15 problems

Let \boldsymbol{p}_{\eta}_{\eta>0} and c_{\eta}_{\eta\geq0} satisfy … Assume $(H_{7})$. Stability is understood as local asymptotic stability with respect to $(E_{KPP}^{0})$. Limit…

Let $e\in\mathbb{S}^{n-1}$ and let $c_e^\star$ denote the critical speed in direction $-e$. A planar pulsating traveling wave solution has speed $c$, propagates in direction $-e$,…

Finite-population traveling-wave conjecture. There exists a propagation speed $\sigma^K$ such that, in an appropriate convergence sense,…

Nonlinear stability conjecture. Under these assumptions, the modulated traveling wave is exponentially and asymptotically orbitally stable.

Subsonic-or-sonic stability conjecture. A standing or traveling wave profile for the QHD systems with linear or nonlinear viscosity is spectrally stable if and only if its velocity…

Let $J$ denote the independent jump-size distribution, and let $v_{**}$ be the minimum selected speed. A mean-field limit (MFL) is the deterministic limiting particle-system dynami…

Let $u$ be the solution of the homogeneous reaction-diffusion equation with initial datum as in the paper, let $X_\lambda(t,x')$ be its $\lambda$-level set, and let $\varphi$ be th…

Let $M$ denote the total myosin mass of a traveling wave solution and let $V$ denote its velocity. Consider parameter ranges in which $M$ decreases as a function of $V$ for small…

Propagation and blocking threshold conjecture. For every $R>0$, there is $\alpha_R\in(0,\pi/2]$ such that complete propagation holds for all $\alpha\in(0,\alpha_R)$, while blocking…

The systems under consideration admit traveling waves, including trivial traveling waves and, in some parameter regimes, nontrivial traveling waves with profiles $N$ and $P$. The n…

Let $f^n(\cdot,t)$ denote the empirical distribution of the locations of an $n$-particle stochastic system, and let $f^n_*(\cdot,t)$ be its recentered version whose median is at…

Let $\varphi$ be the profile of a highest cusped periodic traveling-wave solution of Whitham's equation, with speed $\mu>0$, satisfying … where $L$ has Fourier symbol…

Full decay-characterization conjecture. For any solution $0<u<1$, the limit $\lambda$ exists in $[0,\sqrt{f'(0)}]$, and $u$ is a transition front connecting $0$ and $1$ if and only…

Asymptotic-speed and profile conjecture. Any such transition front has asymptotic past and future speeds $c_\pm$ satisfying

Let $u^R(t,x,y):=\mathbb{P}_y(M^R_t\leq x)$, where $\mathbb{P}_y$ is the law of branching reflected Brownian motion starting at $y\geq 0$. Let $q^R_{\delta}(y,t)$ be defined as in…