Fractional heat equations with subcritical absorption having a measure as initial data

الأصل · EN

We study existence and uniqueness of weak solutions to (F) ∂_t u+ (-Δ)+h(t, u)=0 in (0,∞)×ⁿ,with initial condition u(0,·)=ν in ⁿ, where N≥2, the operator (-Δ)αis the fractional Laplacian with α∈(0,1), ν isa bounded Radon measure and h:(0,∞)×→ is a continuous function satisfying a subcritical integrability condition.In particular, if h(t,u)=tβuᵖ with β-1 and 0 p p*_β:=1+2α(1+β)/N, we prove that there exists a unique weak solution u_k to (F) with ν=kδ_0, where δ_0 is the Dirac mass at the origin. We obtain that u_k→∞ in (0,∞)×ⁿ as k→∞ for p∈(0,1] and the limit of u_k exists as k→∞ when 1 p p*_β, we denote it by u_∞.When 1+2α(1+β)/N+2α:=p**_β p p*_β,u_∞ is the minimal self-similar solution of (F)_∞ ∂_t u+ (-Δ)αu+tβuᵖ=0 in (0,∞)×ⁿ with the initial condition u(0,·)=0 in ⁿ{0} and it satisfies u_∞(0,x)=0 for x≠ 0.While if 1 p p**_β, then u_∞≡ U_p, where U_p is the maximal solution of the differential equation y'+tβyᵖ=0 on _+.