Decay estimates for time-fractional porous medium flow with nonlocal pressure

2023/04/25 by Dao, Nguyen Anh, Tien, Anh Nguyen Vu
#35A23 #35R11 #45K05 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.12580

Abstract

The main purpose of this paper is to study weak solutions of time-fractional of porous medium equation with nonlocal pressure: ∂^α_t u=div( |u|^m∇ (-Δ)^-s u) in ℝ^N× (0,T) , with m≥ 1, N≥ 2, (1)/(2)≤ s<1, and α∈(0,1). We first prove an existence of weak solutions to the equation with initial data in L¹(ℝ^N)∩ L^∞(ℝ^N) (possibly mixed sign). After that, we establish the L^q-L^∞ decay estimate of weak solutions: ‖u(t)‖_L^∞(ℝ^N) ≤ C t^-\fracαq(1-λ₀)+ m ‖u₀‖_L^q(ℝ^N)^{(q(1-λ₀))/(q(1-λ₀) + m)} , for t∈(0,∞), with λ₀=(N-2(1-s))/(N).

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