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 uₜ=Δ uᵐ + uᵖ, posed in dimension N≥3, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if p>pₛ=m(N+2)/(N-2). In particular, this establishes non-optimality of the Lepin critical exponent introduced in {Le90} in the semilinear case $m=1$ and extended for $m>1$ in {GV97, GV02}, 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 with the semilinear case, where the Lepin critical exponent has been proved to be optimal.