Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption \[ ∂_t u=Δ u^m-|x|σu^q, \] posed for (t,\,x)∈ (0,\,∞ )× RN , with $m>1$ , q∈ (0,\,1) and σ =σ c:=2(1-q)/ (m-1) is proved. Looking for radially symmetric solutions of the form \[ u(t,x)={ e}-α tf(|x|\,{ e}β t), α=2/m-1β, \] we show that there exists a unique exponent β ^*∈ (0,\,∞ ) for which there exists a one-parameter family (uA)A>0 of solutions with compactly supported and non-increasing profiles (fA)A>0 satisfying fA(0)=A and fA'(0)=0 . An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when σ ∈ (0,\,σ c) .