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arXiv:2212.10463v1 [math.AP] 20 Dec 2022

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH

POWER-LAW MEMORY

N. FAUSTINO AND J. MARQUES

Abstract. We study space-time-fractional analogues of evolution equations

arising in applications toward the theory of structural damping equations of

plate/wave type. On the generalization to be adopted, we consider an integro-

diﬀerential counterpart of the σ−evolution equation of the type

∂2

tu(t, x) + µ(−∆) σ

2∂tu(t, x) + (−∆)σu(t, x) = f(t, x),

with σ > 0 and µ > −2, that encodes memory of power-law type. To do so,

we replace the time derivatives ∂tand ∂2

tby the so-called Caputo-Djrbashian

derivatives ∂β

tof order β=αand β= 2α, respectively, and the inhomo-

geneous term f(t, x) by the Riemann-Liouville integral I1−α

0+f(t, x), whereby

0< α ≤1. For the solution representation of the underlying Cauchy prob-

lems of parabolic type (case of 0 < α ≤1

2) and hyperbolic type (case of

1

2< α ≤1) on the space-time [0, T ]×Rn, we then consider the wide class of

pseudo-diﬀerential operators, endowed by the fractional Laplacian −(−∆) σ

2

and the two-parameter Mittag-Leﬄer functions Eα,β. On our results, we use

the fact that the Fourier multipliers of the aforementioned pseudodiﬀerential

operators are radially symmetric to represent the convolution kernels of both

Cauchy problems in terms of the Hankel transform. In doing so, dispersive and

Strichartz estimates yield from the sharp control of the decay of the Fourier

multipliers of such operators.

Contents

1. Introduction 2

1.1. State of Art 2

1.2. Model Problems 3

1.3. Organization of the Paper 5

2. Preliminaries 5

2.1. Laplace Transform and Mittag-Leﬄer functions 5

2.2. Fourier analysis, function spaces and associated operators 8

3. Dispersive Estimates 11

3.1. Proof Strategy 11

3.2. Lp−Lqand Lp−˙

Wσ,q decay estimates 12

Date: December 21, 2022.

2020 Mathematics Subject Classiﬁcation. 26A33, 33C10, 33E12, 42B15, 44A20.

Key words and phrases. Fractional diﬀerential equations, Mittag-Leﬄer functions, structural

damping.

N. Faustino was supported by The Center for Research and Development in Mathematics and

Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT),

references UIDB/04106/2020 and UIDP/04106/2020.

J. Marques was supported by Centre for Business and Economics Research (CeBER) through

the Portuguese Foundation for Science and Technology (FCT), reference UIDB/05037/2020.

1

2 N. FAUSTINO AND J. MARQUES

3.3. ˙

Hγ−Lqand ˙

Hγ−˙

Wσ,q decay estimates 15

4. Main Results 19

4.1. Notations 19

4.2. Solution of the Cauchy Problem 1.1 20

4.3. Solution of the Cauchy Problem 1.2 23

4.4. Strichartz estimates 24

Acknowledgments 32

Appendix A. Proof of Lemma 3.2 35

Appendix B. Proof of Lemma 3.3 36

Appendix C. Proof of Theorem 4.7 36

References 38

1. Introduction

1.1. State of Art. Let us begin with a few facts concerning the theory of struc-

turally damped equations. After Russell’s fundamental paper [36] on mathematical

models of structural mechanics describing elastic beams, much more attention has

been paid rightly to the strongly damped plate/wave problems and to its further ap-

plications on the crossroads of control theory (cf. [3]), dynamical systems (cf. [41])

and transmition problems (cf. [32]). The current trends on struturally damped

equations, exhibited on the contribution of Pham et al [34], and more recently

on D’Abbicco-Ebert’s series of papers [11, 13, 14], paved the way for considering

space-fractional model problems such as the inhomogeneous σ−evolution equation

(1.1) ∂2

t(t, x) + µ(−∆)σ

2∂tu(t, x) + (−∆)σu(t, x) = f(t, x).

We refer the reader to the works of Carvalho et al (cf. [5]) and Denk-Schnaubelt

(cf. [16]) just to mention a few papers on analysis of PDEs towards struturally

damped plate equations (case of σ= 2) and damped wave equations (case of σ= 1).

Notice that the σ−evolution equation (1.1) describes the scenario where the

higher frequencies exhibit a strongly damped behavior in comparison with the lower

frequencies. Here, the fractional diﬀerential operator (−∆) γ

2(that will be deﬁned

later on Subsection 2.2) is used to model the damping term (γ=σ) and the elastic

term as well (γ= 2σ). Noteworthy, in the view of Carvalho et al (cf. [5]) and Chill-

Srivastava (cf. [6]), the model problem (1.1) brings also the possibility to study

the well-posedness of problems carrying Lpdata (1 < p < ∞), in stark constrast

with the weaker damping case so that µ(−∆) σ

2∂tu(t, x) (µ6= 0) resembles to a

regularization of the weak damping term µ∂tu(t, x), in the limit σ→0+.

Our aim here is to investigate a space-time-fractional counterpart of (1.1) that

encodes memory eﬀects, ubiquitous in natural phenomena (cf. [17, 39]). In our

setting it will be replaced the time derivatives ∂tresp. ∂2

tby the so-called Caputo-

Djrbashian derivatives ∂β

tof order β, deﬁned as

∂β

tu(t, x) =

Zt

0

(t−τ)m−β−1

Γ(m−β)

∂mu(τ, x)

∂τ mdτ , m −1< β < m

∂mu(t, x)

∂tm, β =m

,(1.2)

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 3

where Γ(·) denotes the Euler’s Gamma function (see eq. (2.7) of Subsection 2.1)

and m=⌊β⌋+ 1 (⌊β⌋stands for the integer part of β).

For the time-fractional counterpart of the inhomogeneous term f(t, x), it will be

adopted the Riemann-Liouville integral

Im−β

0+f(t, x) :=

Zt

0

(t−τ)m−β−1

Γ(m−β)f(τ, x)dτ , m −1< β < m

f(t, x), β =m

.(1.3)

We observe that for values of β6∈ N, (1.2) and (1.3) are operators with power-law

memory. The underlying memory function, given by gm−β(t) := tm−β−1

Γ(m−β), is a

probability density function over (0,∞) converging to the delta function δ(t), in

the limit β→m−.

Some of the physical reasons for the choice of the Caputo-Djrbashian derivative

(1.2) instead of the Riemann-Liouville derivative RL∂β

t:= ∂m

tIm−β

0+stems on its

ubiquity in the generalized Langevin equations (cf. [28]) and on certain limiting

processes (cf. [22]). Despite their usefulness on real-world applications (cf. [1, 2]),

the Caputo-Djrbashian derivative is suitable for modelling Cauchy problems for the

two amongst many reasons:

(1) they have similar properties to the time-derivatives;

(2) they remove the singularities at t= 0.

We refer to the book of Samko et al. (cf. [37]) for further details on the the-

ory of Caputo-Djrbashian derivatives and Riemann-Liouville operators and Stinga’s

preprint [39] for an abridged overview of it. For its applications on Cauchy prob-

lems, we refer to the books of Podlubny (cf. [35]) and Kilbas et al. (cf. [26]). We

also refer to [12, 27] for recent applications on fractional diﬀusion equations, closely

related to our approach.

1.2. Model Problems. The main focus in this paper are the following two Cauchy

problems of parabolic type (case of 0 < α ≤1

2) and hyperbolic type (case of 1

2<

α≤1), carrying the initial condition(s) u0(x) [and u1(x)] and the inhomogeneous

term I1−α

0+f(t, x), fulﬁlling certain regularity conditions to be presented a posteriori.

Here and elsewhere I1−α

0+stands for the Riemann-Liouville integral deﬁned through

eq. (1.3).

Cauchy Problem 1.1. Given 0 < α ≤1

2,σ > 0 and µ > −2, the function

u: [0, T ]×Rn→Ris a (weak) solution of

∂2α

tu(t, x) + µ(−∆)σ

2∂α

tu(t, x) + (−∆)σu(t, x) = I1−α

0+f(t, x),in (0, T ]×Rn

u(0, x) = u0(x),in ∈Rn.

4 N. FAUSTINO AND J. MARQUES

Cauchy Problem 1.2. Given 1

2< α ≤1, σ > 0 and µ > −2, the function

u: [0, T ]×Rn→Ris a (weak) solution of

∂2α

tu(t, x) + µ(−∆)σ

2∂α

tu(t, x) + (−∆)σu(t, x) = I1−α

0+f(t, x),in (0, T ]×Rn

u(0, x) = u0(x),in Rn

∂tu(0, x) = u1(x),in Rn.

The programme of studying Cauchy problems, similar to Cauchy Problem

1.1 &Cauchy Problem 1.2, has been started by Fujita in the former papers

[19, 20], where it has been shown that fractional diﬀusion and wave equations

can be reformulated as integro-diﬀerential equations of D’Alembert type. From an

operational calculus perpective, these Cauchy problems share the same features of

Cauchy problems underlying to time-fractional telegraph equations so that one can

exploit, as in [27], Orsingher-Beghin approach [33] to obtain the underlying solution

representations (see also ref. [29]).

Apart from the solution of these Cauchy problems being interesting from the

point of view of spectral analysis and stochastic processes, we want to push forward

the investigation of dispersive and Strichartz estimates to measure the size and the

decay of the solutions of Cauchy Problem 1.1 &Cauchy Problem 1.2. In our

framework, such a study is possible by taking into account the class of pseudo-

diﬀerential operators Eα,β −λ(−∆) σ

2tαand (−∆)σ

2Eα,β −λ(−∆)σ

2tα, en-

dowed by the fractional Laplacian −(−∆) σ

2and the two-parameter Mittag-Leﬄer

functions Eα,β (see eq. (3.1) on Section 3).

The study of such type of estimates have become a cornerstone tool for many

years now, mainly due to the Keel-Tao’s breakthrough contribution to the topic

(cf. [23]); see also [40, Chapter 2.] for an overview. In case of α= 1 our framework is

comparable to Pham et al and D’Abbicco-Ebert approaches (cf. [34, 13]), who have

investigated Lp−Lqestimates for Cauchy problems similar to Cauchy Problem

1.2. We note also that Cordero-Zucco’s approach (cf. [7, 8]) on Strichartz estimates

for the vibrating plate equation on Sobolev and modulation spaces (case of µ= 0

&σ= 2) also falls into the framework of the class of function spaces and pseudo-

diﬀerential operators to be investigated in depth on Section 3. The investigation of

such overlap will be left as future work.

The framework developed in Section 3 will constitute a signiﬁcant part of the pa-

per. The major contribution of it stems on the possibility of proving decay estimates

for Cauchy problems of heat type and wave type without studying the asymptotic

behaviour of the fundamental solution. In particular, we note that the fundamental

solutions encoded by higher dimensional fractional diﬀerential equations – that is,

the convolution kernels that yield from Fourier convolution formula – are mostly

Fox-H functions (cf. [26, Subsection 1.12]). And, as it was highlighted on the paper

[25], obtaining optimal decay estimates for it revealed to be a non-trivial task.

Our approach is somewhat diﬀerent from [25] and is aimed the exploitation of

the [optimal] decay estimates for the Mittag-Leﬄer functions Eα,β (−z), available on

the book [35] (case of z∈C) and on the papers [38, 4] (case of z≥0) to our setting,

through the aid of the Hankel transform (cf. [15]). Such framework allowed us to

control, in Section 4, the decay of the underlying solution representions of Cauchy

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 5

Problem 1.1 &Cauchy Problem 1.2, in terms of the Fourier multipliers of

Eα,β −λ(−∆)σ

2tαand (−∆)σ

2Eα,β −λ(−∆)σ

2tα.

The results enclosed on this paper can certainly be rewritten in terms of the

language of fractional resolvent families (see e.g. [29] and the references therein) in

a way that the Banach space framework, considered e.g. in the papers [6, 5, 16],

can be faithfully generalized. But we have decided to use mostly pseudo-diﬀerential

calculus, rather than operator theory, to keep the approach as self-contained as

possible. Last but not least, it should be stressed that the study of the size and

the decay of the solutions of Cauchy problems of heat type, possessing memory

terms, has been extensively studied during the last years (see e.g. [24, 9] and the

references given there). However, there seems that the study of the size and decay

of the solutions of Cauchy problems, similar to Cauchy Problem 1.1 &Cauchy

Problem 1.2, has not yet been fully investigated, up to authors’s knowledge. The

present paper aims also to ﬁll this gap.

1.3. Organization of the Paper. We have organized this paper as follows:

In Section 2 we collect some results to be employed in the subsequent sections.

Namely, some of the properties involving the Laplace transform and the generalized

Mittag-Leﬄer functions Eα,β and Eγ

α,β are recalled in Subsection 2.1. And in Sub-

section 2.2 we introduce some relevant deﬁnitions and properties to deﬁne properly

the space-fractional diﬀerential operator (−∆) γ

2and the associated function spaces.

Special emphasize will be given to the homogeneous Sobolev spaces ˙

Hγand to sharp

Sobolev inequalities that in turn results on the Sobolev embedding of ˙

Hγonto Lp.

In Section 3 we deduce dispersive estimates for the wide class of pseudo-diﬀerential

operators (−∆) η

2Eα,β −λ(−∆)σ

2tα. The propositions proved in Subsection 3.2

& Subsection 3.3 describe the decay properties of (−∆) η

2Eα,β −λ(−∆)σ

2tα.

Here, we make use of the reformulation of the Fourier transform for radial symmet-

ric functions in terms of the Hankel transform to obtain, in case of λ≥0, a sharp

control of decay of the underlying convolution kernel, after application of Young’s

inequality.

Finally, in Section 4 we provide the main results of this paper. More precisely, we

obtain in Subsection 4.2 & Subsection 4.3 explicit solution representation formulae

for the Cauchy Problem 1.1 &Cauchy Problem 1.2, respectively. And on

Subsection 4.4 we turn to the study of Strichartz estimates on the mixed-normed

Lebesgue spaces Ls

tLq

x([0, T ]×Rn) for both Cauchy problems, from the dispersive

estimates obtained in Section 3.

2. Preliminaries

2.1. Laplace Transform and Mittag-Leﬄer functions. For a real-valued func-

tion g: [0,∞)→Rsatisfying sup

t∈[0,∞)

e−ωt|g(t)|<∞, the Laplace transform of gis

deﬁned as

G(s) := L[g(t)](s) = Z∞

0

e−stg(t)dt (ℜ(s)> ω).(2.1)

Here, we notice that the function G(the Laplace image of g) is analytic on the

right-half plane {s∈C:ℜ(s)> ω},whereby ωis chosen as the inﬁmum of the

values of sfor which the right-hand side of (2.1) is convergent. We refer to [26,

Subsection 1.2] for further details.

6 N. FAUSTINO AND J. MARQUES

For the inverse of the Laplace transform, deﬁned as g(t) = L−1[G(s)](t), the

integration is performed along the strip [c−i∞, c +i∞] (c:= ℜ(s)> ω), and

whence, the resulting formula is independent of the choice of c.

Associated to the Laplace transform is the Laplace convolution formula (cf. [26,

(1.4.10) & (1.4.12) of p. 19] & [35, (2.237) & (2.238) of pp. 103–104]), deﬁned for

two functions gand hby

LZt

0

h(t−τ)g(τ)dτ(s) = L[h](s)· L[g](s).(2.2)

Of foremost importance are also the following operational identities, involving

the Caputo-Djrbashian derivative (1.2) and the Riemann-Liouville integral (1.3)

respectively. Namely, the property

L[∂β

tg(t)] = sβL[g](s)−

m−1

X

k=0

sβ−k−1∂k

tg(0)(2.3)

is fulﬁlled whenever L[g(t)](s), L[∂m

tg(t)](s) exist and

lim

t→∞ (∂t)kg(t) = 0 (k= 0,1,...,m−1),

hold for every g∈Cm(0,∞) such that (∂t)mg∈L1(0, b) for any b > 0 (cf. [26,

Lemma 2.24]). On the other hand, the identity

L[Im−β

0+g(t)] = sβ−mL[g](s) (ℜ(s)> ω)(2.4)

is always satisﬁed in case of g∈L1(0, b), for any b > 0 (cf. [26, Lemma 2.14]).

Next, we turn our attention to some of the special functions to be considered

on the sequel. We introduce the two-parameter/three-parameter Mittag-Leﬄer

functions Eα,β resp. Eγ

α,β , as power series expansions of the form

Eα,β (z) =

∞

X

k=0

zk

Γ(αk +β)(ℜ(α)>0),(2.5)

Eγ

α,β (z) =

∞

X

k=0

(γ)k

k!

zk

Γ(αk +β)(ℜ(α)>0),(2.6)

where Γ(·) stands for the Euler’s Gamma function

Γ(z) = Z∞

0

e−ttz−1dt (ℜ(z)>0)(2.7)

and (γ)k=Γ(γ+k)

Γ(γ)(ℜ(γ)>−k) for the Pochhammer symbol.

The wide class of Mittag-Leﬄer functions, deﬁned viz (2.5) and (2.6), permits

us to represent several transcendental and special functions (cf. [26, Subsection 1.8

& Subsection 1.9] & [21, Chapter 3 -Chapter 5]), such as:

(i) the exponential function

ez=E1,1(z);

(ii) the error function

erf(z) := 2

√πZz

0

e−t2dt = 1 −e−z2E1

2,1(−z);

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 7

(iii) the hyperbolic functions

cosh(z) = E2,1(z2) and sinh(z) = zE2,2(z2);

(iv) the complex exponential function

eiz := cos(z) + isin(z) = E2,1(−z2) + izE2,2(−z2);

(v) the Kummer conﬂuent hypergeometric function

Φ(γ;β;z) := 1F1(γ;β , z) = Γ(β)Eγ

1,β (z);

(vi) the generalized hypergeometric function

1Fmγ;β

m,β+ 1

m,...,β+m−1

m;zm

mm= Γ(β)Eγ

m,β (z) (m∈N).

Here, we would like to stress that Eγ

α,β corresponds to a faithful generalization

of Eα,β . Indeed, from (1)k=k! one obtains that E1

α,β (γ= 1) coincides with Eα,β .

For these functions, one has the following Laplace transform identity (cf. [21, eq.

(5.1.26) of p. 102] & [26, eq. (1.9.13) of p. 47])

L[tβ−1Eγ

α,β (λtα)](s) = sαγ −β

(sα−λ)γ(ℜ(s)>0; ℜ(β)>0; |λs−α|<1).(2.8)

In the remainder part of this subsection we list some of the properties required

for the sake of the rest of the paper. We start to point out the following decay

estimates, involving the two-parameter Mittag-Leﬄer functions (2.5), that will be

employed on the sequel of results to be proved in Section 3.

Theorem 2.1 (cf. [35], Theorem 1.6).If α≤2,βis an arbitrary real number, θ

is such that πα

2< θ < min{π, πα}and Cis a real constant, then

|Eα,β (z)| ≤ C

1 + |z|(z∈C;θ≤ |arg(z)| ≤ π)

Theorem 2.2 (cf. [38], Theorem 4. & [4], Proposition 4.).The following

optimal estimates are fulﬁlled by Eα,β(−z):

1

1 + Γ(1 −α)z≤Eα,1(−z)≤1

1 + 1

Γ(1+α)z(z≥0 ; 0 < α ≤1);

1

1 + qΓ(1−α)

Γ(1+α)z2≤Γ(α)Eα,α(−z)≤1

1 + qΓ(1+α)

Γ(1+2α)z2(z≥0 ; 0 < α ≤1);

1

1 + Γ(β−α)

Γ(β)z≤Γ(β)Eα,β (−z)≤1

1 + Γ(β)

Γ(β+α)z(z≥0; 0 < α ≤1; β > α).

Last but not least, the next lemma will be applied in Section 4 on the proof

of the solution representation for both Cauchy problems, Cauchy Problem 1.1

resp. Cauchy Problem 1.2.

Lemma 2.3 (cf. [21], p. 99).Let z∈Cbe given.

(i) If α, β, γ ∈Care such that ℜ(α)>0,ℜ(β)>0,ℜ(β−α)>0, then

zEγ

α,β(z) = Eγ

α,β−α(z)−Eγ−1

α,β−α(z).

8 N. FAUSTINO AND J. MARQUES

(ii) If α, β, γ ∈Care such that ℜ(α)>0,ℜ(β)>0,α−β /∈N0, then

zEα,β (z) = Eα,β −α(z)−1

Γ(β−α).

(iii) If α, β ∈Care such that ℜ(α)>0,ℜ(β)>1, then

αE2

α,β (z) = Eα,β−1(z)−(1 + α−β)Eα,β (z).

2.2. Fourier analysis, function spaces and associated operators. Let us now

denote by S(Rn) the Schwartz space over Rnand by S′(Rn) the space of tempered

distributions (dual space of S(Rn)). Throughout this paper we also denote by Lp

(1 ≤p≤ ∞) the standard Lebesgue spaces and by k · kpits norm. We adopt the

bracket notation h·,·i as the inner product underlying to the Hilbert space L2. The

Fourier transform of ϕ∈ S(Rn) is deﬁned as (cf. [40, Chapter A])

(2.9) bϕ(ξ) := (Fϕ)(ξ) = ZRn

ϕ(x)e−ix·ξdx

and the corresponding inverse Fourier transform as

(2.10) ϕ(x) := (F−1bϕ)(x) = 1

(2π)nZRnbϕ(ξ)eix·ξdξ ,

where x·ξdenotes the standard Euclidean inner product between x, ξ ∈Rn.

We note that the action of the isomorphism F:S(Rn)−→ S(Rn) can be ex-

tended to Lp−spaces, using the fact that S(Rn) is a dense subspace of Lp(Rn), for

values of 1 ≤p < ∞, and to S′(Rn) via the duality relation

hbϕ, ψi=hϕ, b

ψi, ϕ ∈ S(Rn), ψ ∈ S′(Rn).

In particular, the Plancherel identity

1

(2π)n

2kbϕk2=kϕk2

(2.11)

yields from the fact that the automorphism ϕ7→ 1

(2π)n

2bϕunderlying to S(Rn)

yields an unitary operator over L2(Rn).

The Fourier transform (2.9) and the convolution product over Rn

(ϕ∗ψ)(x) = ZRn

ϕ(x−y)ψ(y)dy,(2.12)

intertwined by the Fourier convolution formula (cf. [31, Theorem 5.8])

F(ϕ∗ψ)(ξ) = bϕ(ξ)b

ψ(ξ)(2.13)

share many interesting features of the Lp−spaces. Of special interest so far is the

Young’s inequality (cf. [31, Theorem 4.2])

kψ∗ϕkq≤κpκr

κqn

kψkrkϕkp,(2.14)

that holds for every 1 ≤p, q, r ≤ ∞ such that 1

r+1

p=1

q+ 1.Here and elsewhere,

κs=qs1

s(s′)−1

s′(1

s+1

s′= 1) stands for the sharp constant appearing on the

right-hand side of (2.14).

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 9

Despite the Caputo-Djrbashian derivative (1.2), the fractional diﬀerential oper-

ator (−∆)γ

2(γ≥0), deﬁned viz the spectral formula

F(−∆)γ

2ϕ(ξ) = |ξ|γbϕ(ξ),(2.15)

or equivalenty, as (−∆) γ

2=F−1|ξ|γF, is also required on the formulation of

Cauchy Problem 1.1 &Cauchy Problem 1.2. Underlying to (−∆) γ

2are the

homogeneous Sobolev spaces of order γ≥0

˙

Wγ,p (Rn) := nϕ∈ S′(Rn) : (−∆) γ

2ϕ∈Lp(Rn)o,

induced by the seminorm ϕ7→ k(−∆) γ

2ϕkp.

From now on, we will adopt the standard notation ˙

Hγ(Rn) when we are refer-

ing to ˙

Wγ,2(Rn). Also, the so-called Riesz fractional integral operator of order γ,

deﬁned for values of 0 < γ < n by (cf. [37, Section 25])

(−∆)−γ

2ϕ(x) = (Rγ∗ϕ)(x),with Rγ(x) = 2−γπ−n

2Γn−γ

2

Γγ

21

|x|n−γ

(2.16)

can also be represented in terms of its Fourier multiplier. Namely, one has (cf. [37,

Theorem 25.1])

F(−∆)−γ

2ϕ(ξ) = |ξ|−γbϕ(ξ),for 0 < γ < n.

In our approach, it is also of foremost interest to apply Sobolev embedding results

to obtain dispersive estimates under some Sobolev type regularity conditions on the

initial datum. Going into details, the existence of Sobolev inequalities follow from

a breakthrough result on fractional integral operators, coined in a series of books

and papers by Hardy-Littlewood-Sobolev (HLS) inequality (cf. [37, Theorem 3.5]).

The duality between the sharp HLS inequality and the sharp Sobolev inequality

brought to light ﬁrstly by Lieb in [30]. Such inequality, established in the paper

[10] of Cotsiolis-Tavoularis for the fractional operators (−∆) γ

2, by combining the

original argument of [30], already noted before on [31, Theorem 8.3 & Theorem

8.4] for sharp Sobolev inequalities involving the gradient operator (see also [18] for

further comparisons).

Speciﬁcally, for 0 < γ < n

2and p=2n

n−2γthe sharp fractional Sobolev inequality

(cf. [31, cf. Theorem 5.7]))

kϕkp≤(Sn,γ )1

2k(−∆)γ

2ϕk2, Sn,γ =Γn−2γ

2

22γπγΓn+2γ

2 Γ(n)

Γn

2!2γ

n

(2.17)

yields from the sharp HLS inequality

ZZRn×Rn

ϕ(x)ϕ(y)

|x−y|λdxdy≤πλ

2Γn−λ

2

Γn−λ

2 Γ(n)

Γn

2!1−λ

n

kϕk2

p,(2.18)

where 0 < λ < n and p=2n

2n−λ. In other words, the technique employed in [10, 18]

starts from the observation that, for each 0 < γ < n

2, the Riesz kernel of (−∆)−γ,

R2γ(x) (see eq. (2.16)), corresponds to the fundamental solution of (−∆)γso that

the substitution λ=n−2γin the inequality (2.18) gives rise to

ZRn

ϕ(x).(−∆)−γϕ(x)dx≤Sn,γkϕk2

p,

10 N. FAUSTINO AND J. MARQUES

˙

H0(Rn) = L2(Rn) (γ= 0)

˙

H0(Rn)

Figure 1. Function space equality involving the L2−space and

the homogeneous Sobolev space ˙

H0.

p=2n

n−2γ(2 < p < ∞)

Lp(Rn)

˙

Hγ(Rn) (0 < γ < n

2)

Figure 2. Function space embedding obtained from the sharp

fractional Sobolev inequality (2.17).

where Sn,γ stands for the sharp constant appearing in eq. (2.17).

On the sequel of main results to be proved in the end of Section 4, one also needs

the notion of mixed-normed Lebesgue spaces Ls

tLq

x. For values of 1 ≤s < ∞, we

deﬁne Ls

tLq

x([0, T ]×Rn), with 0 < T < ∞, as the Banach space with norm

kukLs

tLq

x([0,T ]×Rn)= ZT

0ku(t, ·)ks

qdt!1

s

(1 ≤q≤ ∞).(2.19)

For s=∞, the space L∞

tLq

x([0, T ]×Rn) is deﬁned in terms of the norm

kukL∞

tLq

x([0,T ]×Rn)= ess sup

t∈[0,T ]ku(t, ·)kq(1 ≤q≤ ∞).(2.20)

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 11

3. Dispersive Estimates

3.1. Proof Strategy. In this section we shall deduce estimates for pseudo-diﬀerential

operators of the type (−∆) η

2Eα,β −λ(−∆)σ

2tα, represented through the pseudo-

diﬀerential formula

(−∆)η

2Eα,β −λ(−∆)σ

2tαϕ(x) =

=1

(2π)nZRn|ξ|ηEα,β (−λ|ξ|σtα)bϕ(ξ)eix·ξdξ,(3.1)

where 0 < α ≤1, β, σ > 0, λ∈Cand η∈R.

Recall that the Fourier convolution formula (2.13) together with eq. (2.12) states

that eq. (3.1) can be rewritten as

(−∆)η

2Eα,β −λ(−∆)σ

2tαϕ(x) = (Kη

σ,n(t, ·|α, β, λ)∗ϕ)(x),(3.2)

with

Kη

σ,n(t, x|α, β, λ) = 1

(2π)nZRn|ξ|ηEα,β (−λ|ξ|σtα)eix·ξdξ.(3.3)

Then, for every 1 ≤p, q, r ≤ ∞ satisfying 1

r+1

p=1

q+ 1, the condition

Eα,β −λ(−∆)σ

2tαϕ∈˙

Wη,q (Rn) is satisﬁed for every ϕ∈Lp(Rn) whenever

Kη

σ,n(t, ·|α, β, λ)∈Lr(Rn).

On the other hand, due to the fact that ξ7→ |ξ|ηEα,β (−λ|ξ|σtα) is radially

symmetric one can recast eq. (3.3) in terms of the Hankel transform of order ν,e

Hν

(cf. [15]), deﬁned for values of ν > −1 by

e

Hνφ(τ) = Z∞

0

φ(ρ) (τρ)−νJν(τρ)ρ2ν+1 dρ (τ > 0),

where Jνstands for the Bessel function of order ν(cf. [26, Subsection 1.7]).

Namely, from the Fourier inversion formula for radial symmetric functions (cf.

[37, p. 485, Lemma 25.1])

ZRn

φ(|ξ|)eix·ξdξ =(2π)n

2

|x|n

2−1Z∞

0

φ(ρ)Jn

2−1(ρ|x|)ρn

2dρ

it immediately follows that eq. (3.3) simpliﬁes to

Kη

σ,n(t, x|α, β, λ) = 1

(2π)n

2|x|n

2−1Z∞

0

ρηEα,β (−λρσtα)Jn

2−1(ρ|x|)ρn

2dρ

=1

(2π)n

2e

Hn

2−1φ(|x|),with φ(|x|) = |x|ηEα,β(−λ|x|σtα).(3.4)

Thereby, the representation formula (3.4) shows in turn that the condition

Kη

σ,n(t, ·|α, β, λ)∈Lr(Rn) is assured by the boundedness of the Hankel trans-

form e

Hn

2−1on the weighted Lebesgue spaces Lr((0,∞), ρn−1dρ), tactically inves-

tigated on De Carli’s paper [15]. To exploit De Carli’s framework to our scope,

we shall obtain ﬁrstly estimates for ρ7→ ρηEα,β (−λρσtα) w.r.t. the norm of

Lr((0,∞), ρn−1dρ).

On the sequence of results to be proved on the subsequent subsections and else-

where, the constants to be adopted are listed below to avoid cluttering up the

notations.

12 N. FAUSTINO AND J. MARQUES

Notation 3.1. We denote by

(a) ωn−1=2πn

2

Γn

2the (n−1)−dimensional measure of the sphere Sn−1;

(b) κs=qs1

s(s′)−1

s′(with 1

s+1

s′= 1) the sharp constant underlying to

Young’s inequality (2.14);

(c) C(s)

r,σ,n(α, β, λ) the constant that yields from Proposition 3.2;

(d) D(s)

r,σ,n(α, λ) the constant that yields from Proposition 3.3;

(e) Sn,γ the constant provided by the sharp Sobolev inequality (2.17).

3.2. Lp−Lqand Lp−˙

Wσ,q decay estimates. Our ﬁrst aim is to establish under

which conditions Eα,β −λ(−∆)σ

2tαϕbelongs to Lq(Rn) or to ˙

Wσ,q(Rn). In

concrete, the following Mellin’s integral identities

Z∞

0

ρn−rs−1

(1 + bρσ)adρ =Γn−rs

σΓaσ+rs−n

σ

σΓ(a)b−n−rs

σ

(3.5)

0<ℜn−rs

σ<ℜ(a),

Z∞

0

ρn+r(σ−s)−1

(1 + bρσ)2rdρ =

Γn+r(σ−s)

σΓr(σ+s)−n

σ

σΓ(2r)b−n+r(σ−s)

σ

(3.6)

0<ℜn−rs

σ<ℜ(r),

that yield from [26, eq. (1.4.65) of p. 24], together with Theorem 2.1 & Theorem

2.2 will be used on the proof of Lemma 3.2 & Lemma 3.3 to ensure afterwards, on

the proof of Proposition 3.5, that the kernel Kη

σ,n(·, x|α, β, λ), represented through

eq. (3.4), belongs to Lr(Rn). The proof of both results is enclosed on appendix A

and appendix B.

Lemma 3.2 (see appendix A).Let 0< α ≤1,β > 0,σ > 0and s≥0be given.

In case that rsatisﬁes the following condition:

r < n

s(s6= 0) ∧max 1,n

σ+s< r < ∞

one has

Z∞

0ρ−sEα,β (−λρσtα)rρn−1dρ1

r

≤

≤C(s)

r,σ,n(α, β, λ)1

r

|Γ(β)|t−α

σ(n

r−s),(3.7)

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 13

with C(s)

r,σ,n(α, β, λ)equals to

Γn−rs

σΓr(2σ+s)−n

σ

σΓ(2r) sΓ(1 + α)

Γ(1 + 2α)λ!−n−rs

σ

, λ ≥0, β =α

Γn−rs

σΓr(σ+s)−n

σ

σΓ(r)Γ(β)

Γ(β+α)λ−n−rs

σ

, λ ≥0, β ∈ {1}∪ (α, +∞)

CrΓn−rs

σΓr(σ+s)−n

σ

σΓ(r)|λ|−n−rs

σ,πα

2< θ < πα , θ ≤ | arg(λ)| ≤ π.

Here, the constant C > 0– that yields from Theorem 2.1 – is independent of

α, β, r, s, σ, n and θ.

Lemma 3.3 (see appendix B).Let 0< α ≤1,σ > 0and s≥0be given. In case

that rsatisﬁes the following condition:

r < n

s(s6= 0) ∧max 1,n

σ+s< r < ∞

one has

Z∞

0ρσ−sEα,α(−λρσtα)rρn−1dρ1

r

≤

≤D(s)

r,σ,n(α, λ)1

r

Γ(α)t−α

σ(n

r+σ−s),(3.8)

with D(s)

r,σ,n(α, λ)equals to

Γn+r(σ−s)

σΓr(σ+s)−n

σ

σΓ(2r) sΓ(1 + α)

Γ(1 + 2α)λ!−n+r(σ−s)

σ

, λ ≥0

CrΓn+r(σ−s)

σΓr(σ+s)−n

σ

σΓ(2r)|λ|−n+r(σ−s)

σ,πα

2< θ < πα;θ≤ | arg(λ)| ≤ π.

Here, the constant C > 0– that yields from Theorem 2.1 – is independent of

α, β, r, s, σ, n and θ.

Remark 3.4.Due to the fact that the estimates depicted in Theorem 2.2 are indeed

sharp (cf. [38, 4]), one can say that the estimates obtained in lemmata 3.2 and 3.3

are sharp whenever λ≥0. The strategy considered to compute the best upper

bound, in terms of the norm of weighted Lebesgue space Lr((0,∞), ρn−1dρ), relies

essentially on a straightforward application of Mellin’s integral identities (3.5) and

(3.6), respectively.

Proposition 3.5. Let us assume that 0 < α ≤1, β > 0 and σ > n

2. If

max n1,n

σo< r ≤2 and 1 ≤p, q ≤ ∞

14 N. FAUSTINO AND J. MARQUES

are such that 1

r+1

p=1

q+ 1, then

Eα,β −λ(−∆)σ

2tαϕ

q≤

≤κpκr

κqnωn−1C(0)

r,σ,n(α, β, λ)1

r

(2π)n

2|Γ(β)|t−α

σ·n

rkϕkp,(3.9)

(−∆)σ

2Eα,α −λ(−∆)σ

2tαϕ

q≤

≤κpκr

κqnωn−1D(0)

r,σ,n(α, λ)1

r

(2π)n

2Γ(α)t−α−α

σ·n

rkϕkp,(3.10)

whereby ωn−1, κs, C(s)

r,σ,n(α, β, λ) and D(s)

r,σ,n(α, λ) (s= 0) stand for the constants

deﬁned in Notation 3.1.

Proof. By applying Young’s inequality (2.14) to the convolution formula (3.2) we

obtain that

(−∆)η

2Eα,β (−λ(−∆)σ

2tα)ϕ

q≤κpκr

κqn

Kη

σ,n(t, ·|α, β, λ)

rkϕkp

is satisﬁed for every 1 ≤p, q, r ≤ ∞ such that 1

r+1

p=1

q+ 1, whereby κsdenotes

the sharp constant arising from Young’s inequality (see Notation 3.1).

At this stage, we observe that the reformulation of Kη

σ,n(t, ·|α, β, λ), obtained in

eq. (3.4), in terms of the Hankel transform e

Hn

2−1gives rise to the norm equality

Kη

σ,n(t, ·|α, β, λ)

r=1

(2π)n

2ZRne

Hn

2−1[|x|ηEα,β (−λ|x|σtα) ]

r

dx1

r

Also, from eq. (3.4) we infer that x7→ e

Hn

2−1[|x|ηEα,β (−λ|x|σtα) ]

r

is radially

symmetric. Thereby, the change of variables to spherical coordinates gives rise to

Kη

σ,n(t, ·|α, β, λ)

r=(ωn−1)1

r

(2π)n

2Z∞

0e

Hn

2−1[ργEα,β (−λρσtα) ]

r

ρn−1dρ1

r

,

where ωn−1denotes the (n−1)−dimensional measure of the sphere Sn−1(see

Notation 3.1).

Then, for every max n1,n

σo< r ≤2, the optimal inequality

Kη

σ,n(t, ·|α, β, λ)

r≤(ωn−1)1

r

(2π)n

2Z∞

0|ρηEα,β (−λρσtα)|rρn−1dρ1

r

.

yields from the fact that ke

Hn

2−1k= 1 is the operator norm associated to the

following mapping property (cf. [15, Proposition 2.1.]), in case of 1 < r ≤2:

e

Hn

2−1:Lr(0,∞), ρn−1dρ−→ Lr(0,∞), ρn−1dρ.

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 15

This altogether with the norm estimates provided by lemmata 3.2 and 3.3 gives

rise to the inequalities (3.9) and (3.10). Indeed,

1≤max n1,n

σo<2∧max n1,n

σo≥n

σ

ensures that r−n

σ>0 is fulﬁlled for every max n1,n

σo< r ≤2 so that the

boundedness of e

Hn

2−1yield from the estimates obtained in Lemma 3.2 (case of

s= 0 and η= 0) and Lemma 3.3 (case of s= 0 and η=σ).

Remark 3.6.The proof of Proposition 3.5 encompasses part of the Lp−Lqdecay

estimates obtained by Pham et al in [34, Section 5.] (case of α= 1) and Kemppainen

et al in [25, Section 3.1] (case of σ= 2). We note here that our case roughly works

with the same arguments as in Pham et al. with exception of a slightly technical

condition: it is not necessary to assume, a priori, parity arguments encoded by the

dimension of the Euclidean space Rn(see [34, pp. 575-576] for further comparisons).

Remark 3.7.Although our technique of proof, depicted throughout Subsection 3.2,

is complementary to Kemppainen et al technique (cf. [25, Section 5]), it allows

us to rid one of the ma jors stumbling blocks on the computation of Lp−decay

estimates for the fundamental solution (the kernel function (3.3) in our case), whose

technicality of proofs heavily relies on the asymptotics of Fox-H functions (see [25,

Section 3]).

3.3. ˙

Hγ−Lqand ˙

Hγ−˙

Wσ,q decay estimates. Now, we turn our attention to

the homogeneous Sobolev spaces discussed in the end of Subsection 2.2. Although

the next proposition follows mutatis mutandis the technique of proof considered in

Proposition 3.5, we provide below a self-contained proof of it for reader’s conve-

nience.

Proposition 3.8. Let 0 < α ≤1, β > 0, σ > 0 and 0 < γ < n be given. We

assume that one of the following conditions is fulﬁlled for every 1 ≤p, q, r ≤ ∞,

satisfying 1

r+1

p=1

q+ 1:

(i) Sobolev embedding case

max n1,n

σo< r ≤2∧p=2n

n−2γ

in case of 0 < γ < n

2∧σ > n

2+γ.

(ii) Sobolev embedding cannot be applied in general

max 1,n

σ+γ< r ≤2∧p= 2

in case of 0 < γ < n

2∧σ > n

2−γ.

(iii) No Sobolev embedding

max 1,n

σ+γ< r < n

γ∧p= 2

in case of n

2≤γ < n and σ > 0.

16 N. FAUSTINO AND J. MARQUES

Then, we ﬁnd the following estimates to hold:

(i) Sobolev embedding case

Eα,β (−λ(−∆)σ

2tα)ϕ

q≤

≤κpκr

κqnωn−1C(0)

r,σ,n(α, β, λ)1

r(Sn,γ )1

2

(2π)n

2|Γ(β)|t−αn

σr

(−∆)γ

2ϕ

2,(3.11)

(−∆)σ

2Eα,α −λ(−∆)σ

2tαϕ

q≤

≤κpκr

κqnωn−1D(0)

r,σ,n(α, λ)1

r(Sn,γ )1

2

(2π)n

2Γ(α)t−α−αn

σr

(−∆)γ

2ϕ

2.(3.12)

(ii) Sobolev embedding cannot be applied in general

+

(iii) No Sobolev embedding

Eα,β (−λ(−∆)σ

2tα)ϕ

q≤

≤κpκr

κqnωn−1C(γ)

r,σ,n(α, β, λ)1

r

(2π)n

2|Γ(β)|t−α

σ(n

r−γ)

(−∆)γ

2ϕ

2,(3.13)

(−∆)σ

2Eα,α −λ(−∆)σ

2tαϕ

q≤

≤κpκr

κqnωn−1D(γ)

r,σ,n(α, λ)1

r

(2π)n

2Γ(α)t−α−α

σ(n

r−γ)

(−∆)γ

2ϕ

2.(3.14)

Hereby ωn−1, κs, C(s)

r,σ,n(α, β, λ), D(s)

r,σ,n(α, λ) (s= 0 or s=γ) and Sn,γ stand for

the constants deﬁned in Notation 3.1.

Proof. For the proof of (i), we note that for each 0 < γ < n

2the substitution

p=2n

n−2γpermits us to obtain an upper bound for the right-hand side of the

inequalities (3.9) and (3.10) provided by Proposition 3.5 . Thereafter, the set of

inequalities (3.11) and (3.12) are thus immediate from the sharp Sobolev inequality

(2.17).

For the proof of (ii) and (iii), we recall that the convolution representation of

(3.1) – provided by eq. (3.2) – can be reformulated as follows:

(−∆)η

2Eα,β −λ(−∆)σ

2tαϕ(x) = Kη−γ

σ,n (t, ·|α, β, λ)∗(−∆) γ

2ϕ(x).

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 17

Then, by a straightforward application Young’s inequality (2.14) there holds

(−∆)η

2Eα,β (−λ(−∆)σ

2tα)ϕ

q≤

≤κpκr

κqn

Kη−γ

σ,n (t, ·|α, β, λ)

rk(−∆)γ

2ϕk2.(3.15)

On the other hand, to prove the boundedness of

Kη−γ

σ,n (t, ·|α, β, λ)

rwe ob-

serve that in the view of lemmata 3.2 and 3.3, the optimal inequality

Kη−γ

σ,n (t, ·|α, β, λ)

r≤(ωn−1)1

r

(2π)n

2Z∞

0ρη−γEα,β (−λρσtα)rρn−1dρ1

r

,

obtained from [15, Proposition 2.1.], is attained for s=γ, with 0 < γ < n, whenever

one of the following conditions hold:

(ii) max 1,n

σ+γ< r ≤2 in case of 0 < γ < n

2∧σ+γ > n

2;

(iii) max 1,n

σ+γ< r < n

γin case of n

2≤γ < n.

Thus, the estimates (3.13) and (3.14) are thus immediate from the combination

of the sharp inequality (3.15) with the estimates provided by lemmata 3.2 and 3.3,

respectively.

Remark 3.9.Despite the estimates obtained in lemmata 3.2 and 3.3 (which are

sharp in case of λ≥0), we also have adopted the sharp estimates obtained by Lieb

and Loss in [31, Theorem 4.2] (Young’s inequality), De Carli in [15] (Hankel trans-

form sharp estimates) and Cotsiolis and Tavoularis in [10] (sharp Sobolev inequality

in terms of (−∆) γ

2) to derive Proposition 3.5 and, subsequently, Proposition 3.8.

Remark 3.10.Worthy of mention, from the estimates obtained in lemmata 3.2

and 3.3 it was not necessary to investigate the Mellin integral representation of

the kernel (3.3) as a Fox-H function (see, for instance, the proof of [25, Lemma

3.3] for further comparisons). In our case, a sharp estimate result on the Hankel

transform, obtained by De Carli in [15], allowed us to downsize the computation of

the [optimal] estimates from the decay of the generalized Mittag-Leﬄer functions

Eα,β (cf. Theorem 2.2). Such approach goes towards the optimal bound argument

considered by Kemppainen et al in [24, Remark 3.1].

STRUCTURALLY DAMPED σ−EVOLUTION EQUATIONS WITH POWER-LAW MEMORY 19

σ

γ

On

n

n

2

n

2

Figure 4. Geometric interpretation of the parameter constraints

considered in Proposition 3.8, for values of 0 < σ < n and

0< γ < n. Case (i) corresponds to the dark gray triangle; case

(ii) to the union of the two triangular regions – including the line

segment connecting the points n

2,0and n, n

2; case (iii) to the

rectangular region and to the line