Preprint No.
A-00-05
S. R. Umarov, Yu. F. Luchko, R. Gorenflo
Partial pseudo-differential equations of fractional order: well-posedness of the Cauchy and multi-point value problems.
Abstract: This paper is devoted to the Cauchy and multi-point value problems for
partial pseudo-differential equations of fractional order. The used
pseudo-differential operators are associated with the sym\-bols which
may have singularities. The solvability theorems for these problems in
the space $\Psi_{G,p}(\re^n),\ 1<p<\infty,$ of functions in $L_p$ whose
Fourier transforms are compactly supported in a domain $G\subset
\re^n$ and in its dual space $\Psi'_{-G,q}(\re^n),\ q=p/(p-1),$ are
proved. The representations of the solutions in terms of
pseudo-differential operators are constructed and some of their
properties are given. The obtained results are then used to get
solvability theorems in the Sobolev spaces $H^s(\re^n),\ s\in
\re$.
Keywords: Pseudo-differential equations, Cauchy problem, multi-point value problem, Caputo fractional derivative, Mittag-Leffler function, fractional diffusion-wave equation
Mathematics Subject Classification (MSC2000): 26A33 Fractional derivatives and integrals
, 45K05 Integro-partial differential equations
Language: ENG
Available: Pr-A-00-05.ps
Contact: Luchko, Yu. F.; Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (luchko@math.fu-berlin.de)
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