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http://dx.doi.org/10.4134/JKMS.j150343

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

Kim, Kyeong-Hun (Department of Mathematics Korea University)
Lim, Sungbin (Department of Mathematics Korea University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.4, 2016 , pp. 929-967 More about this Journal
Abstract
Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, 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 $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.
Keywords
fractional diusion; Levy process; asymptotic behavior; fundamental solution; space-time fractional dierential equation;
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