Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source

Abstract

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source ∂tu = Δum + |x| σup, (x, t) ∈ RN × (0, ∞) with mc = (N − 2)+ N ≤ m < 1, σ ∈ (max{−2, −N}, ∞), p > max 1 + σ(1 − m) 2 , 1 are considered. It is proved that global self-similar solutions with the specific tail behavior u(x, t) ∼ C(m)|x| −2/(1−m), as |x|→∞ exist exactly for p ∈ (pF (σ), ps(σ)), where pF (σ) = m + σ + 2N , ps(σ) = m(N+2σ+2) N−2 , N ≥ 3, ∞, N ∈ {1, 2}, are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any σ ∈ (−2, 0) and p as above, but do not exist for any σ ≥ 0 and p ∈ (pF (σ), ps(σ)). We stress that all these results are new also in the homogeneous case σ = 0.