9 problems

Low-index nonexistence conjecture. There are no solutions of $\mathrm{HSE}$ satisfying $\mathrm{HBC1}$--$\mathrm{HBC2}$ when $\nu(n,\sigma)\leq2$.

Slow blow-up existence conjecture. The analysis of He and Velázquez can be used to prove the existence of slow blow-up solutions for $n>4$.

Exponential-case blow-up conjecture. The blow-up behavior of solutions is close, in some sense, to the blow-up behavior in the exponential case $f(u)=e^u$. This is motivated by the…

Existence conjecture. There exist sign-changing type II blow-up solutions in the lower dimensions

Global existence conjecture. The problem $$ admits a global solution whenever

Critical-exponent conjecture. The numbers $1+\frac{2}{\alpha}$ and $\frac{Q-\alpha}{Q-2-\alpha}$ are the critical exponents for local and global existence of a weak solution of $$,…

Let $f$ satisfy (F1) with $q\geq 1$, suppose that $f'(u)F(u)\leq q$ for all sufficiently large $u>0$, and let $\phi\geq 0$. Set … If … then (S1E1) has a local-in-time solution. Exi…

Consider a solution of the semilinear heat equation in the supercritical case, and let its singular set at a fixed time slice be the set of spatial points where the solution is sin…

Critical-growth non-existence conjecture. If