Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity

Abstract

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation${\partial }_{t}u-\Delta u=f\left(u\right),$ for nonlinearities which are genuinely non scale invariant, unlike in the standard case $f\left(u\right)=u^p$. Indeed, our results apply to a large class of nonlinearities of the form $f\left(u\right)=u^{p}L\left(u\right)$, where $p>1$ is Sobolev subcritical and L is a slowly varying function at infinity (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions).

More precisely, denoting by ψ the unique positive solution of the corresponding ODE $y^{\prime }\left(t\right)=f\left(y\right(t\left)\right)$ which blows up at the same time T, we show that if $a\in \Omega$ is a blowup point of u, then$\underset{t\to T}{\lim }\frac{u(a+y\sqrt{T-t},t)}{\psi \left(t\right)}=1,\text{uniformly for }y\text{ bounded.}$ Additional blow-up properties are obtained, including the compactness of the blow-up set for the Cauchy problem with decaying initial data.