8 problems

Consider an ODE of the form of equation with a bistable function $f(y)$. Five-solution conjecture. Such an ODE has at most five $1$-periodic solutions. This conjecture concerns the…

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…

Consider a bounded initial function $u_0$ for the bistable reaction diffusion equation, and let $\alpha$ denote the intermediate unstable equilibrium between the stable equilibria…

Let $u$ be an entire solution of the bistable reaction diffusion equation referred to as, and let the asymptotic stable states of this equation be the stable states defined in the…

Let $f$ be a generic bistable function, let $C$ be the wave-speed parameter, and let $L_*(C)$ be the threshold appearing in the paper. A barrier is a blocking configuration in the…

Let $f$ be a generic bistable function, and let $lpha_*$ and $\beta_*$ denote the quantities associated with Lemma, viewed as functions of the relevant parameter $C$. Monotonicity…

Let $\Omega$ be a ball or an annulus, and let $u$ be a global minimizer of the mixed-diffusion bistable energy, not necessarily radial. A global minimizer should not change sign, e…