^{1}

^{1}

In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem
* D*
_{t}
^{ α}
*u(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x*
_{0} on a Banach space
*X* with order a ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if
*A* generates an analytic α-times resolvent family on
*X* and
* f ∈ L*
^{p}
* ([0,T];X*) for some
*p* > 1/α, then the mild solution to the above equation is in
*C*
^{α-1/p}
*[ò,T]* for every ò > 0. Moreover, if f is H?lder continuous, then so are the
*D*
_{t}
^{ α}
*u(t)* and
*Au(t)*.

Recently there are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al. For more details we refer to [

Many evolution equations can be rewritten as an abstract Cauchy problem, and then they can be studied in an unified way. For example, a heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C_{0}-semigroup, and then the solution is given by the operation of this semigroup on the initial data. See for instance [_{0}-semigroup, we can invoke an operator described by the C_{0}-semigroup and a probability density function to solve this problem, for more details we refer to [

There are some papers devoted to the fractional differential equations in many different respects: the connection between solutions of fractional Cauchy problems and Cauchy problems of first order [

Pazy [

u ′ ( t ) = A u ( t ) + f ( t ) , t ∈ [ 0 , T ] u ( 0 ) = x 0 (1.1)

where A is the infinitesimal generator of an analytic C_{0}-semigroup. He showed that if f ∈ L p [ 0, T ] for some 1 < p < ∞ , then u ( t ) is Hölder continuous with

exponent p − 1 p in [ ϵ , T ] ; if moreover x ∈ D ( A ) , then u is Hölder continuous

with the same exponent in [ 0, T ] . If in addition f is Hölder continuous, then

Pazy showed that there are some further regularity of A u ( t ) and d u d t . Li [

gave similar results for fractional differential equations with order α ∈ ( 1,2 ) . In this paper we will extend their results to fractional Cauchy problems with order in ( 0,1 ) .

Our paper is organized as follows. In Section 2 there are some preliminaries on fractional derivatives, fractional Cauchy problems and fractional resolvent families. In Section 3 we give the regularity of the mild solution under the condition that f ∈ L p ( [ 0, T ] , X ) . And some further continuity results are given in Section 4.

Let A be a closed densely defined linear operator on a Banach space X. In this paper we consider the following equation:

D t α u ( t ) = A u ( t ) + f ( t ) , t ∈ ( 0 , T ] u ( 0 ) = x 0 (2.1)

where u and f are X-valued functions, 0 < α < 1 , and D t α is the Caputo fractional derivative defined by

D t α f ( t ) : = ∫ 0 t g 1 − α ( t − s ) f ′ ( s ) d s ,

in which for α > 0 ,

g α ( t ) : = { t α − 1 Γ ( α ) , t > 0 0 , t ≤ 0 ,

and g 0 ( t ) is understood as the Dirac measure δ at 0. The convolution of two functions f and g is defined by

( f ∗ g ) ( t ) = ∫ 0 t f ( t − s ) g ( s ) d s = ∫ 0 t f ( s ) g ( t − s ) d s

when the above integrals exist.

The classical (or strong) solution to (2.1) is defined as:

Definition 2.1. If 0 < α ≤ 1 , u ∈ C ( [ 0, T ] , X ) is called a solution of (2.1) if

1) u ∈ C ( [ 0, T ] , D ( A ) ) .

2) ( g 1 − α ∗ ( u − x 0 ) ) ( t ) ∈ C 1 ( [ 0, T ] , X ) .

3) u satisfies (2.1) on [ 0, T ] .

By integration (2.1) for α-times, we are able to define a kind of weak solutions.

Definition 2.2. If 0 < α ≤ 1 , u ∈ C ( [ 0, T ] , X ) is called a mild solution of (2.1) if ( g α ∗ u ) ( t ) ∈ D ( A ) for every t ∈ [ 0, T ] and

u ( t ) = x 0 + A ( g α ∗ u ) ( t ) + ( g α ∗ f ) ( t ) .

And it is therefore natural to give the following definition of α-resolvent family for the operator A.

Definition 2.3. A family { S α ( t ) } t ≥ 0 ⊂ B ( X ) is called an α-resolvent family for the operator A if the following conditions are satisfied:

1) S α ( t ) x : ℝ + → X is continuous for every x ∈ X and S α ( 0 ) = I ;

2) S α ( t ) D ( A ) ⊂ D ( A ) and A S α ( t ) x = S α ( t ) A x for all x ∈ D ( A ) and t ≥ 0 ;

3) the resolvent equation

S α ( t ) x = x + ( g α ∗ S α ) ( t ) A x

holds for every x ∈ D ( A ) .

If there is an α-times resolvent family S α ( t ) for the operator A, then the mild solution of (2.1) is given by the following lemma.

Lemma 2.4. [

u ( t ) = S α ( t ) x 0 + d d t ( g α ∗ S α ∗ f ) ( t ) , t ≥ 0.

For the strong solution of (2.1), we have

Lemma 2.5. [

(a) (2.1) has a strong solution on [ 0, T ] .

(b) S α ∗ f is differentiable on [ 0, T ] .

(c) d d t ( g α ∗ S α ∗ f ) ( t ) ∈ D ( A ) for t ∈ [ 0, T ] and A ( d d t ( g α ∗ S α ∗ f ) ( t ) ) is

continuous on [ 0, T ] .

If in addition, the α-times resolvent family S α ( t ) admits an analytic extension to some sector Σ θ + π / 2 : = { λ ∈ ℂ : | arg ( λ ) | < θ + π / 2 } , and | | S α ( t ) | | ≤ M e ω R e t for all t ∈ Σ θ + π / 2 , we will then denote it by A ∈ A α .

If A ∈ A α , then there exists constants C, ω and θ 0 such that λ α ∈ ρ ( A ) and

| | λ α − 1 R ( λ α , A ) | | ≤ C | λ − ω | (2.2)

for each λ ∈ ω + Σ θ 0 + π / 2 . The α-times resolvent family generated by A can be given by

S α ( t ) = 1 2 π i ∫ Γ e λ t λ α − 1 R ( λ α , A ) d λ , t > 0

where

Γ : = { ω + r e − i ( 2 / π ± δ ) : ρ ≤ r < ∞ } ∪ { ω + ρ e i ϕ : | ϕ | ≤ π / 2 + δ }

is oriented counter-clockwise. And the corresponding operators P α ( t ) are defined by

P α ( t ) = 1 2 π i ∫ Γ e λ t R ( λ α , A ) d λ , t > 0.

Lemma 2.6. Let 0 < α < 1 and A ∈ A α . We have

(1) P α ( t ) ∈ B ( X ) for every t > 0 and | | P α ( t ) | | ≤ C e ω t ( 1 + t α − 1 ) for t > 0 ;

(2) for every x ∈ X , P α ( t ) x ∈ D ( A ) and | | A P α ( t ) | | ≤ C e ω t ( 1 + t − 1 ) for t > 0 ;

(3) S ′ α ( t ) = − A ( g α − 1 ∗ S α ) ( t ) = A P α ( t ) for t > 0 , R ( P α ( l ) ( t ) ) ⊂ D ( A ) for any integer l ≥ 0 and | | A k P α ( l ) ( t ) | | ≤ C α e ω t ( 1 + t − l − 1 − α ( k − 1 ) ) for t > 0 , where k = 0 , 1 .

Proof. (1) By the definition of P α ( t ) and (2.2),

| | P α ( t ) | | ≤ 1 2 π ∫ Γ e R e ( λ t ) ⋅ | λ | 1 − α | λ − ω | | d λ | ≤ 1 2 π ∫ Γ e R e ( λ t ) ⋅ c ( | λ − ω | 1 − α + 1 ) | λ − ω | | d λ | ≤ 1 2 π ( ∫ Γ e R e ( λ t ) | d λ | | λ − ω | α + ∫ Γ e R e ( λ t ) | d λ | | λ − ω | ) .

Since

∫ Γ e R e ( λ t ) | d λ | | λ − ω | α ≤ 2 ∫ ρ ∞ e ω t e − r t s i n δ d r r α + ∫ 0 π e ω t e ρ t c o s ϕ ρ 1 − α d ϕ ,

taking ρ = 1 / t , we can obtain that the above integral is bounded by

2 e ω t ∫ 1 ∞ e − r s i n δ t α − 1 d r r α + e ω t t α − 1 ∫ 0 π e c o s φ d ϕ ≤ C e ω t t α − 1 .

Analogously one can show the estimate

∫ Γ e R e ( λ t ) | d λ | | λ − ω | ≤ C e ω t .

It thus follows the estimate for | | P α ( t ) | | .

(2) By the identity A R ( λ α , A ) = λ α R ( λ α , A ) − I , we have

∫ Γ e λ t A R ( λ α , A ) d λ = ∫ Γ e λ t λ α R ( λ α , A ) d λ − ∫ Γ e λ t d λ = ∫ Γ e λ t λ α R ( λ α , A ) d λ ,

since ∫ Γ e λ t d λ = 0 . Moreover,

| | ∫ Γ e λ t λ α R ( λ α , A ) d λ | | ≤ ∫ Γ e R e ( λ t ) | λ | α | d λ | | λ | α − 1 | λ − ω | = ∫ Γ e R e ( λ t ) | λ | | d λ | | λ − ω | ≤ ∫ Γ e R e ( λ t ) | d λ | + ∫ Γ e R e ( λ t ) | ω | | d λ | | λ − ω | ≤ C e ω t ( t − 1 + 1 ) .

By the closedness of the operator A, the assertion of (2) follows.

(3) By the proof of (2) and the closedness of A,

S ′ α ( t ) = ∫ Γ e λ t λ α R ( λ α , A ) d λ = ∫ Γ e λ t A R ( λ α , A ) d λ = A ∫ Γ e λ t R ( λ α , A ) d λ = A P α ( t ) .

And the second part of (3) can be proved similarly. □

Remark 2.7. Similar results for α ∈ ( 1,2 ) were given in [

P α ( t ) = ( g α − 1 ∗ S α ) ( t )

if 1 < α < 2 and

S α ( t ) = ( g 1 − α ∗ P α ) ( t )

if 0 < α < 1 .

In this section we consider the mild solution of (2.1) with 0 < α < 1 . Suppose that the operator A generates an analytic α-resolvent family, then by Lemma 2.4 and Remark 2.7 the mild solution of (2.1) is given by

u ( t ) = S α ( t ) x 0 + ( P α ∗ f ) ( t ) . (3.1)

Theorem 3.1. Let 0 < α < 1 , A ∈ A α , and f ∈ L p ( [ 0, T ] , X ) with p > 1 / α .

Then for every x 0 ∈ X and ϵ > 0 , u ∈ C α − 1 p ( [ ϵ , T ] , X ) , where u ( t ) is given

by (3.1). If moreover x 0 ∈ D ( A n ) such that n α ≥ 1 , then u ∈ C α − 1 p ( [ 0, T ] , X ) .

Proof. Since S α ( t ) is analytic, we only need to show that ( P α ∗ f ) ( t ) ∈ C α − 1 p .

Let h > 0 and t ∈ [ 0, T − h ] , then

( P α ∗ f ) ( t + h ) − ( P α ∗ f ) ( t ) = ∫ 0 t + h P α ( t + h − s ) f ( s ) d s − ∫ 0 t P α ( t − s ) f ( s ) d s = ∫ t t + h P α ( t + h − s ) f ( s ) d s + ∫ 0 t ( P α ( t + h − s ) − P α ( t − s ) ) f ( s ) d s = I 1 + I 2 .

By Hölder’s inequality and Lemma 2.6,

| | I 1 | | ≤ | | f | | L p ( ∫ t t + h | | P α ( t + h − s ) | | p p − 1 d s ) p − 1 p ≤ C | | f | | L p | | ( ∫ 0 h s p ( α − 1 ) p − 1 d s ) p − 1 p = C | | f | | L p h p α − 1 p .

We remark that the constant C here and in the sequel may be vary line by line, but not depending on t and h. Next, we estimate I 2 . For h > 0 , t ∈ [ 0, T ] , first assume that t > h ,

| | I 2 | | = | | ∫ 0 t ( P α ( t + h − s ) − P α ( t − s ) ) f ( s ) d s | | ≤ ∫ 0 t | | P α ( s + h ) − P α ( s ) | | ⋅ | | f ( t − s ) | | d s ≤ [ ∫ 0 h ( | | P α ( s + h ) | | + | | P α ( s ) | | ) p p − 1 d s ] p − 1 p ⋅ | | f | | L p + [ ∫ h t ( | | P α ( s + h ) − P α ( s ) | | ) p p − 1 d s ] p − 1 p ⋅ | | f | | L p

≤ C | | f | | L p { [ ∫ 0 h ( 2 s α − 1 ) p p − 1 d s ] p − 1 p + [ ∫ h t ( h s α − 2 ) p p − 1 d s ] p − 1 p } ≤ C | | f | | L p [ h p α − 1 p + ( ∫ 1 t h h α p − 1 p − 1 τ ( α − 2 ) p p − 1 d τ ) p − 1 p ] ≤ C | | f | | L p h α p − 1 p [ 1 + ( ∫ 1 ∞ τ ( α − 2 ) p p − 1 d τ ) p − 1 p ] ≤ C | | f | | L p h α p − 1 p ,

since

from which it follows also that

If

If we put more conditions on

Proposition 3.2. Let

If

Proof. If

Since

that

proof. □

Motivated by the results in [_{0}-semigroups, we first give the following proposition.

Proposition 4.1. Let

Proof. By Lemma 2.5 we only need to show that

where

belongs to

and

for

for all

By our assumption and Lemma 2.6 there exists a constant

consequently, the function

The continuity of the function

This completes the proof. □

We will then give the regularity of such classical solutions.

Lemma 4.2. Let

Then for any

Proof. For fixed

By the closedness of A, we obtain

Since

We can estimate

And it is easy to show that

The following theorem extends [[

Theorem 4.3. Let

(1) For every

(2) If moreover

(3) If

Proof. (1) If u is the classical solution of (2.1) on

By Lemma 4.2,

thus we have

(2) We only need to show that

as

(3) We show that

In this paper, we proved the Hölder regularity of the mild and strong solutions to the α-order abstract Cauchy problem (2.1) with

Li, C.Y. and Li, M. (2018) Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1). Journal of Applied Mathematics and Physics, 6, 310-319. https://doi.org/10.4236/jamp.2018.61030