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Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^{\alpha}u=-(-\Delta)^{\beta}u, \quad \alpha\in (0,2), \, \beta\in (0,\infty). $$ If $\alpha,\beta\in (0,1)$, then the kernel $p(t,x)$ becomes the transition density of a L\'evy 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 $$ D_{x}^{n}(-\Delta_x)^{\gamma}D_{t}^{\sigma}I_{t}^{\delta}p(t,x), \quad \forall\,\, n\in\mathbb{Z}_{+}, \,\, \gamma\in[0,\beta],\,\, \sigma, \delta \in[0,\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-\Delta_x)^{\gamma}$ is a fractional Laplace operator and $D_{t}^{\sigma}$ and $I_{t}^{\delta}$ are Riemann-Liouville fractional derivative and integral respectively.
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참고문헌 (37)
참고문헌 신청
M. Abramowitz / 1972 / Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables / Courier Dover Publications
G. E. Andrews / 1999 / Special Functions / Cambridge University Press
V. V. Anh / 2001 / Spectral analysis of fractional kinetic equations with random data / J. Statist. Phys 104(5-6):1349~1387
B. L. J. Braaksma / 1964 / Asymptotic expansions and analytic continuations for a class of barnes-integrals / Compositio Math 15:239~341
Z.-Q. Chen / 2015 / Fractional time stochastic partial differential equations / Stochastic Process. Appl 125(4):1470~1499
G. E. Andrews / 1999 / Special Functions / Cambridge University Press
V. V. Anh / 2001 / Spectral analysis of fractional kinetic equations with random data / J. Statist. Phys 104(5-6):1349~1387
B. L. J. Braaksma / 1964 / Asymptotic expansions and analytic continuations for a class of barnes-integrals / Compositio Math 15:239~341
Z.-Q. Chen / 2015 / Fractional time stochastic partial differential equations / Stochastic Process. Appl 125(4):1470~1499
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