Some probability densities and fundamental solutions of fractional evolution equations
Introduction
In this paper, we treat the Cauchy problem of the fractional evolution equationin 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, u0∈S1, {B(t):t∈J} is a family of linear closed operators defined on a dense set S2⊃S1 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 t∈J, for every g∈S2 with exponent β∈(0,1]. We suppose also that there exists a number γ∈(0,1) such thatwhere t1∈(0,∞), g∈E and k is a positive constant [2], [3], [4]. (Notice that Q(t)g∈S1 for each g∈E and each t>0.)
We shall first obtain the Green function for the fractional abstract differential equationwhereFinally 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 by
If 0<α⩽1, we can define the derivative of order α by(see [11],
Fractional evolution equation
Let us solve the following Cauchy problemwhere 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 bywhere Proof If v and g are the Laplace transform of u and f, respectively, thenConsequently, we get formally
An application
Consider the integro-partial differential equation of fractional orderwhere t∈J, x∈Rn, Rn is the n-dimensional Euclidean space,q=(q1,…,qn) is an n-dimensional multi-index, |q|=q1+⋯+qn and S is an open subset of Rn.
We assume the initial conditionLet 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 be the completion of the space with respect to the normwhere is an open bounded set in Rn with smooth boundary . Let be the completion of the space with respect to the norm (5.1). Let with smooth boundary ∂S. We assume the following conditions:
- 1.
The operator is uniformly parabolic on ;
- 2.
, if |q|=2m, and , if |q|<2m;
- 3.
, |q|⩽2m−1, ;
- 4.
The coefficients aq
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