Abstract
We consider the Cauchy problem (∂/∂t)α u(t,x)=(∂/∂x)β u(t,x) for 1≤α, β≤2. The aim of this paper is threefold. First, we show the existence and the uniqueness of the solution, and determine its structure. Second, we give a representation of the solution by the distributions of several stable processes. Third, we show the positivity of the fundamental solution for β=2. These results offer an interpretation to phenomena between the heat equation (α=1, β=2) and the wave equation (α=β=2).
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Fujita, Y. Cauchy problems of fractional order and stable processes. Japan J. Appl. Math. 7, 459–476 (1990). https://doi.org/10.1007/BF03167854
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DOI: https://doi.org/10.1007/BF03167854