In this article, we study the existence and uniqueness of a local mild solution for a class of semilinear differential equations involving the Ca- puto fractional time derivative of orderα(0&lt

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE AND UNIQUENESS OF MILD SOLUTIONS FOR FRACTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS

BAMBANG HENDRIYA GUSWANTO, TAKASHI SUZUKI

Abstract. In this article, we study the existence and uniqueness of a local mild solution for a class of semilinear differential equations involving the Ca- puto fractional time derivative of orderα(0< α <1) and, in the linear part, a sectorial linear operatorA. We put some conditions on a nonlinear term f and an initial datau0 in terms of the fractional power ofA. By applying Banach’s Fixed Point Theorem, we obtain a unique local mild solution with smoothing effects, estimates, and a behavior attclose to 0. An example as an application of our results is also given.

1. Introduction

Some existing researches showed that, in diffusion process, there are particle’s movements that can be no longer modelled by the (normal) diffusion equation. To see these phenomenons, one can refer to [1, 3, 8, 16] observing the dispersion in a heterogeneous aquifer, the transport of contaminants in geological formations, the dispersive transport of ions in column experiments, and the diffusion of water in sand, respectively. All of these processes follow the pattern

hx²(t)i ∼t^α, 0< α <1, (1.1) wherehx²(t)iis the mean square displacement at timet. These processes are called subdiffusion and can be modelled by the equation

D^α_tu(x, t) =Dα∆u(x, t), x∈Rⁿ, t >0, (1.2) where 0< α <1,D_α is a subdiffusion coeficient, andD_t^α is the Caputo fractional derivative of orderα. Reaction subdiffusion equation was also derived (see [2, 9, 10, 11, 17, 18, 22, 27, 28]). Subdifusion model can also be a formula for memory phenomenon (see [13, 21]). In [5], Du et al. also found that the order of fractional derivative is an index of memory. Thus a study to investigate a solution to this model is very useful. Recently, there are some researches studying a solution to fractional evolution equations, for instance, see [4, 6, 19, 20, 24, 26, 29, 31, 32, 33].

2010Mathematics Subject Classification. 34A08, 34A12.

Key words and phrases. Fractional semilinear differential equation; sectorial operator;

Caputo fractional derivative; fractional power; mild solution.

c

2015 Texas State University - San Marcos.

Submitted April 2, 2015. Published June 18, 2015.

1

In this article, we show the existence and uniqueness of a local mild solution to the fractional abstract Cauchy problem

D^α_tu=Au+f(u), t >0, 0< α <1,

u(0) =u₀, (1.3)

where H is a Banach space, D_t^α is the Caputo fractional derivative of order α, A : D(A) →H is a sectorial linear operator, u₀ ∈ H, and f : H → H. We use some conditions onf andu₀in terms of the fractional power ofA. The conditions are

(i) f(0) = 0,

(ii) there existC₀>0,ϑ >1, and 0< β <1 such that

kf(u)−f(v)k ≤C0(kA^βuk+kA^βvk)^ϑ−1kA^βu−A^βvk for allu, v∈D(A^β),

(iii) u0∈D(A^ν) for some 0< ν <1.

These conditions are used to study the solvability and smoothing effect for some class of semilinear parabolic equations (see [14]). As in [14], we apply Banach’s Fixed Point Theorem to construct a local mild solution to the problem (1.3) by employing the properties of solution operators generated by A and the fractional power ofA. In this paper, we obtain the existence and uniqueness of the local mild solution with smoothing effects, estimates, and a behaviour at t close to 0 as the advantages of our results compared with the preceding related results.

This article is composed of four sections. In section 2, we introduce briefly the fractional integration and differentiation of Riemann-Liouville and Caputo opera- tors. In this section, we also provides some properties of analytic solution operators for fractional evolution equations including some estimates involving the fractional power of sectorial operators. In next section, our main results are showed. Finally, in the last section, an application of our main results is given.

2. Preliminaries

2.1. Fractional time derivative. Let 0< α <1,a≥0 andI = (a, T) for some T >0. TheRiemann-Liouville fractional integral of orderαis defined by

J_a,t^α f(t) = Z t

a

(t−s)^α−1

Γ(α) f(s)ds, f ∈L¹(I), t > a. (2.1) We setJ_a,t⁰ f(t) =f(t). The fractional integral operator (2.1) obeys the semigroup property

J_a,t^α J_a,t^β =J_a,t^α+β, 0≤α, β <1. (2.2) TheRiemann-Liouville fractional derivative of orderαis defined by

D_a,t^α f(t) =Dt

Z t

a

(t−s)^−α

Γ(1−α)f(s)ds, f ∈L¹(I), t^−α∗f ∈W^1,1(I), t > a, (2.3) where∗ denotes the convolution of functions

(f∗g)(t) = Z t

a

f(t−τ)g(τ)dτ, t > a,

and W^1,1(I) is the set of all functions u ∈ L¹(I) such that the distributional derivative of uexists and belongs to L¹(I). The operator D_a,t^α is a left inverse of J_a,t^α ; that is,

D^α_a,tJ_a,t^α f(t) =f(t), t > a, but it is not a right inverse, that is

J_a,t^α D_a,t^α f(t) =f(t)−(t−a)^α−1

Γ(α) J_a,t^1−αf(a), t > a.

TheCaputo fractional derivative of orderαis defined by D_a,t^α f(t) =D_t

Z t

a

(t−s)^−α

Γ(1−α)(f(s)−f(0))ds, t > a, (2.4) iff ∈L¹(I), t^−α∗f ∈W^1,1(I), or

D^α_a,tf(t) = Z t

a

(t−s)^−α

Γ(1−α)Dsf(s)ds, t > a, (2.5) iff ∈W^1,1(I). The operatorD^α_a,tis also a left inverse ofJ_a,t^α , that is

D_a,t^α J_a,t^α f(t) =f(t), t > a, (2.6) but it is not also a right inverse, that is

J_a,t^α D^α_a,tf(t) =f(t)−f(a), t > a. (2.7) The relation between the Riemann-Liouville and Caputo fractional derivative is

D^α_a,tf(t) =D^α_a,tf(t)−(t−a)^−α

Γ(1−α)f(a), t > a. (2.8) For a = 0, we set J_a,t^α = J_t^α, D_a,t^α = D^α_t, and D^α_a,t = D_t^α. We refer to Kilbas et al [15] or Podlubny [25] for more details concerning the fractional integrals and derivatives.

2.2. Analytic solution operators. In this section, we provide briefly some results concerning solution operators for the fractional Cauchy problem

D^α_tu(t) =Au(t) +f(t), t >0,

u(0) =u₀. (2.9)

For more details, we refer to Guswanto [7].

Henceforth, we assume that the linear operatorA:D(A)⊂H →H satisfies the properties that there is a constantθ∈(π/2, π) such that

ρ(A)⊃Sθ={λ∈C:λ6= 0,|arg(λ)|< θ}, (2.10) kR(λ;A)k ≤ M

|λ|, λ∈S_θ, (2.11)

where R(λ;A) = (λ−A)⁻¹ and ρ(A) are the resolvent operator and resolvent set of A, respectively. We call A as a sectorial operator. Every operator satisfying this property is closed since its resolvent set is not empty. The linear operator A generates solution operators for the problem (2.9), those are

Sα(t) = 1 2πi

Z

Γr,ω

e^λtλ^α−1R(λ^α;A)dλ, t >0, (2.12) Pα(t) = 1

2πi Z

Γr,ω

e^λtR(λ^α;A)dλ, t >0, (2.13)

wherer >0,π/2< ω < θ, and

Γ_r,ω={λ∈C:|arg(λ)|=ω,|λ| ≥r} ∪ {λ∈C:|arg(λ)| ≤ω,|λ|=r}

is oriented counterclockwise. By the Cauchy’s theorem, the integral form (2.12) and (2.13) are independent ofr >0 andω∈(π/2, θ).

LetB(H) be the set of all bounded linear operators onH. The properties of the families{Sα(t)}t>0 and{Pα(t)}t>0are given in the following theorems.

Theorem 2.1. Let A be a sectorial operator and Sα(t) be an operator defined by (2.12). Then the following statements hold.

(i) Sα(t)∈B(H) and there exists a constantC1=C1(α)>0 such that kS_α(t)k ≤C₁, t >0,

(ii) Sα(t)∈B(H;D(A))fort >0, and if x∈D(A) thenASα(t)x=Sα(t)Ax.

Moreover, there exists a constantC2=C2(α)>0 such that kAS_α(t)k ≤C₂t^−α, t >0,

(iii) The functiont7→Sα(t)belongs toC^∞((0,∞);B(H))and it holds that S_α⁽ⁿ⁾(t) = 1

2πi Z

Γr,ω

e^tλλ^α+n−1R(λ^α;A)dλ, n= 1,2, . . . and there exist constants Mn =Mn(α)>0, n= 1,2, . . .such that

kS_α⁽ⁿ⁾(t)k ≤Mnt⁻ⁿ, t >0,

Moreover, it has an analytic continuation Sα(z) to the sectorS_θ−π/2 and, forz∈S_θ−π/2,η∈(π/2, θ), it holds that

Sα(z) = 1 2πi

Z

Γr,η

e^λzλ^α−1R(λ^α;A)dλ.

Theorem 2.2. Let A be a sectorial operator and Pα(t) be an operator defined by (2.13). Then the following statements hold.

(i) P_α(t)∈B(H)and there exists a constant L₁=L₁(α)>0such that kPα(t)k ≤L1t^α−1, t >0,

(ii) Pα(t) ∈ B(H;D(A)) for all t > 0, and if x ∈ D(A) then APα(t)x = Pα(t)Ax. Moreover, there exists a constantL2=L2(α)>0 such that

kAP_α(t)k ≤L₂t⁻¹, t >0,

(iii) The functiont7→Pα(t)belongs to C^∞((0,∞);B(H))and it holds that P_α⁽ⁿ⁾(t) = 1

2πi Z

Γr,ω

e^tλλⁿR(λ^α;A)dλ, n= 1,2, . . . and there exist constants Kn =Kn(α)>0, n= 1,2, . . .such that

kP_α⁽ⁿ⁾(t)k ≤Knt^α−n−1, t >0,

Moreover, it has an analytic continuation Pα(z) to the sectorS_θ−π/2 and, forz∈S_θ−π/2,η∈(π/2, θ), it holds that

Pα(z) = 1 2πi

Z

Γr,η

e^λzR(λ^α;A)dλ.

The following theorem states some identities concerning the operatorsSα(t) and Pα(t) including the semigroup-like property.

Theorem 2.3. LetAbe a sectorial operator,Sα(t)andPα(t)be operators defined by (2.12)and (2.13), respectively. Then the following statements hold.

(i) Forx∈H andt >0,

Sα(t)x=J_t^1−αPα(t)x, DtSα(t)x=APα(t)x, (ii) Forx∈D(A) ands, t >0,

D^α_tSα(t)x=ASα(t)x, Sα(t+s)x=Sα(t)Sα(s)x−A

Z t

0

Z s

0

(t+s−τ−r)^−α

Γ(1−α) Pα(τ)Pα(r)x dr dτ.

The next theorem shows us the behavior of the operatorS_α(t) att close to 0⁺. Theorem 2.4. Let A be a sectorial operator and S_α(t) be an operator defined by (2.12). Then the following statements hold.

(i) If x∈D(A)then lim_t→0+Sα(t)x=x.

(ii) For everyx∈D(A)andt >0, Z t

0

(t−τ)^α−1

Γ(α) Sα(τ)xdτ ∈D(A), Z t

0

(t−τ)^α−1

Γ(α) ASα(τ)xdτ =Sα(t)x−x, (iii) If x∈D(A)andAx∈D(A)then

lim

t7→0⁺

S_α(t)x−x

t^α = 1

Γ(α+ 1)Ax.

The representation of the solution to (2.9) in term ofSα(t) andPα(t) is given in the following theorem.

Theorem 2.5. Letu∈C¹((0,∞);H)∩L¹((0,∞);H),u(t)∈D(A)fort∈[0,∞), Au ∈ L¹((0,∞);H), f ∈ L¹((0,∞);D(A)), and Af ∈ L¹((0,∞);H). If u is a solution to the problem (2.9)then

u(t) =S_α(t)u₀+ Z t

0

P_α(t−s)f(s)ds, t >0. (2.14) Now, we consider the fractional power of operatorA

A^−βx= 1 2πi

Z

Γ_r,ω

λ^−βR(λ;A)xdλ, x∈H, β >0, and

A^βx=A(A^β−1x) = 1 2πi

Z

Γr,ω

λ^β−1R(λ;A)Axdλ, x∈D(A), 0< β <1.

Some estimates involvingA^βand the operators families{Sα(t)}t>0,{Pα(t)}t>0gen- erated by the sectorial operator A are provided by the following theorem. These estimates are analogous to those as stated in [23, Theorem 6.13] for analytic semi- groups.

Theorem 2.6. For each0< β <1, there exist positive constantsC₁⁰ =C₁⁰(α, β), C₂⁰ =C₂⁰(α, β), andC₃⁰ =C₃⁰(α, β)such that for all x∈H,

kA^βSα(t)xk ≤C₁⁰t^−α(t^{−α(β−1)}+ 1)kxk, t >0, (2.15) kA^βPα(t)xk ≤C₂⁰t^{−α(β−1)−1}kxk, t >0. (2.16) Moreover, for all x∈D(A^β),

kSα(t)x−xk ≤C₃⁰t^αβkA^βxk, t >0. (2.17) Now, letξ_ζ =α(ζ−1) + 1, for 0< ζ <1, andx⁺= max{0, x}, forx∈R. Thus we have the following result.

Corollary 2.7. For eachβ >(2−1/α)⁺ orβ= 2−1/α >0 andx∈H,

t^ξβkA^βS_α(t)xk ≤2C₁⁰kxk, 0< t≤1, (2.18) t^ξβkA^βSα(t)xk ≤2C₁⁰t^1−αkxk, t >1, (2.19) t^ξβkA^βP_α(t)xk ≤C₂⁰kxk, t >0, (2.20) t^ξβkA^βSα(t)xk →0, ast→0⁺. (2.21) Furthermore, we have the same result as Theorem 2.3 (ii) with weaker condition.

Theorem 2.8. Let 0< β <1. Then, forx∈D(A^β)ands, t >0,

D^α_tS_α(t)x=AS_α(t)x, (2.22) Sα(t+s)x=Sα(t)Sα(s)x−A

Z t

0

Z s

0

(t+s−τ−r)^−α

Γ(1−α) Pα(τ)Pα(r)x dr dτ. (2.23) 3. Main results

In this section, we show the existence and uniqueness of a mild solution for the problem (1.3) under certain conditions by applying Banach’s Fixed Point Theorem.

Based on Theorem 2.5, we define a mild solution to the problem (1.3) as follows.

Definition 3.1. A continuous function u : (0, T] → H is a mild solution to the problem (1.3) if it satisfies

u(t) =S_α(t)u₀+ Z t

0

P_α(t−s)f(u(s))ds, 0< t≤T.

The conditions onf are:

(i) f(0) = 0,

(ii) there existC0>0,ϑ >1, and 0< β <1 such that

kf(u)−f(v)k ≤C₀(kA^βuk+kA^βvk)^ϑ−1kA^βu−A^βvk, for allu, v∈D(A^β).

Let BC((0, T];D(A^β)) be the set of all bounded and continuous functions w : (0, T]→D(A^β). Under the conditions onf above, we obtain the following results.

Theorem 3.2. Let u0∈D(A^ν)with

β−ν >(2−1/α)⁺, 1−αν−ϑξ_β−ν ≥0, 0< ϑξ_β−ν <1, (3.1) where

ξ_ζ =α(ζ−1) + 1, 0< ζ <1; x⁺= max{0, x}, x∈R.

Then there exits T >0 sufficiently small such that the problem (1.3)has a unique mild solutionusatisfying

t^ξη−νu∈BC((0, T];D(A^η)), lim

t→0⁺t^ξη−νA^ηu(t) = 0, kA^ηu(t)k ≤Ct^−ξη−νkA^νu0k, t∈(0, T], for everyη ∈(ν+ (2−1/α)⁺, β].

Theorem 3.3. Let ube the mild solution to the problem (1.3)in Theorem 3.2. If f(u(t))∈D(A), for t∈(0,∞), then

t^ξ1−νu∈BC((0, T];D(A)) with

kAu(t)k ≤Ct^−ξ1−νkA^νu0k, t∈(0, T].

3.1. Proof of Theorem 3.2. We define first the Banach space Eβ,T ={u: [0, T]→H :t^ξβ−νu∈BC((0, T];D(A^β))}

equipped with the norm

k|u|kβ,T = sup

0<t≤T

t^ξβ−νkA^βu(t)k, (3.2) and define a closed ballB_β,T in E_β,T by

Bβ,T ={u∈Eβ,T :k|u|kβ,T ≤K}, whereT andK are some constants which will be specified later.

Next, we define a mappingF onBβ,T by F u(t) =Sα(t)u0+

Z t

0

Pα(t−s)f(u(s))ds.

First, we prove the continuity ofA^βF u(t) with respect totin (0, T]. SinceA^β is a bounded operator onD(A) and, for eachx∈H,Sα(t)xis continuous with respect to t in (0,∞), then, for each x∈H, A^βSα(t)xis continuous with respect to t in (0,∞). Thus it remains to show the continuity of

A^β Z t

0

P_α(t−s)f(u(s))ds, 0< t≤T.

Note that A^β

Z t+h

0

Pα(t+h−s)f(u(s))ds−A^β Z t

0

Pα(t−s)f(u(s))ds

=A^β Z t

−h

P_α(t−s)f(u(s+h))ds−A^β Z t

0

P_α(t−s)f(u(s))ds

=A^β Z t

0

P_α(t−s)(f(u(s+h))−f(u(s)))ds +A^β

Z h

0

Pα(t+h−s)f(u(s))ds.

Observe that, foru∈E_β,T,

kf(u(t+h))−f(u(t))k ≤C₀2^ϑ−1K^ϑ−1t^{−(ϑ−1)ξβ−ν}kA^βu(t+h)−A^βu(t)k (3.3)

and

kf(u(t))k ≤C0kA^βu(t)k^ϑ≤C0t^{−ϑξβ−ν}k|u|k^ϑ_β,T ≤C0K^ϑt^{−ϑξβ−ν}, (3.4) for 0< t≤T. Next, we have

Z t

0

kA^βPα(t−s)(f(u(s+h))−f(u(s)))kds

≤2^ϑ−1C0C2K^ϑ−1 Z t

0

(t−s)^−ξβs^{−(ϑ−1)ξβ−ν}kA^βu(s+h)−A^βu(s)kds.

Now, consider that, for 0< s < t≤T,

(t−s)^−ξβs^{−(ϑ−1)ξβ−ν}kA^βu(s+h)−A^βu(s)k ≤2K(t−s)^−ξβs^{−ϑξβ−ν}, s7→2K(t−s)^−ξβs^{−ϑξβ−ν} ∈L¹((0, t);H), 0< t≤T,

kA^βu(s+h)−A^βu(s)k →0,ash→0.

Hence, by the Dominated Convergence theorem, Z t

0

(t−s)^−ξβs^{−(ϑ−1)ξβ−ν}kA^βu(s+h)−A^βu(s)kds→0, as h→0.

This implies Z t

0

kA^βPα(t−s)(f(u(s+h))−f(u(s)))kds→0, as h→0.

Next, observe that Z h

0

kA^βPα(t+h−s)kkf(u(s)k)ds

≤C₀C₂⁰(α, β)K^ϑ Z h

0

(t+h−s)^−ξβs^{−ϑξβ−ν}ds

=C0C₂⁰(α, β)K^ϑ(t+h)^{1−ξβ−ϑξβ−ν} Z _t+h^h

0

(1−r)^−ξβr^{−ϑξβ−ν}dr

=C0C₂⁰(α, β)K^ϑ(t+h)^{1−ξβ−ϑξβ−ν} 1 1−ϑξ_β−ν

× h t+h

1−ϑξβ−ν

H

1−ϑξ_β−ν, ξ_β; 2−ϑξ_β−ν; h t+h

= C₀C₂⁰(α, β)K^ϑ

1−ϑξ_β−ν h^{1−ϑξβ−ν}(t+h)^−ξβH

1−ϑξβ−ν, ξβ; 2−ϑξβ−ν; h t+h

, where

H(a, b;c;x) = Γ(c) Γ(b)Γ(c−b)

Z 1

0

t^b−1(1−t)^c−b−1

(1−xt)^a , c−b−a >0, |x| ≤1 is hypergeometric function (see [15]). Thus

Z h

0

kA^βPα(t+h−s)kkf(u(s)kds→0, as h→0.

Therefore the continuity ofA^βF u(t) with respect totin (0, T] is obtained.

Next, we prove that the mapping F is well-defined and maps Bβ,T into itself.

Consider Z t

0

kA^βPα(t−s)kkf(u(s))kds

≤C₀C₂⁰(α, β)K^ϑ−1k|u|kβ,T

Z t

0

(t−s)^−ξβs^{−ϑξβ−ν}ds

≤C₀C₂⁰(α, β)K^ϑ−1B(1−ϑξ_β−ν,1−ξ_β)k|u|kβ,Tt^{1−ξβ−ϑξβ−ν}, where

B(a, b) = Z 1

0

r^a−1(1−r)^b−1dr, a, b >0, is Beta function. Therefore

t^ξβ−νkA^βF u(t)k ≤t^ξβ−νkA^βSα(t)u0k+C4K^ϑ−1k|u|kβ,Tt^{1−ξβ−ϑξβ−ν+ξβ−ν}, (3.5) whereC₄=C₀C₂⁰(α, β)B(1−ξ_β,1−ϑξ_β−ν), implying

k|F u|k_β,T ≤ sup

0<t≤T

t^ξβ−νkA^βS_α(t)u₀k+C₄K^ϑ−1T^{1−ξβ−ϑξβ−ν+ξβ−ν}k|u|k_β,T. (3.6) Note that 1−ξβ−ϑξβ−ν+ξβ−ν = 1−αν−ϑξβ−ν ≥0 by (3.1). By (2.18), we can find 0< T≤1 such that

t^ξβ−νkA^β−νSα(t)A^νu0k ≤2C₁⁰(α, β−ν)kA^νu0k, 0< t≤T.

Then, foru∈Bβ,T, we have k|F u|kβ,T ≤ sup

0<t≤T

t^ξβ−νkA^β−νS_α(t)A^νu₀k+C₄K^ϑT^{1−αν−ϑξβ−ν}

≤2C₁⁰(α, β−ν)kA^νu0k+C4K^ϑT^{1−αν−ϑξβ−ν}.

(3.7) Next, we chooseK >0 such that

2C₁⁰(α, β−ν)kA^νu0k+C4K^ϑT^{1−αν−ϑξβ−ν} ≤K. (3.8) For the case 1−αν−ϑξβ−ν>0, we can get such aKby takingT sufficiently small.

For the case 1−αν−ϑξβ−ν = 0, we chooseK >0 sufficiently small such that C₄K^ϑ< K,

and then takeT such that sup

0<t≤T

t^ξβ−νkA^β−νSα(t)A^νu0k ≤K−C4K^ϑ. (3.9) Note that in both cases, we can findC=C(α, β)>0 such that

K≤CkA^νu₀k. (3.10)

Hence k|F u|kβ,T ≤ K. Thus the mapping F is well-defined and maps Bβ,T into itself.

Next, we show that the mappingF :Bβ,T →Bβ,T is a strict contraction. Note that, ifu, v∈Bβ,T, we have

kA^βF u(t)−A^βF v(t)k

≤ Z t

0

kA^βPα(t−s)kkf(u(s))−f(v(s))kds

≤C0C₂⁰(α, β) Z t

0

(t−s)^−ξβ k|u|kβ,T+k|v|kβ,T

ϑ−1

×s^{−(ϑ−1)ξβ−ν}k|u−v|kβ,Ts^−ξβ−νds

≤C0C₂⁰(α, β)2^ϑ−1K^ϑ−1 Z t

0

(t−s)^−ξβs^{−ϑξβ−ν}dsk|u−v|kβ,T

≤C42^ϑ−1K^ϑ−1k|u−v|kβ,Tt^{1−ξβ−ϑξβ−ν}. Then

t^ξβ−νkA^βF u(t)−A^βF v(t)k ≤C42^ϑ−1K^ϑ−1k|u−v|kβ,Tt^{1−αν−ϑξβ−ν}

≤C42^ϑ−1K^ϑ−1k|u−v|kβ,TT^{1−αν−ϑξβ−ν}. Note that we can selectK >0 andT >0 sufficiently small such that

C5=C42^ϑ−1K^ϑ−1T^{1−αν−ϑξβ−ν} <1. (3.11) Consequently,

k|F u−F v|k ≤C5k|u−v|kβ,T.

It means the mappingF :Bβ,T →Bβ,T is a strict contraction. Thus, by Banach’s Fixed Point Theorem, we can get a uniqueu∈Bβ,T which is a mild solution to the problem (1.3). Furthermore, by (3.7), (3.8), (3.9), and (3.10), for thisu, we have

k|u|kβ,T ≤ sup

0<t≤T

t^ξβ−νkA^β−νSα(t)A^νu0k+C4K^ϑT^{1−ξβ−ϑξβ−ν+ξβ−ν} ≤CkA^νu0k.

Then, by (3.2),

kA^βu(t)k ≤Ct^−ξβ−νkA^νu₀k, 0< t≤T.

Now, we check the continuity ofuat t= 0. Note that t^ξβ−νkA^βu(t)k ≤t^ξβ−νkA^βSα(t)u0k+t^ξβ−ν

Z t

0

kA^βPα(t−s)kkf(u(s))k

≤t^ξβ−νkA^β−νSα(t)A^νu0k+C4K^ϑt^{1−αν−ϑξβ−ν}.

(3.12)

Thus, if 1−αν−ϑξ_β−ν>0, lettingt→0⁺ on both sides of (3.12), we obtain lim

t→0⁺t^ξβ−νA^βu(t) = 0.

For the case 1−αν−ϑξ_β−ν = 0, consider first that, from (3.6), we have k|u|k_β,T⁰ ≤ sup

0<t≤T⁰

t^ξβ−νkA^β−νS_α(t)A^νu₀k+C₄K^ϑ−1k|u|k_β,T⁰,

for any 0< T⁰ ≤T. Since C₅<1, then C₄K^ϑ−1 <1. Hence there existsC₆ >0 such that

k|u|kβ,T⁰ ≤C6 sup

0<t≤T⁰

t^ξβ−νkA^β−νSα(t)A^νu0k.

By takingT⁰ →0, thus we also have lim

t→0⁺t^ξβ−νA^βu(t) = 0,

for the case 1−αν−ϑξ_β−ν = 0. We can also conclude that the results above also hold for everyη∈(ν+ (2−1/α)⁺, β) since such aη satisfies the condition (3.1).

Remark 3.4. From (2.19), forT >1, we have

t^ξβ−νkA^βSα(t)u0k ≤2C₁⁰(α, β−ν)t^1−αkA^νu0k, t∈(0, T].

Then, it follows that (3.8) becomes

2C₁⁰(α, β−ν)kA^νu₀kT^1−α+C₄K^ϑT^{1−αν−ϑξβ−ν} ≤K. (3.13) Observe that we can not get K >0 satisfying (3.11) and (3.13) for T sufficiently large although K is taken to be sufficiently small. Thus the problem (1.3) has no a global mild solutionuon (0,∞).

Remark 3.5. If we assume thatf is a nonlinear operator inH satisfying (i) f(0) = 0,

(ii) there existC₀>0,ϑ >1, and 0< β <1 such that

kf(u)−f(v)k ≤C0(1 + (kA^βuk+kA^βvk)^ϑ−1)kA^βu−A^βvk, for allu, v∈D(A^β),

then Theorem 3.2 remains valid.

3.2. Proof of Theorem 3.3. We verify first the following lemma.

Lemma 3.6. Let u ∈ B_β,T be a mild solution to (1.3). Then, by the condition (3.1),A^βu(t)is H¨older continuous in[ε, T]for each ε >0.

Proof. First, consider that, by (2.23), A^βS_α(t+h)u₀−A^βS_α(t)u₀

=A^β(S_α(h)−I)S_α(t)u₀−A^β Z t

0

Z h

0

(t+h−τ−r)^−α

Γ(1−α) AP_α(τ)P_α(r)u₀dr dτ and

A^β Z t+h

0

P_α(t+h−s)f(u(s))ds−A^β Z t

0

P_α(t−s)f(u(s))ds

=A^β Z t

−h

Pα(t−s)f(u(s+h))ds−A^β Z t

0

Pα(t−s)f(u(s))ds

=A^β Z t

0

Pα(t−s)(f(u(s+h))−f(u(s)))ds +A^β

Z h

0

P_α(t+h−s)f(u(s))ds.

Now, letε≤t < t+h≤T withε >0. Observe that Z h

0

(t+h−τ−r)^−α

Γ(1−α) τ^−ξ1−δdτ

= h^1−ξ1−δ(t+h−r)^−α Γ(1−α)(1−ξ1−δ) H

1−δ, α; 2−δ; h t+h−r

≤ Γ(2−ξ1−δ)B(α,1−α)

(1−ξ_1−δ)Γ(1−α)Γ(α)Γ(2−ξ_1−δ−α)h^1−ξ1−δ(t+h−r)^−α implying

Z t

0

Z h

0

(t+h−τ−r)^−α

Γ(1−α) τ^−ξ1−δr^{−ξβ+δ−ν}dr dτ

≤ Γ(2−ξ1−δ)B(α,1−α)h^1−ξ1−δ (1−ξ_1−δ)Γ(1−α)Γ(α)Γ(2−ξ_1−δ−α)

Z t

0

(t+h−r)^−αr^{−ξβ+δ−ν}dr

=Γ(2−ξ_1−δ)B(α,1−α)h^1−ξ1−δ(t+h)^{1−α−ξβ+δ−ν} (1−ξ_1−δ)Γ(1−α)Γ(α)Γ(2−ξ_1−δ−α)

Z _t+h^t

0

(1−s)^−αs^{−ξβ+δ−ν}ds

≤C₇h^1−ξ1−δ(t+h)^{1−α−ξβ+δ−ν} where

C7= Γ(2−ξ_1−δ)B(α,1−α)B(1−ξ_β+δ−ν,1−α) (1−ξ_1−δ)Γ(1−α)Γ(α)Γ(2−ξ_1−δ−α) . Then, for every 0< δ <1−β,

kA^βSα(t+h)u0−A^βSα(t)u0k

≤ k(S_α(h)−I)A^βS_α(t)u₀k +

Z t

0

Z h

0

(t+h−τ−r)^−α

Γ(1−α) kA^1−δP_α(τ)A^β+δ−νP_α(r)A^νu₀kdr dτ

≤C₃⁰(α, δ)h^αδkA^β+δ−νSα(t)A^νu0k+C₂⁰(α,1−δ)C₂⁰(α, β+δ−ν)

× Z t

0

Z h

0

(t+h−τ−r)^−α

Γ(1−α) τ^−ξ1−δr^{−ξβ+δ−ν}dr dτkA^νu0k

≤C₁⁰(α, β+δ−ν)C₃⁰(α, δ)h^1−ξ1−δt^−α(t^{1−ξβ+δ−ν} + 1)kA^νu0k +C₂⁰(α,1−δ)C₂⁰(α, β+δ−ν)C7h^1−ξ1−δt^{1−α−ξβ+δ−ν}kA^νu0k

≤C₁⁰(α, β+δ−ν)C₃⁰(α, δ)h^1−ξ1−δt^−α(t^{1−ξβ+δ−ν} + 1)kA^νu₀k +C8h^1−ξ1−δt^−α(t^{1−ξβ+δ−ν} + 1)kA^νu0k

≤C₉h^1−ξ1−δt^−α(t^{1−ξβ+δ−ν} + 1)kA^νu₀k, for some constantsC8, C9>0. Next, note that

Z t

0

kA^βPα(t−s)(f(u(s+h))−f(u(s)))kds

≤2^ϑ−1C0C₂⁰(α, β)K^ϑ−1 Z t

0

(t−s)^−ξβs^{−(ϑ−1)ξβ−ν}kA^βu(s+h)−A^βu(s)kds and

Z h

0

kA^βP_α(t+h−s)f(u(s))kds

≤C0C₂⁰(α, β)K^ϑ Z h

0

(t+h−s)^−ξβs^{−ϑξβ−ν}ds

≤C₁₀(t+h)^{1−ξβ−ϑξβ−ν} Z _t+h^h

0

(1−r)^−ξβr^{−ϑξβ−ν}dr

≤ C10

1−ϑξ_β−ν(t+h)^{1−ξβ−ϑξβ−ν} h t+h

1−ϑξβ−ν

H

1−ϑξ_β−ν, ξβ; 2−ϑξ_β−ν; h t+h

≤C11h^{1−ϑξβ−ν}t^−ξβ,

for some constantsC₁₀,C₁₁>0. Thus we obtain kA^βu(t+h)−A^βu(t)k

≤C₉h^1−ξ1−δt^−α(t^{1−ξβ+δ−ν} + 1)kA^νu₀k+C₁₁h^{1−ϑξβ−ν}t^−ξβ

+ 2C0C2K^ϑ−1 Z t

0

(t−s)^−ξβs^{−(ϑ−1)ξβ−ν}kA^βu(s+h)−A^βu(s)kds By the Gronwall’s inequality, it implies thatA^βu(t) is H¨older continuous on [ε, T]

for anyε >0.

Next, by the Lemma 3.6,f(u(t)) is also H¨older continuous on [ε, T] for anyε >0;

that is,

kf(u(t+h))−f(u(t))k ≤C12

h^1−ξ1−δt^{−α−(ϑ−1)ξβ−ν}(t^{1−ξβ+δ−ν} + 1)kA^νu0k +h^{1−ϑξβ−ν}t^{−ξβ−(ϑ−1)ξβ−ν} ,

for some constant C₁₂ > 0. Note that the assumption (3.1) assures that 0 <

1−ϑξ_β−ν and, for each 0< δ <1−β, it holds that 0<1−ξ_1−δ. Furthermore, consider that, fort∈(0, T], we have

t^ξ1−νkASα(t)u0k ≤C₁⁰(α,1−ν)t^{ξ1−ν−α}(t^1−ξ1−ν + 1)kA^νu0k with

ξ_1−ν−α >0, ξ_1−ν−α+ 1−ξ_1−ν = 1−α >0.

It follows, forT sufficiently small, that

t^ξ1−νkASα(t)u0k ≤2C₁⁰(α,1−ν)kA^νu0k.

Now, observe that A

Z t

0

P_α(t−s)f(u(s))ds

= Z t/2

0

AP_α(t−s)f(u(s))ds +

Z t

t/2

APα(t−s)(f(u(s))−f(u(t)))ds+ (Sα(t/2)−I)f(u(t))

=I₁+I₂+I₃. Next, we note that

t^ξ1−νk(S_α(t/2)−I)f(u(t))k ≤C₁₃t^ξ1−νkf(u(t))k ≤C₀C₁₃K^ϑt^{ξ1−ν−ϑξβ−ν}, for some constantC13>0, and

ξ_1−ν−ϑξ_β−ν = 1−αν−ϑξ_β−ν. Therefore, fort∈(0, T] withT >0 sufficiently small, we have

t^ξ1−νk(Sα(t/2)−I)f(u(t))k ≤2C₀C₁₄K^ϑt^{1−αν−ϑξβ−ν}, for some constantC₁₄>0. Hence, by (3.10), we obtain

t^ξ1−νkI3k ≤C15kA^νu0k, for some constantC15>0. Furthermore,

t^ξ1−νkI1k ≤L2(α)C0K^ϑt^ξ1−ν Z t/2

0

(t−s)⁻¹s^{−ϑξβ−ν}ds

≤C16t^{ξ1−ν−ϑξβ−ν} ≤C16t^{1−αν−ϑξβ−ν},

for some constantC16>0. Thus, forT sufficiently small, we find that t^ξ1−νkI1k ≤C₁₇kA^νu₀k,

for some constantC17>0. Now, consider kAPα(t−s)(f(u(s))−f(u(t)))k

≤L₂(α)C₁₂n

(t−s)^−ξ1−δs^{−α−(ϑ−1)ξβ−ν}(s^{1−ξβ+δ−ν} + 1)kA^νu₀k + (t−s)^{−ϑξβ−ν}s^{−ξβ−(ϑ−1)ξβ−ν}o

. Therefore,

Z t

t/2

kAPα(t−s)(f(u(s))−f(u(t)))kds

≤C18

t^{1−ξ1−δ−α−(ϑ−1)ξβ−ν}(t^{1−ξβ+δ−ν} + 1)kA^νu0k+t^{1−ϑξβ−ν−ξβ−(ϑ−1)ξβ−ν} , for some constantC18>0. Furthermore, by using the assumption (3.1),

ξ_1−ν+ 1−ξ_1−δ−α−(ϑ−1)ξ_β−ν >1−αν−ϑξ_β−ν≥0, 1−ϑξβ−ν−ξβ−(ϑ−1)ξβ−ν = 2(1−αν−ϑξβ−ν)≥0.

Note also that 1−ξ_β+δ−ν >0. Then, forT sufficiently small, t^ξ1−νkI2k ≤C19kA^νu0k,

for some constantC₁₉>0. Thus we conclude that

kAu(t)k ≤C20t^−ξ1−νkA^νu0k, t∈(0, T], for some constantC₂₀>0.

4. Applications We consider the parabolic initial-value problem

D_t^αu= ∆u+|u|^p−1u, in Ω×(0, T) u|_∂Ω= 0,

u(0) =u0, in Ω

(4.1)

where Ω ∈ R^N with C² boundary and p > 1. The abstract formulation of the problem (4.1) is

D_t^αu=Au+f(u), in Ω×(0, T)

u(0) =u₀, in Ω, (4.2)

where

A= ∆, f(u) =|u|^p−1u.

Here, we setH =L²(Ω) andD(A) =H_D² ={u∈H²(Ω) :u= 0 on∂Ω}. Note that Ais sectorial inH.

Next, forβ ≥N(1−1/p)/4 andp >1, we have kuk2p≤CkA^βuk2, u∈D(A^β)

(see [12] for more details). By the mean value theorem and the H¨older inequality, foru, v∈D(A^β), one can obtain that

kf(u)−f(v)k²₂≤p²(kuk_(p−1)q+kvk_(p−1)q)^2(p−1)ku−vk²_r where 2/p+ 2/r= 1. It implies

kf(u)−f(v)k2≤p(kuk2p+kvk2p)^p−1ku−vk2p

by takingr= 2psuch that (p−1)q= 2p. Thus we get

kf(u)−f(v)k2≤p(kA^βuk2+kA^βvk2)^p−1kA^βu−A^βvk2

foru, v∈D(A^β). We find that

D(A^β) =H_D^2β, H_D^2β={u∈H^2β(Ω) :u|∂Ω= 0}, 1/4< β <1 (see [30] for more details). Thus, for

1

4 < β <1, ifN 1−1 p

≤1, N

4 1−1 p

≤β <1, if 1< N 1−1 p

<4, andu0∈D(A^ν) with

pξ_β−1

α(p−1) ≤ν < β−(2− 1

α)⁺, ifpξ_β>1, 0< ν < β−(2− 1

α)⁺, ifpξβ≤1,

by Theorem 3.2, problem (4.1) has a unique mild solutionusatisfying t^ξη−νu∈BC((0, T];D(A^η)), lim

t→0⁺t^ξη−νA^ηu(t) = 0, kA^ηu(t)kH ≤Ct^−ξη−νkA^νu0kH, t∈(0, T] for everyη∈(ν+ (2−1/α)⁺, β] withT sufficiently small.

Acknowledgments. The first author would like to thank the Directorate General of Higher Education, the Ministry of Research, Technology, and Higher Education of the Republic of Indonesia for their support.

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Bambang Hendriya Guswanto

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Jenderal Soedirman University (UNSOED), Purwokerto, Indonesia

E-mail address:bambanghg unsoed@yahoo.com; bambang.guswanto@unsoed.ac.id

Takashi Suzuki

Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka, Japan

E-mail address:suzuki@sigmath.es.osaka-u.ac.jp