Refined blow-up behavior for reaction-diffusion equations with non scale invariant exponential nonlinearities

Abstract

We consider positive radial decreasing blow-up solutions of the semilinear heat equation \begin{equation*} u_t-\Delta u=f(u):=e^{u}L(e^{u}),\quad x\in \Omega,\ t>0, \end{equation*} where $\Omega=\mathbb{R}^n$ or $\Omega=B_R$ and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating unbounded functions). We characterize the aymptotic blow-up behavior and obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$. Namely, assuming for instance $u_t\ge 0$, we have \begin{equation*} u(x,t)=G^{-1}\bigg(T-t+\frac{1}{8}\frac{|x|^2}{|\log |x||}\bigg)+o(1)\quad \ \hbox{as $(x,t)\to (0,T)$, where } \quad G(X)=\int_{X}^{\infty} \frac{ds}{f(s)}ds. \end{equation*} This estimate in particular provides the sharp final space profile and the refined space-time profile. For exponentially growing nonlinearities, such results were up to now available only in the scale invariant case $f(u)=e^u$. Moreover, this displays a universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the ODE composed with a fixed time-space building block, which is robust with respect to the factor $L(e^u)$.