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Regularity theory for a nonlinear fractional reaction–diffusion equation
Nonlinear Analysis xxx (xxxx) xxx
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Regularity theory for a nonlinear fractional reaction–diffusion equation Bruno de Andrade a ,1 ,∗, Thamires Santos Cruz b a b
Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, Brazil Universidade Federal Rural de Pernambuco, 52171-900, Recife- PE, Brazil
article
info
abstract
Article history: Received 12 February 2019 Accepted 15 November 2019 Communicated by Enrico Valdinoci MSC: primary 35K57, 45K05 secondary 35R11
This paper is dedicated to the study of a nonlinear fractional reaction–diffusion equation. We analyze the behavior of the resolvent family associated with the problem in the scale of fractional power spaces associated with the Laplace operator. We ensure the existence and uniqueness of regular mild solutions to the problem in the Lq setting. Furthermore, we consider global existence or non-continuation by a blow-up of such solutions. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Fractional diffusion equations Regularity theory of solutions Blow-up alternative
1. Introduction Fractional diffusion equations ∂t u(t, x) = ∂t (gα ∗ ∆u(t, x)) + r(t, x) t > 0, x ∈ Rn ,
(1.1)
α−1
where gα (t) = tΓ (α) , 0 < α < 1, have attracted much interest mostly due to their applications in the modeling of anomalous diffusion, since this subject involves a large variety of natural sciences such as physics, chemistry, biology, geology and their interfacial disciplines, see e.g. [2,6,14,22] and the references therein. One of the main characteristics of an anomalous diffusion of this kind is the non-Markovian nature of the subdiffusive process defined by (1.1). Indeed, in the fractional diffusion case the mean squared displacement is given by m(t) = ctα , t > 0, ∗ Corresponding author. E-mail addresses: [email protected] (B. de Andrade), [email protected] (T.S. Cruz). 1 Bruno de Andrade is partially supported by CNPQ, Brazil and CAPES/FAPITEC, Brazil under grants 308931/2017-3, 88881.157450/2017-01 and88887.157906/2017-00.
https://doi.org/10.1016/j.na.2019.111705 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
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with some constant c > 0, which shows that the diffusion is slower than in the classical case of Brownian motion, see [14]. From the mathematical point of view, the study of these equations was initiated by Schneider and Wyss [20] and has been of interest of many researchers since then. For example, Kemppainen et al. [8] prove optimal estimates for the decay in time of solutions to a class of non-local in time subdiffusion equations by using estimates based on the fundamental solution and Young’s inequality, see also [24]. In [5], de Andrade and Viana consider the nonlinear fractional diffusion equation ∫ t ρ−1 ∂t u(t, x) = ∂t gα (s)∆u(t − s, x)ds + |u(t, x)| u(t, x), in (0, ∞) × Rn , 0
u(x, 0) = u0 (x), in Rn , and prove a global well-posedness result for initial data u0 ∈ Lp (Rn ) in the critical case p = αn 2 (ρ − 1). They also provide sufficient conditions to obtain self-similar solutions and study spatial decays to the problem. We recommend [3,9–13,15,17,18,21] and the references therein for more informations on the development of this theory and its applications. Stimulated by these works, in this paper we will investigate further the problem ⎧ ∫ t ρ−1 ⎪ ⎪ gα,γ (s)ds∆u(t − s, x) + |u(t, x)| u(t, x) + h, in (0, ∞) × Ω , ⎨∂t u(t, x) = ∂t 0 (1.2) ⎪ ⎪ ⎩u(t, x) = 0, in (0, ∞) × ∂Ω , u(0, x) = u0 (x), on Ω , where ρ > 1, Ω ⊂ Rn is a bounded smooth domain, h : [0, ∞) × Ω → R is a given function and u0 ∈ Lq (Ω ), 1 < q < ∞. For γ ≥ 0 and 0 < α ≤ 1, the function gα,γ is defined by gα,γ (t) = e−γt gα (t),
t > 0.
Note that if γ = 0 then gα,γ = gα and the above problem comes to a nonlinear fractional diffusion equation. We remark that this kind of kernel was previously considered in the literature. Indeed, Pruss et al. [19] consider a temperature dependent phase field model with memory and, among others things, they guarantee global well-posedness for the non-isothermal Cahn–Hilliard equation. By using an appropriate Lyapunov functional, some compactness properties and the Lojasiewicz–Simon inequality, Vergara [23] investigates the long-time behavior of bounded solutions of a nonlinear integral evolution equation with such a kernel. In [3], de Andrade consider a nonlinear Volterra equation coming from the theory of viscoelasticity. Firstly, using Laplace transform theory, the author ensures existence of a resolvent for the problem and establishes the behavior of this resolvent in the scale of fractional power spaces associated to the Stokes operator. Posteriorly, employing these results, he proves the well-posedness of the problem. In some sense, ideas used in that work provide a good mechanism to approach such equations and for this reason we will apply this same schedule to (1.2). We discuss well-posedness of problem (1.2) in the Lq (Ω ) setting, 1 < q < ∞. Our general strategy will be to rewrite problem (1.2) as an abstract evolutionary integral equation on Lq (Ω ) given by ∫ t ∫ t ∫ t u(t) = u0 + gα,γ (t − s)∆u(s)ds + f (u(s))ds + h(s)ds, t ≥ 0, (1.3) 0 ρ−1
where f (u) = |u|
0 q
0
q
u and ∆ : D(∆) ⊂ L (Ω ) → L (Ω ) is the Laplace operator with domain D(∆) = W 2,q (Ω ) ∩ W01,q (Ω ).
The Laplace transform applied to (1.3) yields ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s))ds + S(t − s)h(s)ds, t ≥ 0, 0
0
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−1
where S(t) is the inverse Laplace transform of λ−1 (λ + γ)α ((λ + γ)α − ∆) , if t > 0, and S(0) := I, where I denotes the identity operator. From now we will call resolvent for (1.3) the family {S(t)}t≥0 associated to the Laplace operator. Motivated by this discussion and the related literature, see for example [16], we have adopted the following concept for a solution to the problem (1.3). Definition 1.1. Let τ > 0. (i) A function u : [0, τ ] → Lq (Ω ) is said to be a mild solution to (1.3) in [0, τ ] if u ∈ C([0, τ ]; Lq (Ω )) and for all t ∈ [0, τ ] ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s))ds + S(t − s)h(s)ds. 0
0
(ii) A function u : [0, τ ) → L (Ω ) is said to be a mild solution to (1.3) in [0, τ ) if for any τ ′ ∈ [0, τ ), u is a mild solution to (1.3) in [0, τ ′ ]. q
In the next sections we ensure sufficient conditions for existence, uniqueness and regularity of mild solutions to (1.3), as well as analyze the possible continuation of this solution to a maximal interval of existence. 2. On the resolvent associated to the Laplace operator The Laplace operator ∆ with Dirichlet boundary conditions in a bounded and smooth domain Ω ⊂ Rn can be seen as a sectorial in Eq0 = Lq (Ω ), for 1 < q < ∞, with domain Eq1 := W 2,q (Ω ) ∩ W01,q (Ω ). ( π operator ) That is, for each η0 ∈ , π there exists a constant C > 0 such that 2 ∥(λ − ∆)−1 ∥ ≤
C , for all λ ∈ Ση0 , |λ|
where Ση0 is the sector {λ ∈ C − {0}/ |arg(λ)| < η0 }. Furthermore, its scale of fractional powers spaces {Eqβ }β∈R verifies Eqβ ↪→ Lr (Ω ), for r ≤
nq , n − 2qβ
0≤β<
n , 2q
Eq0 = Lq (Ω ), n , Eqβ ←↩ Lr (Ω ), for r ≥ n − 2qβ
(2.4) n − ′ < β ≤ 0. 2q
To a proof of these embeddings we recommend [1]. The main scope of this section is to study the behavior of the family {S(t)}t≥0 in these spaces. We start proving the following result. Proposition 2.1.
Consider α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. The function ⎧ ∫ ⎨ 1 −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ, S(t) := 2πi Ha ⎩I,
t > 0,
(2.5)
t = 0,
where Ha is the path given by (2.6) and I is the identity operator, is well defined in Lq (Ω ) and there exists a constant M ≥ 1 such that ∥S(t)ψ∥Lq (Ω) ≤ M ∥ψ∥Lq (Ω) , for all t ≥ 0 and ψ ∈ Lq (Ω ). Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Fig. 1. The path Ha.
Proof . Consider ϕ ∈ (π/2, π) and let Σϕ be the sector associated to the sectorial operator ∆ and choose arbitrary values r > 0 and η ∈ (π/2, ϕ]. Let Ha = Ha(r, η) be the Hankel path given by Ha = Ha1 + Ha2 − Ha3 (see Fig. 1), where Hai are such that ⎧ ⎨Ha1 := {seiη : r ≤ s < ∞}, Ha2 := {reis : |s| ≤ η}, (2.6) ⎩ Ha3 := {se−iη : r ≤ s < ∞}. We will estimate the function ∥S(t)∥ on each Hai , according to definition (2.6), for any t > 0. Observe that for all ψ ∈ Lq (Ω ) and for each fixed t ̸= 0, if we assume r = t−1 then • Over Ha1 ∫ 1 ∫ C∥ψ∥Lq (Ω) ∞ ets cos η −1 λt α −1 α ≤ ds e (λ + γ) λ ((λ + γ) − ∆) ψ dλ 1 q 2πi Ha1 2π |seiη | t L (Ω) ∫ C∥ψ∥Lq (Ω) ∞ ts cos η ≤ e t ds 1 2π t ≤
−C∥ψ∥Lq (Ω) ecos η . 2π cos η
• Over Ha2 ∫ 1 2πi
∫ C∥ψ∥Lq (Ω) η ert cos s eλt (λ + γ)α −1 α ((λ + γ) − ∆) ψ dλ ≤ r ds is λ 2π Ha2 −η |re | Lq (Ω) ∫ C∥ψ∥Lq (Ω) η e ds ≤ 2π −η C∥ψ∥Lq (Ω) eη ≤ . π • Over Ha3 we proceed in the same way as in Ha1 . Taking { } −Cecos η Ceη M = 3 max , 2π cos η π we have ∥S(t)ψ∥Lq (Ω) ≤ M ∥ψ∥Lq (Ω) . Thus we deduce that S(t) is a well defined operator for each t > 0. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Fig. 2. The region Dn := D ∩ {z ∈ C : |z| ≤ n}.
We claim that the integral representation (2.5) is independent of r > 0 and η ∈ ( π2 , η0 ). In fact, consider r, r′ ∈ (0, ∞),
π η, η ′ ∈ ( , η0 ) 2
and the integral (2.5) on the paths Ha(r, η) and Ha(r′ , η ′ ). Without loss of generality we suppose η ′ > η and r > r′ . Let D be the region between the curves Ha(r, η) and Ha(r′ , η ′ ) and for every n ∈ N set Dn := D ∩ {z ∈ C : |z| ≤ n}, see Fig. 2. By the Cauchy integral theorem ∫ −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ = 0. ∂Dn
Let R1 and R2 be the two arcs contained in {z ∈ C : |z| = n}. Then ∫ −1 λt α −1 α e (λ + γ) λ ((λ + γ) − ∆) dλ R1
L(Lq (Ω))
∫ ′ η ( is )−1 tneis is α is −1 α is = e (ne + γ) (ne ) (ne + γ) − ∆ n ie ds η
L(Lq (Ω))
∫
η′
≤C η tnL
≤ Ce
etn cos s ds (η ′ − η),
where L = cos η < 0. Therefore the above integral converges to zero as n → ∞. A similar procedure shows that the integral over the arc R2 has the same property. Consequently, ∫ ∫ −1 −1 eλt (λ + γ)α λ−1 ((λ + γ)α − ∆) dλ = eλt (λ + γ)α λ−1 ((λ + γ)α − ∆) dλ Ha(r ′ ,η ′ )
and this proves the statement.
Ha(r,η)
□
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Remark 2.1. The proof of Proposition 2.1 ensures that path (2.6) is independent of α ∈ (0, 1], namely, ∫ 1 −1 S(t) = eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ, 2πi Ha for all α ∈ (0, 1]. Proposition 2.2. Let α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. For all ψ ∈ D(∆) = W 2,q (Ω ) ∩ W01,q (Ω ) and t > 0 we have ∫ t S(t)ψ = ψ + gα,γ (t − s)∆S(s)ψ ds. 0
Proof . By definition of the Laplace transform ∫ ∞ ˆ S(µ) = e−µt S(t) dt 0 ) ∫ (∫ ∞ 1 −1 −(µ−λ)t = e dt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ 2πi Ha 0 = µ−1 (µ + γ)α ((µ + γ)α − ∆)
−1
.
Hence ˆ ˆ S(µ) = µ−1 + (µ + γ)−α ∆S(µ) and consequently ∫
t
gα,γ (t − s)∆S(s) ds. □
S(t) = I + 0
Proposition 2.3. Lq (Ω ).
For α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0, the family {S(t)}t≥0 is strongly continuous on
Proof . For each ψ ∈ D(∆) ∫ ∫ 1 1 −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) ψ dλ − eλt λ−1 ψ dλ 2πi Ha 2πi Ha ∫ 1 −1 = eλt λ−1 ((λ + γ)α − ∆) ∆ψ dλ. 2πi Ha
S(t)ψ − ψ =
Thus
∫ C −1 −α eRe(λt) |λ| |(λ + γ)| ∥∆ψ∥Lq (Ω) |dλ|. 2π Ha Similarly to demonstration of Proposition 2.1 we can show that ∥S(t)ψ − ψ∥Lq (Ω) ≤
∥S(t)ψ − ψ∥Lq (Ω) → 0 when t → 0+ . By taking u(t) = S(t)ψ for t > 0 and ψ ∈ D(∆) we have by Proposition 2.2 ∫ t u(t) = gα,γ (t − r)∆u(r) dr + ψ, t > 0. 0
Consider s ∈ (0, t]. So ∫ t ∫ s gα,γ (t − r)∆u(r) dr − gα,γ (s − r)∆u(r) dr ∥u(t) − u(s)∥Lq (Ω) = q 0 0 L (Ω) ∫ t ∫ s ≤ gα,γ (t − r)∆u(r) dr + (gα,γ (t − r) − gα,γ (s − r))∆u(r) dr . s
Lq (Ω)
0
Lq (Ω)
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Analyzing each part separately we verify ∥u(t) − u(s)∥Lq (Ω) → 0 when s → t− . Analogously we treat the case s > t. Therefore {S(t)}≥0 is strongly continuous in D(∆). Since Lq (Ω ) = D(∆) follows the strong continuity in Lq (Ω ). □ We close this section studying the behavior of the resolvent {S(t)}t≥0 in the scale {Eqβ }β∈(0,1) of fractional power spaces associated to the Laplace operator. Proposition 2.4. Let α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. Given 0 < β < 1 there exist a constant M > 0 such that for all ψ ∈ Lq (Ω ) ∥S(t)ψ∥E β ≤ M t−αβ (1 + γt)αβ ∥ψ∥Lq (Ω) q
for all t > 0. Proof . Fixing t > 0 and setting r = t−1 we have ∫ 1 −1 λt −1 α β α ∥S(t)ψ∥E β = e λ (λ + γ) (−∆) ((λ + γ) − ∆) ψ dλ q q 2πi Ha L (Ω) ∫ ˜ C∥ψ∥ Lq (Ω) −1 β−1 ≤ |eλt | |λ| |(λ + γ)α | |(λ + γ)α | |dλ| 2π Ha ∫ ˜ C∥ψ∥ Lq (Ω) −1 αβ ≤ |eλt | |λ| |λ + γ| |dλ|. 2π Ha Analyzing the above estimate on each part of the path Ha we take M = 3 max
{ ˜ ( cos η ) ˜ } e αβecos η Ceη C − + , 2 2π cos η cos η π
such that ∥S(t)ψ∥E β ≤ M t−αβ (1 + γt)αβ ∥ψ∥Lq (Ω) . □ q
Note that if γ = 0 the resolvent family {S(t)}t≥0 coincides with the Mittag-Leffler family associated to the Laplace operator. Furthermore, if α = 1 this family coincides with the heat semigroup. Remark 2.2.
Given 0 ≤ θ ≤ β ≤ 1, Proposition 2.4 implies ∥S(t)∥L(E β ,
E 1+θ )
≤ M t−α(1+θ−β) (1 + γt)α(1+θ−β) .
Moreover, if u0 ∈ E 1 then ∥tαθ S(t)u0 ∥E 1+θ → 0, as t → 0+ . With the same approach used in the proof of Proposition 2.3 we can prove the following result. Proposition 2.5. For α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0, the resolvent family {S(t)}t≥0 is strongly continuous on the scale {Eqβ }β∈(0,1) . Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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3. Mild solutions to the problem In this section we consider the problem (1.2) with initial data u0 ∈ Lq (Ω ). For this, let (−∆)β be the realization of −∆ in Eqβ , β ∈ R. So, the sectorial operator (−∆)β : D((−∆)β ) = Eqβ+1 ⊂ Eqβ → Eqβ is a isometry of Eqβ+1 in Eqβ . To analyze the existence of solution in Lq (Ω ) we will denote Xqβ := Eqβ−1 , β ∈ R. By using the embeddings (2.4), we observe that the fractional power spaces associated with (−∆)−1 : Xq1 ⊂ Xq0 → Xq0 satisfy Xqβ ↪→ Lr (Ω ), r ≤
2q + n nq , 1≤β< , n + 2q − 2qβ 2q
Xq1 = Lq (Ω ),
(3.7)
Xqβ ←↩ Lr (Ω ), r ≥
nq 2q ′ − n < β ≤ 1. , n + 2q − 2qβ 2q ′ ρ−1
Lemma 3.1. Let 1 < q < ∞. The function f : R → R given by f (u) = |u| Xq1 on Xqβ satisfying the following inequalities:
u induces an application of
+ ∥v∥ρ−1 )∥u − v∥Xq1 ∥f (u) − f (v)∥X β ≤ c(∥u∥ρ−1 X1 X1 q
q
q
and ∥f (u)∥X β ≤ c∥u∥ρX 1 , q
for some c > 0 with max{1 −
n 2q ′ , 0}
q
< β < 1 and 1 < ρ ≤ 1 + n2 (q − βq). n 2q ′ , 0}
Proof . By using the assumption that max{1 −
< β < 1 and (3.7), we obtain
nq
L n+2q−2βq (Ω ) ↪→ Xqβ . Moreover, since 1 < ρ ≤ 1 + n2 [q − βq] implies
ρnq n+2q−2βq
(3.8)
≤ q, we conclude that ρnq
Lq (Ω ) ↪→ L n+2q−2βq (Ω ).
(3.9)
Finally, by taking u, v ∈ Xq1 = Lq (Ω ) and by using (3.8), (3.9) and H¨older’s inequality, we get ∥f (u) − f (v)∥X β ≤ c∥f (u) − f (v)∥ q
≤ c∥u − v∥
nq
L n+2q−2βq (Ω) ρ−1
ρnq
L n+2q−2βq (Ω)
(∥u∥
ρnq
L n+2q−2βq (Ω)
+ ∥v∥ρ−1
ρnq
)
L n+2q−2βq (Ω)
ρ−1 ≤ c(∥u∥ρ−1 + ∥v∥X 1 )∥u − v∥Xq1 X1 q
q
and ∥f (u)∥X β ≤ c∥f (u)∥
nq
L n+2q−2βq (Ω)
q
≤ c∥u∥ρ ≤
nqρ
L n+2q−2βq (Ω) c∥u∥ρX 1 . □ q
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From now on B : (0, ∞) × (0, ∞) → (0, ∞) will represent the Beta function which is defined by ∫ B(a, b) =
1
(1 − s)a−1 sb−1 ds.
0
Theorem 3.1. Consider α ∈ (0, 1], γ ≥ 0, 1 < q < ∞, max{1 − 2qn′ , 0} < β < 1 and 1 < ρ ≤ 1 + n2 (q − βq). Let u0 ∈ Xq1 and suppose h : [0, ∞) → Xq1 a continuous function such that ∥h(s)∥Xq1 ≤ ksφ , for some k > 0 and φ > −1. Then there exist a constant τ0 > 0 and a unique mild solution u ∈ C([0, τ0 ]; Xq1 ) of the problem (1.3). Furthermore, u ∈ C((0, τ0 ]; Xq1+θ ) and tαθ ∥u(t)∥X 1+θ → 0, as t → 0+ , q
for all 0 < θ < β. Moreover, if u0 and h are nonnegative functions, then the same is true to the mild solution u(t, ·), for every t ≥ 0. Proof . Existence: Let µ > 0 and τ0 > 0 be such that ∥u0 ∥Xq1 <
µ , 2
∥S(t)u0 ∥Xq1 ≤
µ , 2
M ktφ+1 B(φ + 1, 1) <
µ 4
and 4M Rct1−α(1−β) (1 + γt)α(1−β) B(1, 1 − α(1 − β)) < min {µ, 1} , for all t ∈ (0, τ0 ], where R = max{µρ , 2µρ−1 }. Consider the set { K(τ0 ) =
C([0, τ0 ]; Xq1 )
u∈
}
: sup ∥u(s)∥Xq1 ≤ µ 0≤s≤τ0
and define the operator ∫
t
∫ S(t − s)f (u(s)) ds +
T u(t) = S(t)u0 + 0
t
S(t − s)h(s) ds. 0
Claim 1. If u ∈ K(τ0 ) then ∥T u(t)∥Xq1 ≤ µ, for all t ∈ [0, τ0 ]. In fact, if u ∈ K(τ0 ) and t ∈ [0, τ0 ] then ∫ t ∫ t ∥T u(t)∥Xq1 ≤ ∥S(t)u0 ∥Xq1 + ∥S(t − s)f (u(s))∥Xq1 ds + ∥S(t − s)h(s)∥Xq1 ds 0 0 ∫ t µ (t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s))∥X β ds ≤ +M q 2 0 ∫ t +M ∥h(s)∥Xq1 ds 0 ∫ t µ α(1−β) ≤ + M c(1 + γt) (t − s)−α(1−β) ∥u(s)∥ρX 1 ds q 2 0 ∫ t + Mk sφ ds 0
µ + M cµρ t1−α(1−β) (1 + γt)α(1−β) B(1, 1 − α(1 − β)) 2 + M ktφ+1 B(φ + 1, 1) ≤ µ.
≤
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Claim 2. If u ∈ K(τ0 ), then T u ∈ C([0, τ0 ]; Xq1 ). In fact consider t ∈ (0, τ0 ] and {tn }n∈N ⊂ [0, t] such that limn→∞ tn = t. Thus ∥T u(tn ) − T u(t)∥Xq1 ≤ ∥(S(tn ) − S(t))u0 ∥Xq1 ∫ tn + ∥(S(tn − s) − S(t − s))f (u(s))∥Xq1 ds 0
∫
tn
∥(S(tn − s) − S(t − s))h(s)∥Xq1 ds
+ 0
∫
t
+ tn ∫ t
+ tn
∥S(t − s)f (u(s))∥Xq1 ds ∥S(t − s)h(s)∥Xq1 ds.
By using the strong continuity of the family {S(t)}t≥0 and the Lebesgue’s dominated convergence theorem we prove the three first terms of the right side of the above inequality go to 0 as n → ∞. For the fourth term note that ∫ t ∫ t α(1−β) (t − s)−α(1−β) ∥u(s)∥ρX 1 ds ∥S(t − s)f (u(s))∥Xq1 ds ≤ M c(1 + γ(t − tn )) q tn tn ∫ t ≤ M cµρ (1 + γ(t − tn ))α(1−β) (t − s)−α(1−β) ds, tn
which goes to 0 as n → ∞. Likewise, the last term goes to 0 as n → ∞. The case t ∈ [0, τ0 ) and limn→∞ tn = t is proved in the same way. Now, consider u, v ∈ K(τ0 ). Then ∫ ∥T u(t) − T v(t)∥Xq1 ≤ M
0
t
∫
≤ M c(1 + γt)α(1−β)
t
0
(t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s)) − f (v(s))∥X β ds q ) (
ρ−1 ρ−1 (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ∥u(s)∥X 1 + ∥v(s)∥X 1
ds
q
q
≤ M c(1 + γt)α(1−β) 2µρ−1 B(1, 1 − α(1 − β)) sup ∥u(t) − v(t)∥Xq1 t∈[0,τ0 ]
1 ≤ sup ∥u(t) − v(t)∥Xq1 . 4 t∈[0,τ0 ] Hence by the Banach contraction principle T has a unique fixed point u ∈ K(τ0 ). Uniqueness: Let v ∈ C([0, τ0 ]; Xq1 ) be another mild solution of (1.3). Then for all t ∈ [0, τ0 ] we have t
∫ ∥u(t) − v(t)∥Xq1 ≤ M ≤ M c(1 + γt)α(1−β)
∫ 0
≤ M c sup t∈[0,τ0 ]
[
0 t
(t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s)) − f (v(s))∥X β ds q ( )
ρ−1 ρ−1 (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ∥u(s)∥X 1 + ∥v(s)∥X 1 q
ds
q
]∫ t ρ−1 ρ−1 (1 + γt)α(1−β) (∥u(t)∥X (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ds. 1 + ∥v(t)∥X 1 ) q
q
0
Thus the result follows from the Singular Gronwall Inequality, see [7, Lemma 7.1.1]. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Regularity: Consider 0 < θ < β. Then for all t ∈ (0, τ0 ] ∥u(t)∥X 1+θ ≤ M t−αθ (1 + γt)αθ ∥u0 ∥Xq1 + M c(1 + γt)α(1+θ−β) q ∫ t αθ (t − s)−αθ sφ ds + M k(1 + γt)
∫ 0
t
(t − s)−α(1+θ−β) ∥u(s)∥ρX 1 ds q
0
≤ M t−αθ (1 + γt)αθ ∥u0 ∥Xq1 + M c(1 + γt)α(1+θ−β) t1−α(1+θ−β) B(1, 1 − α(1 + θ − β)) sup ∥u(t)∥ρX 1 t∈[0,τ0 ]
q
+ M kt1−αθ+φ (1 + γt)αθ B(φ + 1, 1 − αθ) Therefore, u : (0, τ0 ] → Xq1+θ is well defined. A similar argument to the used in the proof of Claim 1 proves u(t) ∈ C((0, τ0 ]; Xq1+θ ). From the above estimate if t > 0 then tαθ ∥u(t)∥X 1+θ → 0, as t → 0+ . q
Nonnegativity: Let {Eα (t)}t≥0 be the Mittag-Leffler family associated with the Laplace operator. In [4], the authors prove that Eα (t) is increasing for every t ≥ 0. Moreover, by taking a(t) = g1−α (t)e
−γt
∫ +γ
t
g1−α (s)e−γs ds,
t > 0,
0 −α
t where g1−α (t) = Γ (1−α) , 0 < α < 1, it follows from the uniqueness of the resolvent family {S(t)}t≥0 associated with (1.2) that
S(t) = (a ∗ Eα )(t) = e
−γt
∫ Eα (t) + γ
t
e−γs Eα (s)ds,
t ≥ 0.
0
Hence, S(t) is increasing for every t ≥ 0. Thereby, considering v0 (t) = u0 , for t ∈ [0, τ0 ], it follows that v0 (·) and f (v0 (·)) are nonnegative functions. Since h : [0, ∞) → Xq1 is also a nonnegative function, the same is true to ∫ t ∫ t v1 (t) = T (v0 (t)) = S(t)u0 + S(t − s)f (v0 (s))ds + S(t − s)h(s)ds, t ∈ [0, τ0 ]. 0
0
Iterating, we obtain a Picard’s sequence of nonnegative functions vn (t) = T (vn−1 (t)),
t ∈ [0, τ0 ],
which converges to the mild solution u ∈ C([0, τ0 ]; Xq1 ). Consequently, u is a nonnegative function and this concludes the proof. □ Definition 3.1. Let u : [0, τ ] → Xq1 be a mild solution of the problem (1.3). If T > 0 and v : [0, τ +T ] → Xq1 is a mild solution of the problem (1.3), then we say that v is a continuation of u in [0, τ + T ]. We continue the study of problem (1.3) proving that the mild solution provided by Theorem 3.1 has a unique continuation to a bigger interval of existence. Theorem 3.2. Under the conditions of Theorem 3.1, let τ0 > 0 and u : [0, τ0 ] → Xq1 the mild solution of problem (1.3). Then there exist T > 0 and a unique continuation u∗ of u in [0, τ0 + T ]. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Proof . For µ > 0 fixed consider T > 0 such that for all t ∈ [τ0 , τ0 + T ] the following conditions are satisfied ∫ τ0 ∫ τ0 µ µ ∥(S(t − s) − S(τ0 − s))f (u(s))∥Xq1 ds ≤ , ∥(S(t − s) − S(τ0 − s))h(s)∥Xq1 ds ≤ , 5 5 0 0 ∫ t µ µ ∥S(t)u0 − S(τ0 )u0 ∥Xq1 ≤ , N k sφ ds ≤ 5 5 τ0 and ∫ t µ (t − s)−α(1−β) ds ≤ , R(1 + γ(t − τ0 ))α(1−β) 5 τ0 [ ] ) + ∥f (u(τ0 ))∥X β . where R = M cµ((µ + ∥u(τ0 )∥Xq1 )ρ−1 + ∥u(τ0 )∥ρ−1 X1 q
q
Let K be the set of all w ∈ C([0, τ0 + T ]; Xq1 ) such that w(t) = u(t) for all t ∈ [0, τ0 ] and ∥w(t) − u(τ0 )∥Xq1 ≤ µ, for all t ∈ [τ0 , τ0 + T ]. Define the operator Λ on K by ∫ t ∫ t Λw(t) = S(t)u0 + S(t − s)f (w(s)) ds + S(t − s)h(s)ds. 0
0
As in the previous theorem our purpose is to show K is Λ-invariant and Λ is a contraction. Similarly to the proof of Theorem 3.1 we have Λw ∈ C([0, τ0 + T ]; Xq1 ) for all w ∈ K. By definition, if w ∈ K then w(t) = u(t), for all t ∈ [0, τ0 ]. Hence, Λw(t) = Λu(t) = u(t),
∀ t ∈ [0, τ0 ].
On the other hand, if t ∈ [τ0 , τ0 + T ] we have ∫ τ0 ∥Λw(t) − u(τ0 )∥Xq1 ≤ ∥S(t)u0 − S(τ0 )u0 ∥Xq1 + (S(t − s) − S(τ − s))f (w(s)) ds 0 1 0 Xq ∫ t ∫ τ0 S(t − s)f (w(s)) ds + (S(t − s) − S(τ − s))h(s) ds + 0 Xq1
0
∫ +
τ0
Xq1
t S(t − s)h(s) ds
τ0
Xq1
t 3µ +M (t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (w(s))∥X β ds q 5 τ0 ∫ t + Nk sφ ds
∫
≤
τ0
4µ ≤ + M (1 + γ(t − τ0 ))α(1−β) 5
∫
4µ + M c(1 + γ(t − τ0 ))α(1−β) 5
∫
≤
t
τ0
(t − s)−α(1−β) [∥f (w(s)) − f (u(τ0 ))∥X β + ∥f (u(τ0 ))∥X β ]ds q
q
t
ρ−1 (t − s)−α(1−β) ∥w(s) − u(τ0 )∥Xq1 (∥w(s)∥ρ−1 + ∥u(τ0 )∥X 1 )ds Xq1 q ∫ t +M (1 + γ(t − τ0 ))α(1−β) ∥f (u(τ0 ))∥X β (t − s)−α(1−β) ds ∫ t q τ0 4µ ≤ + R(1 + γ(t − τ0 ))α(1−β) (t − s)−α(1−β) ds ≤ µ. 5 τ0 τ0
Consequently, K is Λ-invariant. Finally it is not hard to see that Λ is a 25 -contraction and this concludes the proof. The unicity can be shown by Gronwall Lemma. □ Next is our result on global existence or non-continuation by blow-up. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Theorem 3.3. Under the conditions of Theorem 3.1, if u is the mild solution of the problem (1.3) with a maximal time of existence τmax < ∞ then lim sup ∥u(t)∥Xq1 = ∞. − t→τmax
Proof . Suppose that there exists R > 0 such that ∥u(t)∥Xq1 ≤ R for all t ∈ [0, τmax ). Consider − {tj }j∈N ⊂ [0, τmax ) such that tj → τmax , as j → ∞. Given ϵ > 0 fix N ∈ N such that for all n, m ≥ N the following estimates are satisfied ∫ tn ϵ ≤ ϵ , ∥S(tn )u0 − S(tm )u0 ∥Xq1 ≤ , (S(t − s) − S(τ − s))f (u(s)) ds n max 1 5 10 0 Xq ∫ tn ϵ ≤ , (S(τmax − s) − S(tm − s))f (u(s)) ds 10 Xq1 0∫ tn ϵ (S(tn − s) − S(τmax − s))h(s) ds 1 ≤ 10 , ∫ 0tn Xq ϵ (S(τmax − s) − S(tm − s))h(s) ds 1 ≤ 10 , ∫ tm ∫Xtqm 0 ϵ ϵ sφ ds ≤ (t − s)−α(1−β) ds ≤ . Nk and M cRρ (1 + γ(tm − tn ))α(1−β) 5 5 tn tn Without loss of generality we suppose above tn < tm . Hence ∥u(tn ) − u(tm )∥Xq1 ≤ ∥S(tn )u0 − S(tm )u0 ∥Xq1 ∫ tn (S(tn − s) − S(tm − s))f (u(s)) ds + 1 0 Xq ∫ tn + (S(tn − s) − S(tm − s))h(s) ds 0
∫ +
tm
tn
S(tm − s)f (u(s)) ds
Xq1
∫ +
Xq1 tm
tn
S(tm − s)f (u(s)) ds
Xq1
∫ tn ϵ ≤ + (S(tn − s) − S(τmax − s))f (u(s)) ds 1 5 0 Xq ∫ tn (S(τmax − s) − S(tm − s))f (u(s)) ds + Xq1
0
+ + + +
∫ tn (S(t − s) − S(τ − s))h(s) ds n max 1 0 Xq ∫ tn (S(τmax − s) − S(tm − s))h(s) ds 1 0 Xq ∫ tm M (tm − s)−α(1−β) (1 + γ(tm − s))α(1−β) ∥f (u(s))∥X β ds q tn ∫ tm Nk sφ ds ≤ ϵ. tn
This computation shows that (u(tn ))n∈N ⊂ Xq1 is a Cauchy sequence and therefore there is a u ˜ ∈ Xq1 such that limn→∞ u(tn ) = u ˜. Then, we can extend u to [0, τmax ] obtaining the equality ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s)) ds + S(t − s)h(s) ds 0
0
for all t ∈ [0, τmax ]. Therefore, by Theorem 3.2 we can extend the solution to some bigger interval. However this contradicts the maximality of τmax . □ Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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References [1] H. Amann, On abstract parabolic fundamental solutions, J. Math. Soc. Japan 39 (1) (1987) 93–116. [2] J.-P. Bouchaud, A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990) 127–293. [3] B. De Andrade, On the well-posedness of a volterra equation with applications in the navier-stokes problem, Math. Methods Appl. Sci. 41 (2018) 750–768. [4] B. De Andrade, A.N. Carvalho, P.M. Carvalho-Neto, P. Mar´ın-Rubio, Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal. 41 (2015) 439–467. [5] B. De Andrade, A. Viana, On a fractional reaction–diffusion equation, Z. Angew. Math. Phys. 68 (2017) 59. [6] P. Heitjans, J. K¨ arger, Diffusion in Condensed Matter, Springer-Verlag Berlin Heidelberg, Basel, 1993. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlim, 1981. [8] J. Kemppainen, J. Siljander, V. Vergara, R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in Rd , Math. Ann. 366 (2016) 941–979. [9] V. Keyantuo, C. Lizama, H¨ older Continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations 230 (2006) 634–660. [10] M. Krasnoschok, V. Pata, N. Vasylyeva, Solvability of linear boundary value problems for subdiffusuion equations with memory, J. Integral Equations Appl. (2018) (in press). [11] S.O. Londen, An existence result on a volterra equation in a banach space, Trans. Amer. Math. Soc. 235 (1978) 285–304. [12] A. Lunardi, Regular solutions for time dependent abstract integrodifferential equations with singular kernel, J. Math. Anal. Appl. 130 (1988) 1–21. [13] E. Mainini, G. Mola, Exponential and polynomial decay for first order linear Volterra evolution equations, J. Quart. Appl. Math. 67 (2009) 93–111. [14] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen. 37 (2004) R161–R208. [15] J. Pr¨ uss, On linear volterra equations of parabolic type in Banach spaces, Trans. Amer. Math. Soc. 301 (2) (1987) 691–721. [16] J. Pr¨ uss, Evolutionary Integral Equations and Applications, Vol. 87, in: Monographs in Mathematics, Birkh¨ auser, 1993. [17] J. Pr¨ uss, Decay properties for the solutions of a partial differential equation with memory, Arch. Math. 92 (2009) 158–173. [18] J. Pr¨ uss, G. Simonett, R. Zacher, On the qualitative behaviour of incompressible two-phase flows with phase transitions: the case of equal densities, Interfaces Free Bound. 15 (2013) 405–428. [19] J. Pr¨ uss, V. Vergara, R. Zacher, Well-posedness and long-time behaviour for the non-isothermal cahn-hilliard equation with memory, Discrete Contin. Dyn. Syst. 26 (2009) 625–647. [20] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134–144. [21] P. Souplet, Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys. 55 (2004) 28–31. [22] V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I: Background and Theory, Springer-Verlag Berlin Heidelberg, 2013. [23] V. Vergara, Convergence to steady states of solutions to nonlinear integral evolution equations, Calc. Var. 40 (2011) 319–334. [24] V. Vergara, R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal. 47 (2015) 210–239.
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Contents lists available at ScienceDirect
Nonlinear Analysis www.elsevier.com/locate/na
Regularity theory for a nonlinear fractional reaction–diffusion equation Bruno de Andrade a ,1 ,∗, Thamires Santos Cruz b a b
Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, Brazil Universidade Federal Rural de Pernambuco, 52171-900, Recife- PE, Brazil
article
info
abstract
Article history: Received 12 February 2019 Accepted 15 November 2019 Communicated by Enrico Valdinoci MSC: primary 35K57, 45K05 secondary 35R11
This paper is dedicated to the study of a nonlinear fractional reaction–diffusion equation. We analyze the behavior of the resolvent family associated with the problem in the scale of fractional power spaces associated with the Laplace operator. We ensure the existence and uniqueness of regular mild solutions to the problem in the Lq setting. Furthermore, we consider global existence or non-continuation by a blow-up of such solutions. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Fractional diffusion equations Regularity theory of solutions Blow-up alternative
1. Introduction Fractional diffusion equations ∂t u(t, x) = ∂t (gα ∗ ∆u(t, x)) + r(t, x) t > 0, x ∈ Rn ,
(1.1)
α−1
where gα (t) = tΓ (α) , 0 < α < 1, have attracted much interest mostly due to their applications in the modeling of anomalous diffusion, since this subject involves a large variety of natural sciences such as physics, chemistry, biology, geology and their interfacial disciplines, see e.g. [2,6,14,22] and the references therein. One of the main characteristics of an anomalous diffusion of this kind is the non-Markovian nature of the subdiffusive process defined by (1.1). Indeed, in the fractional diffusion case the mean squared displacement is given by m(t) = ctα , t > 0, ∗ Corresponding author. E-mail addresses: [email protected] (B. de Andrade), [email protected] (T.S. Cruz). 1 Bruno de Andrade is partially supported by CNPQ, Brazil and CAPES/FAPITEC, Brazil under grants 308931/2017-3, 88881.157450/2017-01 and88887.157906/2017-00.
https://doi.org/10.1016/j.na.2019.111705 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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with some constant c > 0, which shows that the diffusion is slower than in the classical case of Brownian motion, see [14]. From the mathematical point of view, the study of these equations was initiated by Schneider and Wyss [20] and has been of interest of many researchers since then. For example, Kemppainen et al. [8] prove optimal estimates for the decay in time of solutions to a class of non-local in time subdiffusion equations by using estimates based on the fundamental solution and Young’s inequality, see also [24]. In [5], de Andrade and Viana consider the nonlinear fractional diffusion equation ∫ t ρ−1 ∂t u(t, x) = ∂t gα (s)∆u(t − s, x)ds + |u(t, x)| u(t, x), in (0, ∞) × Rn , 0
u(x, 0) = u0 (x), in Rn , and prove a global well-posedness result for initial data u0 ∈ Lp (Rn ) in the critical case p = αn 2 (ρ − 1). They also provide sufficient conditions to obtain self-similar solutions and study spatial decays to the problem. We recommend [3,9–13,15,17,18,21] and the references therein for more informations on the development of this theory and its applications. Stimulated by these works, in this paper we will investigate further the problem ⎧ ∫ t ρ−1 ⎪ ⎪ gα,γ (s)ds∆u(t − s, x) + |u(t, x)| u(t, x) + h, in (0, ∞) × Ω , ⎨∂t u(t, x) = ∂t 0 (1.2) ⎪ ⎪ ⎩u(t, x) = 0, in (0, ∞) × ∂Ω , u(0, x) = u0 (x), on Ω , where ρ > 1, Ω ⊂ Rn is a bounded smooth domain, h : [0, ∞) × Ω → R is a given function and u0 ∈ Lq (Ω ), 1 < q < ∞. For γ ≥ 0 and 0 < α ≤ 1, the function gα,γ is defined by gα,γ (t) = e−γt gα (t),
t > 0.
Note that if γ = 0 then gα,γ = gα and the above problem comes to a nonlinear fractional diffusion equation. We remark that this kind of kernel was previously considered in the literature. Indeed, Pruss et al. [19] consider a temperature dependent phase field model with memory and, among others things, they guarantee global well-posedness for the non-isothermal Cahn–Hilliard equation. By using an appropriate Lyapunov functional, some compactness properties and the Lojasiewicz–Simon inequality, Vergara [23] investigates the long-time behavior of bounded solutions of a nonlinear integral evolution equation with such a kernel. In [3], de Andrade consider a nonlinear Volterra equation coming from the theory of viscoelasticity. Firstly, using Laplace transform theory, the author ensures existence of a resolvent for the problem and establishes the behavior of this resolvent in the scale of fractional power spaces associated to the Stokes operator. Posteriorly, employing these results, he proves the well-posedness of the problem. In some sense, ideas used in that work provide a good mechanism to approach such equations and for this reason we will apply this same schedule to (1.2). We discuss well-posedness of problem (1.2) in the Lq (Ω ) setting, 1 < q < ∞. Our general strategy will be to rewrite problem (1.2) as an abstract evolutionary integral equation on Lq (Ω ) given by ∫ t ∫ t ∫ t u(t) = u0 + gα,γ (t − s)∆u(s)ds + f (u(s))ds + h(s)ds, t ≥ 0, (1.3) 0 ρ−1
where f (u) = |u|
0 q
0
q
u and ∆ : D(∆) ⊂ L (Ω ) → L (Ω ) is the Laplace operator with domain D(∆) = W 2,q (Ω ) ∩ W01,q (Ω ).
The Laplace transform applied to (1.3) yields ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s))ds + S(t − s)h(s)ds, t ≥ 0, 0
0
Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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−1
where S(t) is the inverse Laplace transform of λ−1 (λ + γ)α ((λ + γ)α − ∆) , if t > 0, and S(0) := I, where I denotes the identity operator. From now we will call resolvent for (1.3) the family {S(t)}t≥0 associated to the Laplace operator. Motivated by this discussion and the related literature, see for example [16], we have adopted the following concept for a solution to the problem (1.3). Definition 1.1. Let τ > 0. (i) A function u : [0, τ ] → Lq (Ω ) is said to be a mild solution to (1.3) in [0, τ ] if u ∈ C([0, τ ]; Lq (Ω )) and for all t ∈ [0, τ ] ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s))ds + S(t − s)h(s)ds. 0
0
(ii) A function u : [0, τ ) → L (Ω ) is said to be a mild solution to (1.3) in [0, τ ) if for any τ ′ ∈ [0, τ ), u is a mild solution to (1.3) in [0, τ ′ ]. q
In the next sections we ensure sufficient conditions for existence, uniqueness and regularity of mild solutions to (1.3), as well as analyze the possible continuation of this solution to a maximal interval of existence. 2. On the resolvent associated to the Laplace operator The Laplace operator ∆ with Dirichlet boundary conditions in a bounded and smooth domain Ω ⊂ Rn can be seen as a sectorial in Eq0 = Lq (Ω ), for 1 < q < ∞, with domain Eq1 := W 2,q (Ω ) ∩ W01,q (Ω ). ( π operator ) That is, for each η0 ∈ , π there exists a constant C > 0 such that 2 ∥(λ − ∆)−1 ∥ ≤
C , for all λ ∈ Ση0 , |λ|
where Ση0 is the sector {λ ∈ C − {0}/ |arg(λ)| < η0 }. Furthermore, its scale of fractional powers spaces {Eqβ }β∈R verifies Eqβ ↪→ Lr (Ω ), for r ≤
nq , n − 2qβ
0≤β<
n , 2q
Eq0 = Lq (Ω ), n , Eqβ ←↩ Lr (Ω ), for r ≥ n − 2qβ
(2.4) n − ′ < β ≤ 0. 2q
To a proof of these embeddings we recommend [1]. The main scope of this section is to study the behavior of the family {S(t)}t≥0 in these spaces. We start proving the following result. Proposition 2.1.
Consider α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. The function ⎧ ∫ ⎨ 1 −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ, S(t) := 2πi Ha ⎩I,
t > 0,
(2.5)
t = 0,
where Ha is the path given by (2.6) and I is the identity operator, is well defined in Lq (Ω ) and there exists a constant M ≥ 1 such that ∥S(t)ψ∥Lq (Ω) ≤ M ∥ψ∥Lq (Ω) , for all t ≥ 0 and ψ ∈ Lq (Ω ). Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Fig. 1. The path Ha.
Proof . Consider ϕ ∈ (π/2, π) and let Σϕ be the sector associated to the sectorial operator ∆ and choose arbitrary values r > 0 and η ∈ (π/2, ϕ]. Let Ha = Ha(r, η) be the Hankel path given by Ha = Ha1 + Ha2 − Ha3 (see Fig. 1), where Hai are such that ⎧ ⎨Ha1 := {seiη : r ≤ s < ∞}, Ha2 := {reis : |s| ≤ η}, (2.6) ⎩ Ha3 := {se−iη : r ≤ s < ∞}. We will estimate the function ∥S(t)∥ on each Hai , according to definition (2.6), for any t > 0. Observe that for all ψ ∈ Lq (Ω ) and for each fixed t ̸= 0, if we assume r = t−1 then • Over Ha1 ∫ 1 ∫ C∥ψ∥Lq (Ω) ∞ ets cos η −1 λt α −1 α ≤ ds e (λ + γ) λ ((λ + γ) − ∆) ψ dλ 1 q 2πi Ha1 2π |seiη | t L (Ω) ∫ C∥ψ∥Lq (Ω) ∞ ts cos η ≤ e t ds 1 2π t ≤
−C∥ψ∥Lq (Ω) ecos η . 2π cos η
• Over Ha2 ∫ 1 2πi
∫ C∥ψ∥Lq (Ω) η ert cos s eλt (λ + γ)α −1 α ((λ + γ) − ∆) ψ dλ ≤ r ds is λ 2π Ha2 −η |re | Lq (Ω) ∫ C∥ψ∥Lq (Ω) η e ds ≤ 2π −η C∥ψ∥Lq (Ω) eη ≤ . π • Over Ha3 we proceed in the same way as in Ha1 . Taking { } −Cecos η Ceη M = 3 max , 2π cos η π we have ∥S(t)ψ∥Lq (Ω) ≤ M ∥ψ∥Lq (Ω) . Thus we deduce that S(t) is a well defined operator for each t > 0. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Fig. 2. The region Dn := D ∩ {z ∈ C : |z| ≤ n}.
We claim that the integral representation (2.5) is independent of r > 0 and η ∈ ( π2 , η0 ). In fact, consider r, r′ ∈ (0, ∞),
π η, η ′ ∈ ( , η0 ) 2
and the integral (2.5) on the paths Ha(r, η) and Ha(r′ , η ′ ). Without loss of generality we suppose η ′ > η and r > r′ . Let D be the region between the curves Ha(r, η) and Ha(r′ , η ′ ) and for every n ∈ N set Dn := D ∩ {z ∈ C : |z| ≤ n}, see Fig. 2. By the Cauchy integral theorem ∫ −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ = 0. ∂Dn
Let R1 and R2 be the two arcs contained in {z ∈ C : |z| = n}. Then ∫ −1 λt α −1 α e (λ + γ) λ ((λ + γ) − ∆) dλ R1
L(Lq (Ω))
∫ ′ η ( is )−1 tneis is α is −1 α is = e (ne + γ) (ne ) (ne + γ) − ∆ n ie ds η
L(Lq (Ω))
∫
η′
≤C η tnL
≤ Ce
etn cos s ds (η ′ − η),
where L = cos η < 0. Therefore the above integral converges to zero as n → ∞. A similar procedure shows that the integral over the arc R2 has the same property. Consequently, ∫ ∫ −1 −1 eλt (λ + γ)α λ−1 ((λ + γ)α − ∆) dλ = eλt (λ + γ)α λ−1 ((λ + γ)α − ∆) dλ Ha(r ′ ,η ′ )
and this proves the statement.
Ha(r,η)
□
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Remark 2.1. The proof of Proposition 2.1 ensures that path (2.6) is independent of α ∈ (0, 1], namely, ∫ 1 −1 S(t) = eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ, 2πi Ha for all α ∈ (0, 1]. Proposition 2.2. Let α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. For all ψ ∈ D(∆) = W 2,q (Ω ) ∩ W01,q (Ω ) and t > 0 we have ∫ t S(t)ψ = ψ + gα,γ (t − s)∆S(s)ψ ds. 0
Proof . By definition of the Laplace transform ∫ ∞ ˆ S(µ) = e−µt S(t) dt 0 ) ∫ (∫ ∞ 1 −1 −(µ−λ)t = e dt λ−1 (λ + γ)α ((λ + γ)α − ∆) dλ 2πi Ha 0 = µ−1 (µ + γ)α ((µ + γ)α − ∆)
−1
.
Hence ˆ ˆ S(µ) = µ−1 + (µ + γ)−α ∆S(µ) and consequently ∫
t
gα,γ (t − s)∆S(s) ds. □
S(t) = I + 0
Proposition 2.3. Lq (Ω ).
For α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0, the family {S(t)}t≥0 is strongly continuous on
Proof . For each ψ ∈ D(∆) ∫ ∫ 1 1 −1 eλt λ−1 (λ + γ)α ((λ + γ)α − ∆) ψ dλ − eλt λ−1 ψ dλ 2πi Ha 2πi Ha ∫ 1 −1 = eλt λ−1 ((λ + γ)α − ∆) ∆ψ dλ. 2πi Ha
S(t)ψ − ψ =
Thus
∫ C −1 −α eRe(λt) |λ| |(λ + γ)| ∥∆ψ∥Lq (Ω) |dλ|. 2π Ha Similarly to demonstration of Proposition 2.1 we can show that ∥S(t)ψ − ψ∥Lq (Ω) ≤
∥S(t)ψ − ψ∥Lq (Ω) → 0 when t → 0+ . By taking u(t) = S(t)ψ for t > 0 and ψ ∈ D(∆) we have by Proposition 2.2 ∫ t u(t) = gα,γ (t − r)∆u(r) dr + ψ, t > 0. 0
Consider s ∈ (0, t]. So ∫ t ∫ s gα,γ (t − r)∆u(r) dr − gα,γ (s − r)∆u(r) dr ∥u(t) − u(s)∥Lq (Ω) = q 0 0 L (Ω) ∫ t ∫ s ≤ gα,γ (t − r)∆u(r) dr + (gα,γ (t − r) − gα,γ (s − r))∆u(r) dr . s
Lq (Ω)
0
Lq (Ω)
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Analyzing each part separately we verify ∥u(t) − u(s)∥Lq (Ω) → 0 when s → t− . Analogously we treat the case s > t. Therefore {S(t)}≥0 is strongly continuous in D(∆). Since Lq (Ω ) = D(∆) follows the strong continuity in Lq (Ω ). □ We close this section studying the behavior of the resolvent {S(t)}t≥0 in the scale {Eqβ }β∈(0,1) of fractional power spaces associated to the Laplace operator. Proposition 2.4. Let α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0. Given 0 < β < 1 there exist a constant M > 0 such that for all ψ ∈ Lq (Ω ) ∥S(t)ψ∥E β ≤ M t−αβ (1 + γt)αβ ∥ψ∥Lq (Ω) q
for all t > 0. Proof . Fixing t > 0 and setting r = t−1 we have ∫ 1 −1 λt −1 α β α ∥S(t)ψ∥E β = e λ (λ + γ) (−∆) ((λ + γ) − ∆) ψ dλ q q 2πi Ha L (Ω) ∫ ˜ C∥ψ∥ Lq (Ω) −1 β−1 ≤ |eλt | |λ| |(λ + γ)α | |(λ + γ)α | |dλ| 2π Ha ∫ ˜ C∥ψ∥ Lq (Ω) −1 αβ ≤ |eλt | |λ| |λ + γ| |dλ|. 2π Ha Analyzing the above estimate on each part of the path Ha we take M = 3 max
{ ˜ ( cos η ) ˜ } e αβecos η Ceη C − + , 2 2π cos η cos η π
such that ∥S(t)ψ∥E β ≤ M t−αβ (1 + γt)αβ ∥ψ∥Lq (Ω) . □ q
Note that if γ = 0 the resolvent family {S(t)}t≥0 coincides with the Mittag-Leffler family associated to the Laplace operator. Furthermore, if α = 1 this family coincides with the heat semigroup. Remark 2.2.
Given 0 ≤ θ ≤ β ≤ 1, Proposition 2.4 implies ∥S(t)∥L(E β ,
E 1+θ )
≤ M t−α(1+θ−β) (1 + γt)α(1+θ−β) .
Moreover, if u0 ∈ E 1 then ∥tαθ S(t)u0 ∥E 1+θ → 0, as t → 0+ . With the same approach used in the proof of Proposition 2.3 we can prove the following result. Proposition 2.5. For α ∈ (0, 1], 1 < q < ∞ and γ ≥ 0, the resolvent family {S(t)}t≥0 is strongly continuous on the scale {Eqβ }β∈(0,1) . Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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3. Mild solutions to the problem In this section we consider the problem (1.2) with initial data u0 ∈ Lq (Ω ). For this, let (−∆)β be the realization of −∆ in Eqβ , β ∈ R. So, the sectorial operator (−∆)β : D((−∆)β ) = Eqβ+1 ⊂ Eqβ → Eqβ is a isometry of Eqβ+1 in Eqβ . To analyze the existence of solution in Lq (Ω ) we will denote Xqβ := Eqβ−1 , β ∈ R. By using the embeddings (2.4), we observe that the fractional power spaces associated with (−∆)−1 : Xq1 ⊂ Xq0 → Xq0 satisfy Xqβ ↪→ Lr (Ω ), r ≤
2q + n nq , 1≤β< , n + 2q − 2qβ 2q
Xq1 = Lq (Ω ),
(3.7)
Xqβ ←↩ Lr (Ω ), r ≥
nq 2q ′ − n < β ≤ 1. , n + 2q − 2qβ 2q ′ ρ−1
Lemma 3.1. Let 1 < q < ∞. The function f : R → R given by f (u) = |u| Xq1 on Xqβ satisfying the following inequalities:
u induces an application of
+ ∥v∥ρ−1 )∥u − v∥Xq1 ∥f (u) − f (v)∥X β ≤ c(∥u∥ρ−1 X1 X1 q
q
q
and ∥f (u)∥X β ≤ c∥u∥ρX 1 , q
for some c > 0 with max{1 −
n 2q ′ , 0}
q
< β < 1 and 1 < ρ ≤ 1 + n2 (q − βq). n 2q ′ , 0}
Proof . By using the assumption that max{1 −
< β < 1 and (3.7), we obtain
nq
L n+2q−2βq (Ω ) ↪→ Xqβ . Moreover, since 1 < ρ ≤ 1 + n2 [q − βq] implies
ρnq n+2q−2βq
(3.8)
≤ q, we conclude that ρnq
Lq (Ω ) ↪→ L n+2q−2βq (Ω ).
(3.9)
Finally, by taking u, v ∈ Xq1 = Lq (Ω ) and by using (3.8), (3.9) and H¨older’s inequality, we get ∥f (u) − f (v)∥X β ≤ c∥f (u) − f (v)∥ q
≤ c∥u − v∥
nq
L n+2q−2βq (Ω) ρ−1
ρnq
L n+2q−2βq (Ω)
(∥u∥
ρnq
L n+2q−2βq (Ω)
+ ∥v∥ρ−1
ρnq
)
L n+2q−2βq (Ω)
ρ−1 ≤ c(∥u∥ρ−1 + ∥v∥X 1 )∥u − v∥Xq1 X1 q
q
and ∥f (u)∥X β ≤ c∥f (u)∥
nq
L n+2q−2βq (Ω)
q
≤ c∥u∥ρ ≤
nqρ
L n+2q−2βq (Ω) c∥u∥ρX 1 . □ q
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From now on B : (0, ∞) × (0, ∞) → (0, ∞) will represent the Beta function which is defined by ∫ B(a, b) =
1
(1 − s)a−1 sb−1 ds.
0
Theorem 3.1. Consider α ∈ (0, 1], γ ≥ 0, 1 < q < ∞, max{1 − 2qn′ , 0} < β < 1 and 1 < ρ ≤ 1 + n2 (q − βq). Let u0 ∈ Xq1 and suppose h : [0, ∞) → Xq1 a continuous function such that ∥h(s)∥Xq1 ≤ ksφ , for some k > 0 and φ > −1. Then there exist a constant τ0 > 0 and a unique mild solution u ∈ C([0, τ0 ]; Xq1 ) of the problem (1.3). Furthermore, u ∈ C((0, τ0 ]; Xq1+θ ) and tαθ ∥u(t)∥X 1+θ → 0, as t → 0+ , q
for all 0 < θ < β. Moreover, if u0 and h are nonnegative functions, then the same is true to the mild solution u(t, ·), for every t ≥ 0. Proof . Existence: Let µ > 0 and τ0 > 0 be such that ∥u0 ∥Xq1 <
µ , 2
∥S(t)u0 ∥Xq1 ≤
µ , 2
M ktφ+1 B(φ + 1, 1) <
µ 4
and 4M Rct1−α(1−β) (1 + γt)α(1−β) B(1, 1 − α(1 − β)) < min {µ, 1} , for all t ∈ (0, τ0 ], where R = max{µρ , 2µρ−1 }. Consider the set { K(τ0 ) =
C([0, τ0 ]; Xq1 )
u∈
}
: sup ∥u(s)∥Xq1 ≤ µ 0≤s≤τ0
and define the operator ∫
t
∫ S(t − s)f (u(s)) ds +
T u(t) = S(t)u0 + 0
t
S(t − s)h(s) ds. 0
Claim 1. If u ∈ K(τ0 ) then ∥T u(t)∥Xq1 ≤ µ, for all t ∈ [0, τ0 ]. In fact, if u ∈ K(τ0 ) and t ∈ [0, τ0 ] then ∫ t ∫ t ∥T u(t)∥Xq1 ≤ ∥S(t)u0 ∥Xq1 + ∥S(t − s)f (u(s))∥Xq1 ds + ∥S(t − s)h(s)∥Xq1 ds 0 0 ∫ t µ (t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s))∥X β ds ≤ +M q 2 0 ∫ t +M ∥h(s)∥Xq1 ds 0 ∫ t µ α(1−β) ≤ + M c(1 + γt) (t − s)−α(1−β) ∥u(s)∥ρX 1 ds q 2 0 ∫ t + Mk sφ ds 0
µ + M cµρ t1−α(1−β) (1 + γt)α(1−β) B(1, 1 − α(1 − β)) 2 + M ktφ+1 B(φ + 1, 1) ≤ µ.
≤
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Claim 2. If u ∈ K(τ0 ), then T u ∈ C([0, τ0 ]; Xq1 ). In fact consider t ∈ (0, τ0 ] and {tn }n∈N ⊂ [0, t] such that limn→∞ tn = t. Thus ∥T u(tn ) − T u(t)∥Xq1 ≤ ∥(S(tn ) − S(t))u0 ∥Xq1 ∫ tn + ∥(S(tn − s) − S(t − s))f (u(s))∥Xq1 ds 0
∫
tn
∥(S(tn − s) − S(t − s))h(s)∥Xq1 ds
+ 0
∫
t
+ tn ∫ t
+ tn
∥S(t − s)f (u(s))∥Xq1 ds ∥S(t − s)h(s)∥Xq1 ds.
By using the strong continuity of the family {S(t)}t≥0 and the Lebesgue’s dominated convergence theorem we prove the three first terms of the right side of the above inequality go to 0 as n → ∞. For the fourth term note that ∫ t ∫ t α(1−β) (t − s)−α(1−β) ∥u(s)∥ρX 1 ds ∥S(t − s)f (u(s))∥Xq1 ds ≤ M c(1 + γ(t − tn )) q tn tn ∫ t ≤ M cµρ (1 + γ(t − tn ))α(1−β) (t − s)−α(1−β) ds, tn
which goes to 0 as n → ∞. Likewise, the last term goes to 0 as n → ∞. The case t ∈ [0, τ0 ) and limn→∞ tn = t is proved in the same way. Now, consider u, v ∈ K(τ0 ). Then ∫ ∥T u(t) − T v(t)∥Xq1 ≤ M
0
t
∫
≤ M c(1 + γt)α(1−β)
t
0
(t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s)) − f (v(s))∥X β ds q ) (
ρ−1 ρ−1 (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ∥u(s)∥X 1 + ∥v(s)∥X 1
ds
q
q
≤ M c(1 + γt)α(1−β) 2µρ−1 B(1, 1 − α(1 − β)) sup ∥u(t) − v(t)∥Xq1 t∈[0,τ0 ]
1 ≤ sup ∥u(t) − v(t)∥Xq1 . 4 t∈[0,τ0 ] Hence by the Banach contraction principle T has a unique fixed point u ∈ K(τ0 ). Uniqueness: Let v ∈ C([0, τ0 ]; Xq1 ) be another mild solution of (1.3). Then for all t ∈ [0, τ0 ] we have t
∫ ∥u(t) − v(t)∥Xq1 ≤ M ≤ M c(1 + γt)α(1−β)
∫ 0
≤ M c sup t∈[0,τ0 ]
[
0 t
(t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (u(s)) − f (v(s))∥X β ds q ( )
ρ−1 ρ−1 (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ∥u(s)∥X 1 + ∥v(s)∥X 1 q
ds
q
]∫ t ρ−1 ρ−1 (1 + γt)α(1−β) (∥u(t)∥X (t − s)−α(1−β) ∥u(s) − v(s)∥Xq1 ds. 1 + ∥v(t)∥X 1 ) q
q
0
Thus the result follows from the Singular Gronwall Inequality, see [7, Lemma 7.1.1]. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Regularity: Consider 0 < θ < β. Then for all t ∈ (0, τ0 ] ∥u(t)∥X 1+θ ≤ M t−αθ (1 + γt)αθ ∥u0 ∥Xq1 + M c(1 + γt)α(1+θ−β) q ∫ t αθ (t − s)−αθ sφ ds + M k(1 + γt)
∫ 0
t
(t − s)−α(1+θ−β) ∥u(s)∥ρX 1 ds q
0
≤ M t−αθ (1 + γt)αθ ∥u0 ∥Xq1 + M c(1 + γt)α(1+θ−β) t1−α(1+θ−β) B(1, 1 − α(1 + θ − β)) sup ∥u(t)∥ρX 1 t∈[0,τ0 ]
q
+ M kt1−αθ+φ (1 + γt)αθ B(φ + 1, 1 − αθ) Therefore, u : (0, τ0 ] → Xq1+θ is well defined. A similar argument to the used in the proof of Claim 1 proves u(t) ∈ C((0, τ0 ]; Xq1+θ ). From the above estimate if t > 0 then tαθ ∥u(t)∥X 1+θ → 0, as t → 0+ . q
Nonnegativity: Let {Eα (t)}t≥0 be the Mittag-Leffler family associated with the Laplace operator. In [4], the authors prove that Eα (t) is increasing for every t ≥ 0. Moreover, by taking a(t) = g1−α (t)e
−γt
∫ +γ
t
g1−α (s)e−γs ds,
t > 0,
0 −α
t where g1−α (t) = Γ (1−α) , 0 < α < 1, it follows from the uniqueness of the resolvent family {S(t)}t≥0 associated with (1.2) that
S(t) = (a ∗ Eα )(t) = e
−γt
∫ Eα (t) + γ
t
e−γs Eα (s)ds,
t ≥ 0.
0
Hence, S(t) is increasing for every t ≥ 0. Thereby, considering v0 (t) = u0 , for t ∈ [0, τ0 ], it follows that v0 (·) and f (v0 (·)) are nonnegative functions. Since h : [0, ∞) → Xq1 is also a nonnegative function, the same is true to ∫ t ∫ t v1 (t) = T (v0 (t)) = S(t)u0 + S(t − s)f (v0 (s))ds + S(t − s)h(s)ds, t ∈ [0, τ0 ]. 0
0
Iterating, we obtain a Picard’s sequence of nonnegative functions vn (t) = T (vn−1 (t)),
t ∈ [0, τ0 ],
which converges to the mild solution u ∈ C([0, τ0 ]; Xq1 ). Consequently, u is a nonnegative function and this concludes the proof. □ Definition 3.1. Let u : [0, τ ] → Xq1 be a mild solution of the problem (1.3). If T > 0 and v : [0, τ +T ] → Xq1 is a mild solution of the problem (1.3), then we say that v is a continuation of u in [0, τ + T ]. We continue the study of problem (1.3) proving that the mild solution provided by Theorem 3.1 has a unique continuation to a bigger interval of existence. Theorem 3.2. Under the conditions of Theorem 3.1, let τ0 > 0 and u : [0, τ0 ] → Xq1 the mild solution of problem (1.3). Then there exist T > 0 and a unique continuation u∗ of u in [0, τ0 + T ]. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Proof . For µ > 0 fixed consider T > 0 such that for all t ∈ [τ0 , τ0 + T ] the following conditions are satisfied ∫ τ0 ∫ τ0 µ µ ∥(S(t − s) − S(τ0 − s))f (u(s))∥Xq1 ds ≤ , ∥(S(t − s) − S(τ0 − s))h(s)∥Xq1 ds ≤ , 5 5 0 0 ∫ t µ µ ∥S(t)u0 − S(τ0 )u0 ∥Xq1 ≤ , N k sφ ds ≤ 5 5 τ0 and ∫ t µ (t − s)−α(1−β) ds ≤ , R(1 + γ(t − τ0 ))α(1−β) 5 τ0 [ ] ) + ∥f (u(τ0 ))∥X β . where R = M cµ((µ + ∥u(τ0 )∥Xq1 )ρ−1 + ∥u(τ0 )∥ρ−1 X1 q
q
Let K be the set of all w ∈ C([0, τ0 + T ]; Xq1 ) such that w(t) = u(t) for all t ∈ [0, τ0 ] and ∥w(t) − u(τ0 )∥Xq1 ≤ µ, for all t ∈ [τ0 , τ0 + T ]. Define the operator Λ on K by ∫ t ∫ t Λw(t) = S(t)u0 + S(t − s)f (w(s)) ds + S(t − s)h(s)ds. 0
0
As in the previous theorem our purpose is to show K is Λ-invariant and Λ is a contraction. Similarly to the proof of Theorem 3.1 we have Λw ∈ C([0, τ0 + T ]; Xq1 ) for all w ∈ K. By definition, if w ∈ K then w(t) = u(t), for all t ∈ [0, τ0 ]. Hence, Λw(t) = Λu(t) = u(t),
∀ t ∈ [0, τ0 ].
On the other hand, if t ∈ [τ0 , τ0 + T ] we have ∫ τ0 ∥Λw(t) − u(τ0 )∥Xq1 ≤ ∥S(t)u0 − S(τ0 )u0 ∥Xq1 + (S(t − s) − S(τ − s))f (w(s)) ds 0 1 0 Xq ∫ t ∫ τ0 S(t − s)f (w(s)) ds + (S(t − s) − S(τ − s))h(s) ds + 0 Xq1
0
∫ +
τ0
Xq1
t S(t − s)h(s) ds
τ0
Xq1
t 3µ +M (t − s)−α(1−β) (1 + γ(t − s))α(1−β) ∥f (w(s))∥X β ds q 5 τ0 ∫ t + Nk sφ ds
∫
≤
τ0
4µ ≤ + M (1 + γ(t − τ0 ))α(1−β) 5
∫
4µ + M c(1 + γ(t − τ0 ))α(1−β) 5
∫
≤
t
τ0
(t − s)−α(1−β) [∥f (w(s)) − f (u(τ0 ))∥X β + ∥f (u(τ0 ))∥X β ]ds q
q
t
ρ−1 (t − s)−α(1−β) ∥w(s) − u(τ0 )∥Xq1 (∥w(s)∥ρ−1 + ∥u(τ0 )∥X 1 )ds Xq1 q ∫ t +M (1 + γ(t − τ0 ))α(1−β) ∥f (u(τ0 ))∥X β (t − s)−α(1−β) ds ∫ t q τ0 4µ ≤ + R(1 + γ(t − τ0 ))α(1−β) (t − s)−α(1−β) ds ≤ µ. 5 τ0 τ0
Consequently, K is Λ-invariant. Finally it is not hard to see that Λ is a 25 -contraction and this concludes the proof. The unicity can be shown by Gronwall Lemma. □ Next is our result on global existence or non-continuation by blow-up. Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
B. de Andrade and T.S. Cruz / Nonlinear Analysis xxx (xxxx) xxx
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Theorem 3.3. Under the conditions of Theorem 3.1, if u is the mild solution of the problem (1.3) with a maximal time of existence τmax < ∞ then lim sup ∥u(t)∥Xq1 = ∞. − t→τmax
Proof . Suppose that there exists R > 0 such that ∥u(t)∥Xq1 ≤ R for all t ∈ [0, τmax ). Consider − {tj }j∈N ⊂ [0, τmax ) such that tj → τmax , as j → ∞. Given ϵ > 0 fix N ∈ N such that for all n, m ≥ N the following estimates are satisfied ∫ tn ϵ ≤ ϵ , ∥S(tn )u0 − S(tm )u0 ∥Xq1 ≤ , (S(t − s) − S(τ − s))f (u(s)) ds n max 1 5 10 0 Xq ∫ tn ϵ ≤ , (S(τmax − s) − S(tm − s))f (u(s)) ds 10 Xq1 0∫ tn ϵ (S(tn − s) − S(τmax − s))h(s) ds 1 ≤ 10 , ∫ 0tn Xq ϵ (S(τmax − s) − S(tm − s))h(s) ds 1 ≤ 10 , ∫ tm ∫Xtqm 0 ϵ ϵ sφ ds ≤ (t − s)−α(1−β) ds ≤ . Nk and M cRρ (1 + γ(tm − tn ))α(1−β) 5 5 tn tn Without loss of generality we suppose above tn < tm . Hence ∥u(tn ) − u(tm )∥Xq1 ≤ ∥S(tn )u0 − S(tm )u0 ∥Xq1 ∫ tn (S(tn − s) − S(tm − s))f (u(s)) ds + 1 0 Xq ∫ tn + (S(tn − s) − S(tm − s))h(s) ds 0
∫ +
tm
tn
S(tm − s)f (u(s)) ds
Xq1
∫ +
Xq1 tm
tn
S(tm − s)f (u(s)) ds
Xq1
∫ tn ϵ ≤ + (S(tn − s) − S(τmax − s))f (u(s)) ds 1 5 0 Xq ∫ tn (S(τmax − s) − S(tm − s))f (u(s)) ds + Xq1
0
+ + + +
∫ tn (S(t − s) − S(τ − s))h(s) ds n max 1 0 Xq ∫ tn (S(τmax − s) − S(tm − s))h(s) ds 1 0 Xq ∫ tm M (tm − s)−α(1−β) (1 + γ(tm − s))α(1−β) ∥f (u(s))∥X β ds q tn ∫ tm Nk sφ ds ≤ ϵ. tn
This computation shows that (u(tn ))n∈N ⊂ Xq1 is a Cauchy sequence and therefore there is a u ˜ ∈ Xq1 such that limn→∞ u(tn ) = u ˜. Then, we can extend u to [0, τmax ] obtaining the equality ∫ t ∫ t u(t) = S(t)u0 + S(t − s)f (u(s)) ds + S(t − s)h(s) ds 0
0
for all t ∈ [0, τmax ]. Therefore, by Theorem 3.2 we can extend the solution to some bigger interval. However this contradicts the maximality of τmax . □ Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.
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Please cite this article as: B. de Andrade and T.S. Cruz, Regularity theory for a nonlinear fractional reaction–diffusion equation, Nonlinear Analysis (2019) 111705, https://doi.org/10.1016/j.na.2019.111705.