45 problems

Let $(u,v)$ solve the reaction-diffusion system of allelopathic phytoplankton with fear function $k(x)$, subject to the parameter restrictions … where $\mathbf{k}$ is either…

Uniqueness conjecture. The problem admits at most one solution.

Let $(\Omega,\varrho)$ be a domain and boundary parameter for the problem … lambda(Omega,varrho)=f'(0), … has no positive bounded solution. This extends the established conclusion…

Let $(\Omega,\varrho)$ be a domain with the boundary conditions and elliptic problem specified by … admits at most one positive bounded solution. The paper's methods do not treat c…

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…

Segregative pulsating fronts exclusivity conjecture. Theorem still holds true in the non-constant case.

Consider the three-species food-chain model given by equations (1.3)–(1.1) on a bounded domain with Neumann boundary conditions. Let the parameters satisfy $c<\frac{w_3}{D_3}$, and…

Consider the diffusive Lotka–Volterra strongly competitive system on an interval $(0,L)$ with boundary-constrained controls, parameters $a>0$ and $b>1$ satisfying $b>a$, initial da…

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 $-e$,…

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…

Finite-graph conjecture. The graph of this system has finitely many nodes, and every node is a fixed point, that is, an equilibrium solution.

Let $c^*(\theta,\Lambda,\alpha)$ denote the velocity of propagation and let $R^*(\theta,\Lambda,\alpha)$ denote the reaction width, where $\theta$ is the ignition temperature,…

Let $u(x,t)$ solve the Neumann problem … with … and initial data $u(x,0)=u_0(x)$, where $u_0(x)>0$ for some $x$. Long-time convergence conjecture. … The claim describes the predict…

Consider the one-dimensional Dirichlet problem … with the Dirichlet boundary conditions and initial data specified in the paper, and let $u$ denote its solution. Large-population w…

Let $(u,v)$ solve the reaction diffusion system referenced as … . Suppose … and the parameters satisfy the restrictions in … . Uniform-convergence conjecture. The solution $(u,v)$…

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…

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…

Nonlinear stability conjecture. All $N$-spot ring patterns are nonlinearly stable unless the spot-spot distance is the smallest binding distance.

Assume that $f_a(u)=u(1-u)(u-a)$ with $0<a<1/2$. Let $\mathcal{K}$ satisfy the stated hypotheses with normalized second moment … Let $\mathcal{R}_j$, $j=1,\ldots,5$, be the regions…

Assume that $\mathcal{K}(x)=\mathrm{e}^{-|x|}/2$ and that $f_a(u)=u(1-u)(u-a)$ with $0<a<1/2$. Let $(a,d)\in \mathcal{R}_2\cup\mathcal{R}_3$ such that $0<\ell_1^*<\infty$ is well d…

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…

Formal front-propagation conjecture. There exists an interval of trait values $J_0$ centered at $1$ such that, for every open interval $J\subset J_0$ centered at $1$, the density…

Let $K(\theta)$ be the directional spreading speed and define the Wulff shape by … Let $\hat{\mathscr{F}}$ be the convex hull of the Frank diagram, let…

Let $\theta$ be the direction of a zigzag front, and let $\theta_{\mathrm{m}}(t)$ and $\theta_{\mathrm{p}}(t)$ determine the two angles forming a peak, namely…

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…