  We consider the semilinear heat equation \begin{eqnarray*} \partial_t u =
\Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} 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 following asymptotic dynamics:
\begin{eqnarray*} u(x,t) \sim \psi(t) \left(1 + \frac{(p-1)|x-a|^2}{4p(T
-t)|\ln(T -t)|} \right)^{-\frac{1}{p-1}} \text{ as } t \to T, \end{eqnarray*}
where $\psi(t)$ is the unique positive solution of the ODE \begin{eqnarray*}
\psi' = \psi^p \ln^{\alpha}(\psi^2 +2), \quad \lim_{t\to T}\psi(t) = + \infty.
\end{eqnarray*} The construction relies on the reduction of the problem to a
finite dimensional one and a topological argument based on the index theory to
get the conclusion. By the interpretation of the parameters of the finite
dimensional 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 non-scale invariant 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.
