8 problems

Energetic criterion for blow-up for $p>2$. If $p>2$, then $u(t)$ should blow up in finite time.

GGMT threshold conjecture. There exists a threshold $c^*$ close to $0.85$ such that, for all $\kappa>1$ and $\delta>0$, one has $G_{c,\delta}(\kappa)>1$ for every $c\in(0,c^*)$, wh…

Codimension-one instability conjecture. The solution $u_T$ is co-dimension one unstable for $c$ sufficiently close to zero.

Global existence conjecture. If

Let $d=1$ and let $\mega=(-1,1)$. For fixed $\psilon>0$, consider the $\lambda$-bifurcation diagram of the corresponding problem, meaning the bifurcation diagram obtained when…

Generic simplicity conjecture. There is a generic subset (a Baire subset) of choices for $\phi$ in $$ such that, if $u$ is a heteroclinic orbit, all of the eigenvalues of $H(t)$ ar…

Let $u$ be an eternal solution of the semilinear parabolic equation \, with coefficients $a_i$ that decay sufficiently fast as $|x|\to\infty$. Heteroclinic-orbit conjecture. All…

Let $u$ be an eternal solution of the semilinear parabolic equation \, with coefficients $a_i$ that decay sufficiently fast as $|x|\to\infty$. Universal bound conjecture. All ete…