Some probability densities and fundamental solutions of fractional evolution equations

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Abstract

In the present paper, if 0<α⩽1, we shall study the Cauchy problem in a Banach space E for fractional evolution equations of the formdαudtα=Au(t)+B(t)u(t),where A is a closed linear operator defined on a dense set in E into E, which generates a semigroup and {B(t):t⩾0} is a family of a closed linear operators defined on a dense set in E into E. The existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the family of operators {B(t):t⩾0}. The solution is given in terms of some probability densities. An application is given for the theory of integro-partial differential equations of fractional orders.

Introduction

In this paper, we treat the Cauchy problem of the fractional evolution equationdαu(t)dtα=Au(t)+B(t)u(t),u(0)=u0,in a Banach space E, where u is an E-valued function on J=[0,T], T<∞, A is a linear closed operator defined on a dense set S1 in E into E, u0S1, {B(t):tJ} is a family of linear closed operators defined on a dense set S2S1 in E into E, 0<α⩽1 and S2 is independent of t.

It is assumed that A generates an analytic semigroup Q(t). This condition implies ∥Q(t)∥⩽K for t⩾0 and ∥AQ(t)∥⩽K/t for t>0, where ∥·∥ is the norm in E and K is a positive constant [1].

Let us suppose that B(t)g is uniformly Hölder continuous in tJ, for every gS2 with exponent β∈(0,1]. We suppose also that there exists a number γ∈(0,1) such that∥B(t2)Q(t1)g∥⩽kt1γ∥g∥,where t1∈(0,∞), gE and k is a positive constant [2], [3], [4]. (Notice that Q(t)gS1 for each gE and each t>0.)

We shall first obtain the Green function for the fractional abstract differential equationdαu(t)dtα=Au(t),whereu(0)=u0∈S1.Finally we prove the existence and uniqueness of the solution of the Cauchy problem , . We give also an application to the theory of integro-partial differential equations of fractional orders. Some of the applications of the theory of fractional calculus can be found in [5], [6], [7], [8], [9].

Section snippets

The fractional abstract differential equations

The theory of fractional calculus is essentially based on the integral convolution between f and the following generalized function, introduced by Gelfand and Shilov [10], φλ(t)=t+λ/Γ(λ), where λ is a complex number, t+λ=tλθ(t), θ(t) being the Heaviside step function, and Γ(λ) is the gamma function.

Following Gelfand and Shilov we can define the integral of order α>0 byIαf(t)=1Γ(α)0t(t−θ)α−1f(θ)dθ.

If 0<α⩽1, we can define the derivative of order α bydαf(t)dtα=1Γ(1−α)ddt0tf(θ)(t−θ)αdθ(see [11],

Fractional evolution equation

Let us solve the following Cauchy problemdαu(t)dtα=Au(t)+f(t),u(0)=u0∈S1,where f is an abstract function defined on [0,∞] and with values in E.

Theorem 3.1

If f satisfies a uniform Hölder condition, with exponent β∈(0,1], then the unique solution of the Cauchy problem , is given byu(t)=∫0ζα(θ)Q(tαθ)u0dθ+F(t),whereF(t)=α∫0t0θ(t−η)α−1ζα(θ)Q((t−η)αθ)f(η)dθdη.

Proof

If v and g are the Laplace transform of u and f, respectively, thenv(p)=pα−1(pαI−A)−1u0+(pαI−A)−1g(p).Consequently, we get formallyu(t)=∫0αζα(θ)Q(tαθ)u

An application

Consider the integro-partial differential equation of fractional orderαu(x,t)tα+∑|q|=2maq(x)Dxqu(x,t)=∑|q|<2maq(x,t)Dxqu(x,t)+∫S|q|<2mbq(ζ,x,t)Dζqu(ζ,t)dζ,where tJ, xRn, Rn is the n-dimensional Euclidean space,Dxq=Dx1q1Dxnqn,Dxi=xi,q=(q1,…,qn) is an n-dimensional multi-index, |q|=q1+⋯+qn and S is an open subset of Rn.

We assume the initial conditionu(x,0)=u0(x),x∈Rn.Let L2(Rn) be the set of all square integrable functions on Rn. We denote by Cm(Rn) the set of all continuous real-valued

An abstract mixed problem

Let Hm(S) be the completion of the space Cm(S) with respect to the norm∥f∥m=|q|⩽mS[Dqf(x)]2dx1/2,where S is an open bounded set in Rn with smooth boundary S. Let H0m(S) be the completion of the space C0m(S) with respect to the norm (5.1). Let S⊂S with smooth boundary ∂S. We assume the following conditions:

  • 1.

    The operator /t+∑|q|=2maq(x)Dq is uniformly parabolic on S=SUS;

  • 2.

    aq∈C2m(S), if |q|=2m, and aq∈C(S×[0,a]), if |q|<2m;

  • 3.

    bq∈C(S̄×S×[0,a]), |q|⩽2m−1, S̄=S∪S;

  • 4.

    The coefficients aq

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