Abstract

We consider the semilinear heat equation

in the whole space ${ℝ}^n$, where $p>1$ and $\alpha \in ℝ$. 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

where $\psi \left(t\right)$ is the unique positive solution of the ODE

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.

Keywords

Mathematical Subject Classification 2010

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