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

2023/07/10 by Iagar, Razvan Gabriel, Sánchez, Ariel · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.04714

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=Δu^m+|x|^σu^p, (x,t)∈ℝ^N×(0,∞) with m_c=((N-2)₊)/(N)≤ mlt;1, σ∈(max\-2,-N\,∞), pgt;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), \rm as |x|→∞ exist exactly for p∈(p_F(σ),p_s(σ)), where p_F(σ)=m+(σ+2)/(N), p_s(σ)=\(m(N+2σ+2))/(N-2), · amp; N≥3,
∞, · amp; 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∈(p_F(σ),p_s(σ)). We stress that all these results are new also in the homogeneous case σ=0.

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