ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP

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1 Dedicated to Professor Gheorghe Bucur on the occasion of his 7th birthday ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP EMIL POPESCU Starting from the usual Cauchy problem, we give a pseudodifferential representation for the solution of the fractional Cauchy problem associated with a Feller semigroup. AMS 2 Subject Classification: 26A33, 47G3, 47D7, 31B1. Key words: fractional derivative, Cauchy problem, Feller semigroup, pseudodifferential operator. 1. INTRODUCTION 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 2 u(x, t) = D u(x, t), u(x, ) = f(x). t x2 The solution u (x, t) depends on location x R and time t 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, < α < 2, α u(x, t) = D u(x, t), u(x, ) = f(x). t xα We observe that α x is a pseudodifferential operator. Thus, we can extend α this equation to u (x, t) = (Au(, t))(x), t u(x, ) = u (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, ) = u (x), MATH. REPORTS 12(62), 2 (21),

2 182 Emil Popescu 2 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 u(x, t) = (Au(, t))(x), tβ u(x, ) = f(x), where β t β 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 < β < 1, t and A is the generator of bounded continuous semigroup {T (t)} t on the Banach space X, can be expressed as an integral transform of the solution to the initial Cauchy problem u(x, t) = (Au(, t))(x), t u(x, ) = 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 is a family of pseudodifferential operators. Their symbols are obtained by transformation of the symbols of the semigroup {T (t)} t, where u(x, t) = T (t)f(x) is the solution to the initial Cauchy problem. 2. INTEGRAL REPRESENTATION OF THE OPERATORS WHICH FORM A FELLER SEMIGROUP Let (X, ) be a Banach space. {T (t)} t is a strongly continuous semigroup on X if for any t, T (t) : X X is a linear operator and there exists M > such that T (t)x M x, T () = I, T (t + s) = T (t)t (s) for t, s, and t T (t)x is continuous in the norm, for all x X. The generator (A, D(A)) of the semigroup {T (t)} t, is defined by { } T (t)x x T (t)x x D(A) = x X lim exists, Ax = lim, t + t t + t where we suppose that the limit exists for at least some nonzero x X. u (t) = T (t) f solves the abstract Cauchy problem d u(t) = Au(t), u() = f, dt for f D(A). In the following, we denote by C (R n ) the Banach space of all continuous functions on R n vanishing at infinity with the supremum norm and by C (Rn ) the set of all C -functions on R n with compact support. S(R n )

3 3 On the fractional Cauchy problem associated with a Feller semigroup 183 will be the Schwartz space, i.e., the set of all functions ϕ C (R n ) such that sup x β α ϕ(x) < for all multi-indices α and β. S(R n ) is dense in C (R n ). x R n A function a : R n C is called negative definite if a() and ξ e ta(ξ) is a positive definite function for all t >. A continuous negative definite function a is described by Lévy-Khinchin formula [ ] a(ξ) = c + ib ξ + q(ξ) + 1 e iξ y ξ y 1 + y 2 i 1 + y 2 y 2 dµ(y) R n \{} with c, b R n, q a continuous non-negative definite quadratic form on R n and µ, called measure Lévy, a σ-finite Borel measure on R n \{} such that min(1, y 2 )dµ(y) <. R n \{} The general form of a pseudodifferential operator is for ϕ C (Rn ), where p(x, D)ϕ(x) = (2π) (n/2) R n e ix ξ p(x, ξ) ϕ(ξ)dξ, ϕ(ξ) = (2π) (n/2) R n e ix ξ ϕ(x)dx is the Fourier transform. p(x, ξ) is called the symbol of the operator p(x, D) (see, for example, [4]). Let A : D(A) C (R n ) be a linear operator, where D(A) is a linear dense subspace of C (R n ). A satisfies the positive maximum principle on D(A) if for all u D(A) and x R n such that it follows that Au(x ). sup u(x) = u(x ) x R n Theorem 2.1 ([3]). Let A : C (Rn ) C b (R n ) be a linear operator satisfying the positive maximum principle. Then Au(x) = (2π) (n/2) R n e ix ξ a(x, ξ)û(ξ)dξ, where a : R n R n C is a locally bounded function such that for any fixed x R n, ξ a(x, ξ) is a continuous negative definite function. The convolution semigroup on C (R n ) generated by a is defined by the formula T (t)u(x) = (2π) (n/2) R n e ix ξ p t (ξ)û(ξ)dξ,

4 184 Emil Popescu 4 for each t > and u S(R n ), where p t (ξ) = e ta(ξ). In this case, we observe that for any t > the symbol is p t (note that there is no x-dependence). The function ξ p t (ξ) is a positive definite function and the infinitesimal generator of {T (t)} t is Au(x) = (2π) (n/2) R n e ix ξ a(ξ)û(ξ)dξ, for all u C (Rn ), x R n. Let {T (t)} t be a strongly continuous semigroup on C (R n ). If T (t)u u for all u C (R n ) and t, then {T (t)} t is a contraction semigroup. A strongly continuous positive contraction semigroup on C (R n ) is called a Feller semigroup on R n. We have an integral representation of the operators which form a Feller semigroup, analogous with the one of convolution semigroup (see [8], [5]). Theorem 2.2. Let {T (t)} t be a Feller semigroup on R n. For any t there exists a unique function p t : R n R n C measurable, locally bounded and such that for any fixed x R n, ξ p t (x, ξ) is a continuous positive definite function with the property that for any u S(R n ), T (t)u(x) = (2π) (n/2) R n e ix ξ p t (x, ξ)û(ξ)dξ. For u S(R n ), the infinitesimal generator A of {T (t)} t is Au(x) = (2π) (n/2) R n e ix ξ a(x, ξ)û(ξ)dξ, where a : R n R n C, a(x, ξ) = d dt p t(x, ξ). t= Moreover, we deduce the following result. Proposition 2.3. Let {T (t)} t be a Feller semigroup on R n. For any t and u C 2 b (Rn ), T (t)u(x) = C(t)u(x) + D(t)u(x), with C(t)u(x) := n i,j=1 ij (x) 2 u (x) + x i x j a (t) n i=1 and { [ D(t)u(x) := N (t) (x, dy) u(y) σ x (t) (y) u(x) + R n b (t) i (x) u (x) + γ (t) (x)u(x) x i n i=1 ] } u (x) (y i x i ), x i

5 5 On the fractional Cauchy problem associated with a Feller semigroup 185 where γ (t) (x) = c (t) (x)+d (t) (x)+1, a (t) a (t) ij = a (t) ji, n i,j=1 ij, b(t) i, c (t), d (t) are continuous functions, a (t) ij (x)ξ iξ j, c (t), σ x (t) is a certain cutt-off function and N (t) (x, dy) is a certain Lévy kernel such that d (t) (x) + N (t) (x, dy) { 1 σ x (t) (y) }, R n for all x R n. Proof. Indeed, T (t) I satisfies the positive maximum principle on C (R n ) for every t. The above formula follows from the last assertion of Lemma 3.3 ([3], p. 2 34) and Corollary 3 ([3], p. 2 1). 3. FRACTIONAL CAUCHY PROBLEM For a function g with g (s) := e st g (t) dt the Laplace transform, we define the Caputo fractional derivative in time β g (t) as the inverse Laplace t β transform of s β g (s) s β 1 g (). On the other hand, D β t g (t) = dm dt m t (t u) m β 1 g (u) du, Γ (m β) m = [β] is the Riemann-Liouville fractional derivative of order β. We have β t β g (t) = t (t u) m β 1 g (m) (u) du, Γ (m β) m = [β]. If β is a positive integer then D β t = β is the usual derivative operator. t We consider the fractional Cauchy β problem β u (x, t) = Au (x, t), u (x, ) = f (x), tβ where < β < 1, t and A is the generator of bounded continuous semigroup {T (t)} t on the Banach space X. We observe that p (x, t) = T (t) f (x) is the unique solution to the abstract Cauchy problem p (x, t) = Ap (x, t), p (x, ) = f (x), t for any f in the domain of A and t >. We note that the fractional Cauchy problem can be written in several equivalent forms (see [1] and [2]). Proposition 3.1. Assume < β < 1. Let A be the generator of a strongly continuous semigroup {T (t)} t on the Banach space X and g

6 186 Emil Popescu 6 C ([, ) X) be Laplace transformable. Then for all f X the following are equivalent: (i) For all t >, the Riemann-Liouville derivative of g exists, g (t) D (A), the Laplace transform of D β t g (t) exists, and D β t g (t) = Ag (t) + t β Γ (1 β) f. (ii) For all t >, the Caputo derivative of g exists, g (t) D (A), the Laplace transform of β g (t) exists, and t β β g (t) = Ag (t), g () = f. tβ (iii) For all t >, the function g is differentiable, g (t) D (A), the Laplace transform of g (t) exists, and t g (t) = D1 β t Ag (t), g () = f. t (iv) The function g (t) is analytic on < t <, satisfies g (t) Me ωt on < t < for some M, ω and t t g (t) = βs 1+1/β g β s 1/β T (s) fds, where g β is such that e λt g β (t) dt = e λβ. In the framework of Proposition 3.1, we define the family of bounded, strongly continuous linear operators on X, t t S (t) h (x) := βs 1+1/β g β s 1/β T (s) f (x) ds, t. On account of (iv) the function g (t) = S (t) f defines a solution to the fractional Cauchy problem for any initial condition f X and this solution depends continuously on the initial condition f. In the sequel we consider X = C (R n ) and A the generator of a Feller semigroup {T (t)} t on R n. Then by Theorem 2.2, for any t there exists a unique function p t : R n R n C measurable, locally bounded and such that for any fixed x R n, ξ p t (x, ξ) is a continuous positive definite function with the property that for any f S (R n ), T (t)f (x) = (2π) (n/2) R n e ix ξ p t (x, ξ) f (ξ) dξ.

7 7 On the fractional Cauchy problem associated with a Feller semigroup 187 We suppose that the integral ( 1 g t s 1+1/β β all x and ξ. The following statement is true. s 1/β ) Proposition 3.2. For any t and f S (R n ), S (t) f (x) = (2π) (n/2) R n e ix ξ q t (x, ξ) f (ξ) dξ, p s (x, ξ) ds is convergent for where q t : R n R n C is measurable, locally bounded and such that for any fixed x R n, ξ q t (x, ξ) is a continuous positive definite function. Proof. In the formula of the definition of S (t) f (x) we apply the above thoughts for the semigroup {T (t)} t on R n. We define q t (x, ξ) := t 1 t β s 1+1/β g β s 1/β p s (x, ξ) ds and we observe that q t satisfies the required properties. Remark 3.3. Using the relation from Proposition 2.3, we obtain the structure of each operator S (t), t. Since T (t) f (x) = C (t) f (x) + D (t) f (x), we have t t S (t) f (x) = βs 1+1/β g β s 1/β C(s)f(x)ds+ t t + βs 1+1/β g β s 1/β D(s)f(x)ds. Thus we can interpret the solution g (t) = S (t) f of the fractional Cauchy problem for the initial condition f as the sum of diffusion part and Lévy part. REFERENCES [1] B. Baeumer and M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (21), [2] B. Baeumer, M.M. Meerschaert and S. Kurita, Inhomogeneous fractional diffusion equations. Fract. Calc. Appl. Anal. 8 (25), [3] P. Courrège, Sur la forme intégro-différentielle des opérateurs de C k dans C satisfaisant au principe du maximum. Sém. Théorie du Potentiel (1965/1966), 38p. [4] N. Jacob, Pseudo-differential operators and Markov processes. Mathematical Research, Vol. 94. Akademie Verlag, Berlin, [5] N. Jacob, Characteristic functions and symbols in the theory of Feller processes. Potential Analysis 8 (1998), [6] J. Klafter and I.M. Sokolov, From diffusion to anomalous diffusion: A century after Einstein s Brownian motion. Chaos 15 (25), [7] A. Pazzy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, Vol. 44. Springer-Verlag, New York, 1983.

8 188 Emil Popescu 8 [8] E. Popescu, A note on Feller semigroups. Potential Analysis 14 (21), [9] E. Popescu, Trotter products and reaction diffusion equations. J. Computational Appl. Math. 233 (21), Received 15 September 29 Technical University of Civil Engineering Department of Mathematics and Informatics Bd. Lacul Tei Bucharest, Romania epopescu@utcb.ro