Subordination principle for space-time fractional evolution equations and some applications

Abstract

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order β ∈ (0, 1) and operator −A α , α ∈ (0, 1), is considered, where −A generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a C 0 -semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation for some special values of the parameters.2010 Mathematics Subject Classification. 26A33, 33E12, 35R11, 47D06.

Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

“…A subordination principle for completely positive measures was discussed in detail in [32] and applied there for constructing new resolvents for the abstract Volterra integral equations based on the known ones. In [1] and [2], this subordination principle was extended and specialized for the abstract fractional evolution equations in the form D β u(t) = Au(t),…”

Probabilistic representation formula for the solution of fractional high-order heat-type equations

“…Anomalous diffusion occurs when the mean square displacement (or time-dependent variance) is stretched by some index, in other words proportional to t α for instance. In the literature, equation (9) is used as a mathematical model of a wide range of important physical phenomena, usually named sub-or super-diffusions, for instance in microelectronics (dielectrics and semiconductors), polymers, transport phenomena in complex systems and anomalous heat conduction in porous glasses and random media (see for instance [62,50,27,28]). Fractional diffusion equations as (9) where N = 2 have been investigated by several researchers.…”

Random time change and related evolution equations. Time asymptotic behavior

“…The subordination principle, see [6], tells that u(x, t) is the solution of the general fractional differential equation…”

“…The appropriate notions of the solutions of (3.1) and (3.3) depend on the specific setting. They were explained (a) in [19] for the case where A is the Laplace operator on R n , (b) in [6,7,5] with abstract semigroup generators for special classes of kernels k, (c) in [25] for abstract Volterra equations.…”