96 problems

Let $u$ solve the Fisher–KPP equation … The initial data are understood in the take-over setting considered for this equation, with traveling-wave states $\phi(-\infty)=1$ and…

Let $Z>0$ and consider the families of traveling-wave solutions of model C bifurcating from the stationary solution at solutions $P_0$ of the bifurcation condition … ordered by inc…

Okubo et al.'s conjecture. The minimal speed $c^*$ is linearly selected, that is,

The tWhitham equation is a model for traveling water waves, and a traveling-wave solution has a wave height, with maximal height meaning the greatest height attained among such sol…

Consider a two-species competition system with homogeneous equilibrium states $(0,0)$, $E_1$, and $E_2$. A stacked traveling front consists of two transition layers propagating at…

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.

A traveling wave has the form … where the profile $\phi$ satisfies … An unbounded traveling wave solution is a function $\phi(\xi)$ defined for $\xi\in\mathbb{R}$ that satisfies th…

Let $m,\alpha,\gamma\geq 1$, let $\chi$ be the chemotaxis sensitivity, and consider the traveling-wave solutions $(u(t,x),v(t,x))=(U(x-ct),V(x-ct))$ of the model equation. Strong-p…

Let $m,\alpha,\gamma\geq 1$ be the parameters of the repulsion/attraction chemotaxis model, and let $c^*_{\chi,m,\gamma}$ denote the lower bound for traveling-wave speeds. Speed-up…

Let $v$ denote the relevant traveling-wave profile, and consider the eigenvalues associated with speed $1$ along the corresponding branch up to the critical speed. Stability conjec…

The preceding theorem introduces an exceptional set $\mathcal N\subset(0,\infty)$, of cardinal at most $8$, whose exclusion ensures that the local bifurcation curve is not of the f…

Fix $k_2>1$. There exist thresholds $k_1^*$ and $k_1^{**}$ with $1<k_1^*<k_1^{**}$ such that the propagation speed $c$ is negative for all $d,r>0$ when $k_1\geq k_1^{**}$, and posi…

Let $k_2>1$ be fixed. Consider the curve in the $(k_1,d)$-plane, for fixed $r$ and $k_2$, on which the propagation speed $c$ is zero. Alzahrani–Davidson–Dodds' monotonicity conject…

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,…

Radius-extension conjecture. We conjecture that these results hold whenever

For each direction $e\in\mathbb{S}^{N-1}$, let $c_e$ denote an admissible speed of a pulsating front. A function of $e$ is sign-unchanging if it is continuous and does not change s…

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

Let $c$ denote the velocity and $\mathcal{H}$ the Hamiltonian along the tertiary branch of two-crested class $\mathrm{II}$ waves, which bifurcates from the primary branch at…

Let $s$ denote wave steepness and consider class $\mathrm{II}$ waves on the secondary branch, with one steeper crest that forms a corner in the limiting shape and another crest of…

Let $s$ denote wave steepness and $c$ the velocity along the tertiary branch of class $\mathrm{II}$ two-crested waves, which bifurcates from the primary branch at $s_2=0.140487$. T…

Let $s$ denote wave steepness and $c$ the velocity along the secondary branch of $2\pi$-periodic class $\mathrm{II}$ Stokes waves, whose two crests have different amplitudes and wh…

Consider a branching Brownian motion with particle positions $X_u(t)$, alive particles $N_t$, maximum … and centering … Let $Z_\infty$ be the almost-sure limit of the derivative ma…

Monotonicity conjecture. The quantity $|d_T|$ is nonincreasing as a function of $T>0$. Consequently, $d_T$ has a fixed sign independent of $T>0$, equal to the sign of

For any $T>0$ and $b1\in (0,1)$, let $X_T^{\alpha}(t)$ be the $\alpha$-level set of the almost periodic traveling wave solution $u_T(t,x)$, defined by … For each $s\in\mathbb{R}$,…