Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

Abstract

We consider the semilinear heat equation

$${\partial }_{t}u=\mathrm{\Delta }u+|u{|}^{p-1}u{ln}^{\alpha }\left(u^2+2\right)$$

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

$$u\left(x,t\right)\sim \psi \left(t\right){\left(1+\frac{\left(p-1\right)|x-a{|}^2}{4p\left(T-t\right)|ln\left(T-t\right)|}\right)}^{-1∕\left(p-1\right)}\text{ as }t\to T,$$

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

$${\psi }^{\prime }={\psi }^p{ln}^{\alpha }\left({\psi }^2+2\right),\underset{t\to T}{lim}\psi \left(t\right)=+\infty .$$

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.