Asymptotic properties of solutions of the fractional diffusion-wave equation

Abstract

For the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

Subordination Principle for a Class of Fractional Order Differential Equations

“…After easily justified differentiation under the integral sign in (41), we obtain from (33) the estimates (18) and, thus, the well-posedness of Problem (1). Then, Identity (16) implies by the uniqueness of the Laplace transform that S(t) is exactly the solution operator of (1).…”

“…For example, the subordination principle for fractional evolution equations with the Caputo derivative (see [16], Ch. 3) has been successfully applied to inverse problems [17], for asymptotic analysis of diffusion wave equations [18], for the study of stochastic solutions [19], semilinear equations of fractional order [20], systems of fractional order equations [21], nonlocal fragmentation models with the Michaud time derivative [22], etc. This gives the author the motivation to present an analogous principle for Problem (1).…”

Asymptotic Behaviors of Fundamental Solution and Its Derivatives to Fractional Diffusion-Wave Equations

“…However, only few of them cover the derivative estimates of the fundamental solutions. In [9,10,17,20], upper bounds of (1.2) were obtained for β = 1, σ = 1 − α, and γ = 0. Also, in [13] asymptotic behavior for the case α = 1, β ∈ (0, 1) and σ = 0 was obtained.…”

“…Also, in [13] asymptotic behavior for the case α = 1, β ∈ (0, 1) and σ = 0 was obtained. Note that it is assumed that either α = 1 or β = 1 in [9,10,13,17,20], and moreover spatial fractional derivative ∆ γ p and time fractional derivative D σ t p are not obtained in [9,10,17,20] and [13] respectively. Our result substantially improves these results because we only assume α ∈ (0, 2) and β ∈ (0, ∞) and we provide two sides estimates of both space and time fractional derivatives of arbitrary order.…”

“…[3,9,10,11,12,13,20,22]). However, only few of them cover the derivative estimates of the fundamental solutions.…”