Fractional-hyperbolic equations and systems. Cauchy problem

The paper is a survey of results on the Cauchy problem for fractionalhyperbolic equations and systems with the Caputo-Djrbashian fractional time derivative. We describe a class of evolution systems of linear partial differential equations with the Caputo-Djrbashian fractional derivative of order α ∈ (0, 1) in the time variable t and the first-order derivatives in spatial variables x = (x1, . . . , xn), which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone {(t, x) : |t−αx| ≤ 1}. We consider an evolution equation of order β ∈ (1, 2) with respect to the time variable, and the second-order uniformly elliptic operator with variable coefficients acting in spatial variables. Properties of such equations are intermediate between those of parabolic and hyperbolic equations. We describe the parametrix method for constructing a fundamental solution of the Cauchy problem, formulate existence and uniqueness theorems for such equations, describe an analog of the principle of limiting amplitude (well known for the wave equation), and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation).

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