ASYMPTOTIC BEHAVIORS OF FUNDAMENTAL SOLUTION AND ITS DERIVATIVES TO FRACTIONAL DIFFUSION-WAVE EQUATIONS

Abstract

Let p(t, x) be the fundamental solution to the problem <TEX>$${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$</TEX>. If <TEX>${\alpha},{\beta}{\in}(0,1)$</TEX>, 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 <TEX>$$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}</TEX><TEX>]</TEX><TEX>,\;{\sigma},{\delta}{\in}[0,{\infty})$$</TEX>, where <TEX>$D^n_x$</TEX> x is a partial derivative of order n with respect to x, <TEX>$(-{\Delta}_x)^{\gamma}$</TEX> is a fractional Laplace operator and <TEX>$D^{\sigma}_t$</TEX> and <TEX>$I^{\delta}_t$</TEX> are Riemann-Liouville fractional derivative and integral respectively.