Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles

Existence of a specific family of eternal solutions in exponential self-similar form is proved for the following porous medium equation with strong absorption ∂ₜ u-Δ uᵐ+|x|σuq = 0 \;\; in \;\; (0,∞)N, with $m>1$, q∈(0,1) and σ=2(1-q)/(m-1). Looking for solutions of the form u(t,x)=e-α tf(|x|eβ t), α=2/m-1β, it is shown that, for $m+q>2$, there exists a unique exponent β_*∈(0,∞) for which there exists a one-parameter family of compactly supported profiles presenting a dead core. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.