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Cauchy problems of fractional order and stable processes. (English) Zbl 0718.35026

We study the Cauchy problem (of the fractional order) \[ (\partial /\partial t)^{\alpha}u(t,x)=(\partial /\partial x)^{\beta}u(t,x)\text{ for } 1\leq \alpha,\quad \beta \leq 2. \] After determining the pair of (\(\alpha\),\(\beta\)) for which the solution exists uniquely, we investigate how the structure of the solution changes as (\(\alpha\),\(\beta\)) changes. We also give the expression of the solution by the stable processes. In particular, for \(\beta =2\), the positivity and the asymptotic behavior of the solution are studied.
Reviewer: Yasuhiro Fujita

MSC:

35G10 Initial value problems for linear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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