Existence and multiplicity of blow-up profiles for a quasilinear diffusion equation with source

We classify radially symmetric self-similar profiles presenting finite time blow-up to the quasilinear diffusion equation with weighted source uₜ=Δ uᵐ+|x|σuᵖ, posed for (x,t)∈N×(0,T), $T>0$, in dimension N≥1 and in the range of exponents -2<σ<∞, 1<m<p<pₛ(σ), where pₛ(σ)=(N+2σ+2)/N-2, & N≥3,\\ +∞, & N∈\1,2\,array. is the renowned Sobolev critical exponent. The most interesting result is the multiplicity of two different types of self-similar profiles for p sufficiently close to m and σ sufficiently close to zero in dimension N≥2, including dead-core profiles. For σ=0, this answers in dimension N≥2 a question still left open in [Section IV.1.4, pp. 195-196]{S4}, where only multiplicity in dimension $N=1$ had been established. Besides this result, we also prove that, for any σ∈(-2,0), N≥1 and m<p<pₛ(σ) existence of at least a self-similar blow-up profile is granted. In strong contrast with the previous results, given any N≥1, σ≥σ^*=(mN+2)/(m-1) and p∈(m,pₛ(σ)), non-existence of any radially symmetric self-similar profile is proved.