Refined blow-up behavior for semilinear heat equations with arbitrarily fast growing nonlinearities

Abstract

We study radially nonincreasing, positive blow-up solutions of a class of superlinear heat equations in the non-scale-invariant setting. This class contains nonlinearities with arbitrarily fast, super-exponential growth. Under suitable structural assumptions on the nonlinearity, we derive the sharp global blow-up profile in the original space-time variables. More precisely, for positive nonlinearities $f$ satisfying $$\underset{X\to \infty }{\lim }{f}^{\prime }(X)G(X)=q\ge 1whereG(X)={\int }_X^{\infty }\frac{ds}{f(s)}<\infty \forall X>0,$$ and assuming either the time monotonicity $u_t\ge 0$ holds or $q>\frac{p_S}{p_S-1}$, we have $$u(x,t)=(1+o(1))G^{-1}(T-t+\frac{|x{|}^2}{4q|\log |x{|}^2|}),as(x,t)\to (0,T).$$ This estimate in particular provides the sharp final space profile and the corresponding refined space-time profile. Moreover, our result recovers the universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the associated ODE evaluated at a time-space building block, which only depends on the limit $q$.