Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations

2015/04/28 by Kyeong-Hun Kim, Kim, Kyeong-Hun, Sungbin Lim +1 · 3 citations
Mathematics · #26A33 #35A08 #35R11 #45K05 #45M05 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1504.07386

openalex publication_date 2015/04/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let p(t,x) be the fundamental solution to the problem ∂_t^αu=-(-Δ)^βu, α∈ (0,2), β∈ (0,∞). In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t,x) and its space and time fractional derivatives D_xⁿ(-Δ_x)^γD_t^σI_t^δp(t,x), ∀ n∈ℤ₊, γ∈[0,β], σ, δ∈[0,∞), where D_xⁿ is a partial derivative of order n with respect to x, (-Δ_x)^γ is a fractional Laplace operator and D_t^σ and I_t^δ are Riemann-Liouville fractional derivative and integral respectively.

Citations

• An L_q(L_p)-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach

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