Numerical blow-up profile conjecture for the rescaling method

Consider the semilinear parabolic problem denoted by equation

, and let $u(x,t)$ be a solution that blows up at the finite time $T$. For $K>0$, set

z=\frac{x}{\sqrt{(T-t)|\log(T-t)|}}.

$$Define$$

\bar{f_\beta}(z)=\left(p-1+b(\beta)|z|^2\right)^{-\frac{1}{p-1}},

with $b(0)=\frac{(p-1)^2}{4p}$. **Numerical blow-up profile conjecture.** Equation

has a solution $u(x,t)$ blowing up at finite time $T$ such that

$$\sup_{|z|<K}\left|(T-t)^{\frac{1}{p-1}}u(x,t)-\bar{f_\beta}(z)\right|\to 0\quad\text{as }t\to T,$$

where $b(\beta)$ is represented by the numerical computations in the source for $p=5$ and $p=7$. This conjecture describes the asymptotic profile suggested by the rescaling computations; the paper provides numerical evidence rather than a proof, and the dependence of $b(\beta)$ is illustrated computationally.