Measurable feller semigroups on rn

Abstract

Fractional derivatives are used to model anomalous diffusion, which occurs when the particles spread in a different manner than the prediction of the classical diffusion equation ∂ ∂tu(x, t) = D ∂ 2 ∂x2 u(x, t), u(x, 0) = f(x). The solution u (x, t) depends on location x ∈ R and time t ≥ 0 and models the dispersion. A known model for an anomalous diffusion (see [6]) is the fractional diffusion equation, where the usual second derivative in space is replaced by a fractional derivative of order α, 0 < α < 2, ∂ ∂tu(x, t) = D ∂ α ∂xα u(x, t), u(x, 0) = f(x). We observe that ∂ α ∂xα is a pseudodifferential operator. Thus, we can extend this equation to ∂u ∂t (x, t) = (Au(· , t))(x), u(x, 0) = u0(x), where A is a pseudodifferential operator. A study of the solutions of a generalized reaction-diffusion equation of the form ∂u ∂t (x, t) = (Au (· , t)) (x) + f (x, u (x, t)), u (x, 0) = u0 (x), MATH. REPORTS 12(62), 2 (2010), 181–188 where A is a pseudodifferential operator which generates a Feller semigroup, was given in [9]. In this paper we consider the fractional Cauchy problem ∂ β ∂tβ u(x, t) = (Au(·, t))(x), u(x, 0) = f(x), where ∂ β ∂tβ u (x, t) is the Caputo fractional derivative in time and A is a pseudodifferential operator which generates a Feller semigroup. In [1] and [2] was shown that the solution of fractional Cauchy problem, where 0 < β < 1, t ≥ 0 and A is the generator of bounded continuous semigroup {T (t)}t≥0 on the Banach space X, can be expressed as an integral transform of the solution to the initial Cauchy problem ∂ ∂tu(x, t) = (Au(·, t))(x), u(x, 0) = f(x). Starting from this integral transform, we give a formula for the solution u(x, t) = S(t)f(x) of the fractional Cauchy problem. We show that {S(t)}t≥0 is a family of pseudodifferential operators. Their symbols are obtained by transformation of the symbols of the semigroup {T(t)}t≥0, where u(x, t) = T(t)f(x) is the solution to the initial Cauchy problem.