…

Supercritical blow-up conjecture. The evolution $u(t)$ blows up at finite time $t^*$ and finite location $(x^*,y^*)$, with

Bulk-energy delta-limit conjecture. The effects of the growing population of moderate-defect equilibria and the upper stability cutoff generate a self-similar distribution of inher…

Bulk self-similarity conjecture. The large-$N$ bulk distribution is independent of $\alpha$ and is self-similar in $N$ about the fixed energy $\bar{E}_*$.

Arias's conjecture. The set $A$ has finite symmetric difference with $E$, and the one-sided difference $A\setminus E$ is a finite subset of

Let $\mathscr{D}^{\ell}$ be the set of addition-chain defects, and let $\mathscr{D}^{\ell}_{\mathrm{st}}$ denote its stable subset. For an ordinal $1\leq\alpha<\omega^\omega$, writ…

Let the self-similar solution of the wave kinetic equation have frequency variable $omega$ and let $x^*$ denote the exponent determined by the associated nonlinear eigenvalue probl…

Self-similarity fractality conjecture. The definition of fractality based on the possession of a nonreal complex dimension, understood as a form of self-similarity, is a step forwa…

Hyperbolic self-similarity conjecture. The geometric reason that the analogue of the parabolic distributional tube-function result fails in the hyperbolic case is an underlying lin…

Scaling hypothesis. The behaviour of solutions $n$ to Smoluchowski's coagulation equation is self-similar as $t\to T_{*}$, perhaps under additional conditions on the initial datum.

Consider the supercritical two-dimensional Zakharov–Kuznetsov equation with $p=4$. Let $u_0\in\mathcal{S}(\mathbb{R}^2)$ have sufficiently large mass and energy and some localizati…