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8 Baeumer, Kovács, Meerschaert solution of (17). A function u : [0, δ) → X is a local mild solution of (17) if u is continuous and satisfies the corresponding integral equation u(t) = T (t)u 0 + ∫ t 0 T (t − s)f(u(s)) ds, 0 ≤ t < δ. (18) We note that the integral in (18) is a Bochner integral [1,24,44], an extension of the Lebesgue integral to the Banach space setting which coincides with a Riemann integral if the integrand is continuous in the Banach space norm. If δ can be chosen arbitrarily large then u is a global mild solution of (17). If f : X → X is Lipschitz continuous, that is; ||f(u) − f(v)|| ≤ M||u − v|| for all u, v ∈ X, then for all u 0 ∈ X there is a unique global mild solution u(t) := W (t)u 0 of (17) with ||W (t)u 0 − W (t)v 0 || ≤ M T ||u 0 − v 0 ||, t ∈ [0, T ] (see, for example, [44, Section 6.1]). Denote the unique mild solution of (17) in the special case A ≡ 0 by u(t) := S(t)u 0 . In this case T (t)u = u for all t ≥ 0, and hence it follows from (18) that u(t) = u 0 + ∫ t 0 f(u(s)) ds (19) for all t > 0. Then u is also a strong solution, since if u and f are continuous, then t ↦→ f(u(t)) is also continuous and t ↦→ ∫ t f(u(s)) ds is differentiable, ∫ 0 and d t dt f(u(s)) ds = f(u(t)) (see, e.g., [24, page 67]). Therefore t → u(t) is 0 differentiable in view of (19), and ˙u(t) = f(u(t)). It also follows from (19) that the collection of nonlinear operators {S(t)} t≥0 forms a semigroup, sometimes called the flow of the abstract differential equation ˙u = f(u). We say that the collection S(·) is generated by f. It is well known that the solution operator W (t)u 0 to the abstract reaction-diffusion equation (17) can be computed by the Trotter Product Formula [ W (t)u 0 = lim T ( t n→∞ n )S( t n )] n [ u0 = lim S( t n→∞ n )T ( t n )] n u0 , u 0 ∈ X, (20) see, for example, [12,18,38]. This result means that the mild solution to the abstract reaction-diffusion equation (17) can be computed as an approximation using the mild solutions of the two sub-problems, the abstract reaction equation ˙u = f(u) and the abstract diffusion equation ˙u = Au. The following proposition is a simplified variant of [18, Theorem 15]. We call a Banach space X an ordered Banach space if it is a real Banach space endowed with a partial ordering ≤ such that 1. u ≤ v implies u + w ≤ v + w for all u, v, w ∈ X. 2. u ≥ 0 implies λu ≥ 0 for all u ∈ X and λ ≥ 0. 3. 0 ≤ u ≤ v implies ||u|| ≤ ||v|| for all u, v ∈ X. 4. The positive cone X + := {x ∈ X : x ≥ 0} is closed. A typical example of an ordered Banach space is C 0 (R), the space of continuous functions u : R → R such that u(x) → 0 as |x| → ∞, endowed with the supremum norm ‖u‖ = sup{|u(x)| : x ∈ R}, and endowed with the partial ordering u ≤ v whenever u(x) ≤ v(x) for all x ∈ R. Another example is L p (R) (1 ≤ p ≤ ∞) endowed with the partial ordering u ≤ v whenever u(x) ≤ v(x)
Fractional reaction-diffusion equation 9 for x ∈ R almost everywhere. An operator A on an ordered Banach space is called positive if 0 ≤ u ≤ v implies 0 ≤ Au ≤ Av. We also write B ≤ A if 0 ≤ Bu ≤ Au for any u ≥ 0. Proposition 1 Let X be an ordered Banach space, and assume that the strongly continuous semigroup T (·) generated by the linear operator A and the nonlinear semigroup S(·) generated by the Lipschitz continuous function f are positive. If T (t)S(t)u 0 ≤ S(t)T (t)u 0 (21) holds for all t ∈ [0, T ] and u 0 ≥ 0, then the unique mild solution W (t)u 0 of the abstract reaction-diffusion equation (17) satisfies [ T ( t n )S( t n )] n u0 ≤ [ T ( t 2n )S( t 2n )] 2n u0 ≤ W (t)u 0 for all u 0 ≥ 0, n ∈ N and t ∈ [0, T ]. ≤ [ S( t 2n )T ( t 2n )] 2n u0 ≤ [ S( t n )T ( t n )] n u0 (22) Proof It follows from our assumption that W (t)u 0 is given by (20). Next we show that [ T ( t n )S( t n )] n u0 ≤ [ T ( t 2n )S( t 2n )] 2n u0 ≤ [ S( t 2n )T ( t 2n )] 2n u0 ≤ [ S( t n )T ( t n )] n u0 , (23) u 0 ≥ 0, n ∈ N and t ∈ [0, T ]. For u 0 ≥ 0 using (21) repeatedly we have T ( t n )S( t n )u 0 = T ( t 2n )T ( t 2n )S( t 2n )S( t 2n )u 0 ≤ T ( t 2n )S( t 2n )T ( t 2n )S( t 2n )u 0 ≤ S( t 2n )T ( t 2n )T ( t 2n )S( t 2n )u 0 ≤ S( t 2n )T ( t 2n )S( t 2n )T ( t 2n )u 0 ≤ S( t 2n )S( t 2n )T ( t 2n )T ( t 2n )u 0 = S( t n )T ( t n )u 0. For any operators A, B : X → X, it is not hard to check that the inequality 0 ≤ B ≤ A implies 0 ≤ B n ≤ A n (n ∈ N) which finishes the proof of (23). Fix n ∈ N and let x k := [ T ( t )S( t ) ] n2 k u n2 k n2 k 0 . By (23), x 0 ≤ x 1 ≤ ... ≤ x k ≤ ... and x k → W (t)u 0 as k → ∞ by (20). Therefore, it follows from the closedness of the positive cone X + that x k ≤ W (t)u 0 for all k = 0, 1, .... This shows the second inequality in (3). A similar argument yields the third inequality and the proof is complete. ⊓⊔ One useful class of strongly continuous semigroups are the infinitely divisible semigroups, which are associated with certain families of probability densities. Suppose that Y is a random variable on R with probability density k(y) and Fourier transform ˆk(λ) = ∫ e −iλy k(y)dy. Let k n = k ∗ · · · ∗ k denote the n−fold convolution of k with itself. We say that Y (or k) is infinitely divisible if for each n = 1, 2, 3, . . . there exist independent random variables Y n1 , . . . , Y nn with the same density k n such that Y n1 + · · · + Y nn is identically distributed with Y . The normal, Cauchy, double-Gamma, Laplace, α-stable, and Student-t densities are all infinitely divisible. The Lévy representation
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