We consider the semilinear heat equation
$${\partial}_{t}u=\Delta u+u{}^{p1}u{ln}^{\alpha}\left({u}^{2}+2\right)$$
in the whole space
${\mathbb{R}}^{n}$,
where
$p>1$ and
$\alpha \in \mathbb{R}$. Unlike the
standard case
$\alpha =0$,
this equation is not scaling invariant. We construct for this equation a solution which blows up in
finite time
$T$ only at
one blowup point
$a$,
according to the asymptotic dynamic
$$u\left(x,t\right)\sim \psi \left(t\right){\left(1+\frac{\left(p1\right)xa{}^{2}}{4p\left(Tt\right)ln\left(Tt\right)}\right)}^{1\u2215\left(p1\right)}\phantom{\rule{1em}{0ex}}\text{as}t\to T,$$
where
$\psi \left(t\right)$ is
the unique positive solution of the ODE
$${\psi}^{\prime}={\psi}^{p}{ln}^{\alpha}\left({\psi}^{2}+2\right),\phantom{\rule{1em}{0ex}}\underset{t\to T}{lim}\psi \left(t\right)=+\infty .$$
The construction relies on the reduction of the problem to a finitedimensional one
and a topological argument based on the index theory to get the conclusion.
By the interpretation of the parameters of the finitedimensional problem
in terms of the blowup time and the blowup point, we show the stability
of the constructed solution with respect to perturbations in initial data.
To our knowledge, this is the first successful construction for a genuinely
nonscaleinvariant PDE of a stable blowup solution with the derivation
of the blowup profile. From this point of view, we consider our result as a
breakthrough.
