Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation$ u_t = \Delta u^m + u^p, $posed in dimension $ N\geq3 $, $ m>1 $. More precisely, we show that there is always at least one solution in backward self-similar form if $ p>p_s = m(N+2)/(N-2) $. In particular, this establishes non-optimality of the Lepin critical exponent for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided $ N $ is sufficiently large. This is in strong contrast to the semilinear case, where the Lepin critical exponent has been proved to be optimal.