38 problems

Let $\nu$ be a unit vector with $|\nu_x|\not=0$, and assume that the Fourier transform $\mathcal{F}J(\xi)$ is maximized at $\mathcal{F}J(\xi_{\perp \hat{\nu}_x})$. Let…

Let $v_N$ denote the asymptotic velocity of the $N$-BBM. Brunet–Derrida velocity conjecture. As $N\rightarrow\infty$, … This conjecture predicts the leading finite-population corre…

Let $g$ be a smooth nonlinearity satisfying hypothesis $(H)$: $g\in C(\mathbb{R}_+,\mathbb{R}_+)$ has one local maximum, $g(0)=0$, $g(x)>0$ for $x>0$, exactly two fixed points $0$…

Consider the delayed reaction-diffusion equation … Assume that $g:\mathbb{R}_+\to\mathbb{R}_+$ has exactly two fixed points, $0$ and $\kappa>0$, is unimodal, and has negative Schwa…

The MMM denotes the model with multiple mutations, and the FMM denotes the model with fixed mutations. A type I tail function is assumed, with the claim restricted at least to…

The paper considers reaction-diffusion equations of the form … where $A$ and $f$ are the operator and reaction term from the deterministic equation, and $N$ is a Hölder continuous…

Fu0dik spectral conjecture. The infimum of the admissible wave speed range can be described by the Fu0dik spectrum of this simple periodic problem; specifically, for every $q>1$,…

Let the quadratic case refer to the corresponding equation in which only classical travelling-wave solutions exist, and let $\tau$ denote the parameter in the travelling-wave probl…

Critical-frequency stability conjecture. For each $p>6$, there exists a critical value $\omega^*(p)$ such that the waves $\phi_{\omega,p}$ are unstable for $\omega>\omega^*(p)$ and…

Let $(u,L)$ solve the free-boundary problem, and define … Let $\Pi_{\min}(x):=\int_x^{\infty}\pi_{\min}(y)\,dy$ denote the integrated minimal travelling wave. Minimal travelling-wa…

The model admits shock-fronted travelling wave solutions, including monotone, nonmonotone, and canard-type families. Geometric destabilization conjecture. Nonmonotonicity may be a…

Consider a stochastic population model with logistic growth and nonlinear diffusion corresponding to the porous medium equation, in which cooperative behaviour is captured by the n…

Let a population expand into new territory as a travelling wave, and consider genealogies in a model with local population density tending to infinity. Bolthausen–Sznitman genealog…

Let $(\lambda,c)$ and the Lagrangian $L$ be chosen so that there is a reversed monotonous mean-field-game travelling wave $({\mathcal V},{\Theta},{\mathcal M})$, and let…

Consider the four-dimensional eigenvalue problem for the viscous relaxation travelling-wave perturbation, whose augmented unstable bundle for $\varepsilon>0$ is a complex $2$-plane…

The paper considers the travelling-wave constructions described above, including the branch $c\mapsto Q_c$ of two-vortex solutions, parametrized for large momentum by…

Let $\kappa>0$ and define … For the tumour-invasion travelling waves connecting $(1,0)$ to $(0,\bar{\mathcal{M}})$, let $c^*_{\kappa}(\bar{\mathcal{M}})$ denote the minimal wave sp…

Fix $\kappa>0$ and define … Let $\bar{m}\in(0,m^*(\kappa)]$ and $c\geq2\sqrt{1-\bar{m}}$. For the travelling-wave system in the source, let $\alpha_0(c)$ and $\alpha_1(c)$ be the s…

Let $\kappa>0$, $\bar{m}\in(0,1)$, and define … For the function … where $M(n)$ is the ECM component expressed as a function of tumour density $n$, Fisher–KPP concavity conjecture.…

Let $c^*_\kappa(\bar{\mathcal{M}})$ denote the minimal wave speed for the travelling waves considered in the model, let $\kappa$ be the ECM degradation rate by tumour cells, and le…

Endpoint separation conjecture. For every $\alpha\in(0,1)$ and $c>0$,

Let $u$ be a radially symmetric spreading solution with free-boundary radius $\eta(t)$, and let $c(\gamma)$ denote the speed of the unique finite wavefront for the reaction-diffusi…

Selection principle for the one-dimensional $N$-BBM. The unique invariant distribution of the system seen from the leftmost particle should converge, as $N\to\infty$, to the centre…

Let $\mathbfcal{A}$ denote the set of speed functions admissible for spatially periodic bistable reaction-diffusion equations, with each speed function assigning a propagation spee…

Let $V_0^{(1)}(t)$ and $V_0^{(2)}(t)$ be the first- and second-order perturbation processes in the stochastic travelling-wave expansion, with values in…