Asymptotic Behaviors of Fundamental Solution and Its Derivatives to Fractional Diffusion-Wave Equations

Let p(t, x) be the fundamental solution to the problem ∂αₜu=-(-Δ)βu,\;α∈(0,2),\;β∈(0,∞) . If α,β∈(0,1) , then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives Dⁿₓ(-Δₓ)γDσₜIδₜp(t,x),\;∀n∈Z₊,\;γ∈[0,β ] ,\;σ,δ∈[0,∞) , where Dⁿₓ x is a partial derivative of order n with respect to x, (-Δₓ)γ is a fractional Laplace operator and Dσₜ and Iδₜ are Riemann-Liouville fractional derivative and integral respectively.