245 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 $u_c$ denote an effective cut-off concentration in the initial boundary value problem for a reaction–diffusion equation, where concentration values represent the density of dis…

A roll wave is a traveling wave train in a reaction-diffusion system. In higher spatial dimensions, consider such roll waves together with their modulational stability properties.…

Let $u\colon\mathbb{R}^{n}\to[-1,1]$ be a solution of the Allen–Cahn equation in $\mathbb{R}^{n}$. Call $u$ stable when the second variation of the Allen–Cahn energy is non-negativ…

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

Let $F(u)=\frac{1}{4}(1-u^2)^2$, so that $F'(u)=u^3-u$, and consider the reaction-diffusion equation … in $\mathbb{R}^n\times[0,\infty)$. Suppose that $u$ satisfies $|u|\leq 1$ and…

Ni's conjecture. In the one-dimensional case, the supremum of the total population over all $d>0$ and $m\in\mathcal{M}_1$ is $3$.

Assume the invasion property, Hypothesis … d{mathcal{H}}(U,Urho)<+infty … . Let $\Omega(u)$ be the omega-limit set of $u$ under time translations. De Giorgi type conjecture. Every…

Consider the two-species competition system in the parameter regime $D>d$, with critical advection value $\alpha_1>0$, semi-trivial equilibrium $(u^*,0)$, and a positive steady sta…

The HHMO model describes concentrations and precipitation with spatially distributed relay variables, and its long-time dynamics are studied in the fast-reaction limit. A precipita…

Let $x(t)$ be a solution of the balanced reaction-network system … with positive initial condition $x_0\in \mathbb{R}_{+}^m$, where $x(t)\in\mathbb{R}_{+}^m$ is the concentration v…

Complete convergence conjecture. Complete convergence should hold more generally when $a=a^\dagger$ and $\Lambda$ is amenable, but not in general on nonamenable lattices. Complete…

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…

Consider the stochastic reaction–diffusion equation on a bounded smooth domain $\mathcal D\subset\mathbb R^d$ with Dirichlet boundary conditions, obtained from (4.10) by replacing…

Consider reaction-diffusion processes of the form … where $m$ and $n$ are the orders of creation and removal, while $k$ and $l$ determine additional symmetries such as parity conse…

Consider the coagulation-fission process with reactions $2A\to 3A$ and $2A\to A$, together with diffusion, including the variants described in the source. Odor's conjecture. The co…

Let the pair-contact process with diffusion (PCPD) be considered at low and high diffusion rates. Odor's conjecture. The PCPD might belong to two different universality classes for…

The pair contact process with diffusion (PCPD) is a one-dimensional reaction-diffusion model in which particles diffuse and pairs can annihilate or create particles; the preceding…

Let $u(t,x)$ be the solution of the shifting KPP equation considered in Theorem (ii), with parameters $\beta$ and $\eta$. In that theorem, conclusion (ii) asserts uniform convergen…

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…

Consider the discrete-time model with $N$ particles on the line, whose positions at time $t$ are $Y_1(t),\ldots,Y_N(t)$ and evolve by … where $j_i$ and $j'_i$ are chosen uniformly…

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…