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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 $\sigma-$evolution equation of the type \begin{equation*} \partial_t^2 u(t,x)+\mu (-\Delta)^{\frac{\sigma}{2}} \partial_t u(t,x)+(-\Delta)^\sigma u(t,x)=f(t,x), \end{equation*} with $\sigma>0$ and $\mu>-2$, that encodes memory of \textit{power-law} type. To do so, we replace the time derivatives $\partial_t$ and $\partial_t^2$ by the so-called Caputo-Djrbashian derivatives $\partial_t^\beta$ of order $\beta=\alpha$ and $\beta=2\alpha$, respectively, and the inhomogeneous term $f(t,x)$ by the Riemann-Liouville integral $I^{1-\alpha}_{0^+}f(t,x)$, whereby $0<\alpha\leq 1$. For the solution representation of the underlying Cauchy problems of parabolic type (case of $0<\alpha\leq \frac{1}{2}$) and hyperbolic type (case of $\frac{1}{2}<\alpha\leq 1$) on the space-time $[0,T]\times \mathbb{R}^n$, we then consider the wide class of pseudo-differential operators, endowed by the fractional Laplacian $-(-\Delta)^{\frac{\sigma}{2}}$ and the two-parameter Mittag-Leffler functions $E_{\alpha,\beta}$. 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.
arXiv:2212.10463v1 [math.AP] 20 Dec 2022
STRUCTURALLY DAMPED σEVOLUTION 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 σevolution equation of the type
2
tu(t, x) + µ(∆) σ
2tu(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 I1α
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. LpLqand 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. ˙
HγLqand ˙
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 σevolution equation
(1.1) 2
t(t, x) + µ(∆)σ
2tu(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 σevolution 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
2tu(t, x) (µ6= 0) resembles to a
regularization of the weak damping term µ∂tu(t, x), in the limit σ0+.
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
(tτ)mβ1
Γ(mβ)
mu(τ, x)
∂τ m , m 1< β < m
mu(t, x)
∂tm, β =m
,(1.2)
STRUCTURALLY DAMPED σEVOLUTION 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 (βstands 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
Imβ
0+f(t, x) :=
Zt
0
(tτ)mβ1
Γ(mβ)f(τ, x) , 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 gmβ(t) := tmβ1
Γ(mβ), is a
probability density function over (0,) converging to the delta function δ(t), in
the limit βm.
Some of the physical reasons for the choice of the Caputo-Djrbashian derivative
(1.2) instead of the Riemann-Liouville derivative RLβ
t:= m
tImβ
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<
α1), carrying the initial condition(s) u0(x) [and u1(x)] and the inhomogeneous
term I1α
0+f(t, x), fulfilling certain regularity conditions to be presented a posteriori.
Here and elsewhere I1α
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 ]×RnRis a (weak) solution of
2α
tu(t, x) + µ(∆)σ
2α
tu(t, x) + (∆)σu(t, x) = I1α
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 ]×RnRis a (weak) solution of
2α
tu(t, x) + µ(∆)σ
2α
tu(t, x) + (∆)σu(t, x) = I1α
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 LpLqestimates 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 zC) and on the papers [38, 4] (case of z0) 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 σEVOLUTION 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 λ0, 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,)
eωt|g(t)|<, the Laplace transform of gis
defined as
G(s) := L[g(t)](s) = Z
0
estg(t)dt ((s)> ω).(2.1)
Here, we notice that the function G(the Laplace image of g) is analytic on the
right-half plane {sC:(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) = L1[G(s)](t), the
integration is performed along the strip [ci, c +i] (c:= (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(tτ)g(τ)(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)
m1
X
k=0
sβk1k
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,...,m1),
hold for every gCm(0,) such that (t)mgL1(0, b) for any b > 0 (cf. [26,
Lemma 2.24]). On the other hand, the identity
L[Imβ
0+g(t)] = sβmL[g](s) ((s)> ω)(2.4)
is always satisfied in case of gL1(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)((α)>0),(2.5)
Eγ
α,β (z) =
X
k=0
(γ)k
k!
zk
Γ(αk +β)((α)>0),(2.6)
where Γ(·) stands for the Euler’s Gamma function
Γ(z) = Z
0
ettz1dt ((z)>0)(2.7)
and (γ)k=Γ(γ+k)
Γ(γ)((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
et2dt = 1 ez2E1
2,1(z);
STRUCTURALLY DAMPED σEVOLUTION 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 (z);
(vi) the generalized hypergeometric function
1Fmγ;β
m,β+ 1
m,...,β+m1
m;zm
mm= Γ(β)Eγ
m,β (z) (mN).
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[tβ1Eγ
α,β (λtα)](s) = sαγ β
(sαλ)γ((s)>0; (β)>0; |λsα|<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 α2,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|(zC;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)zEα,1(z)1
1 + 1
Γ(1+α)z(z0 ; 0 < α 1);
1
1 + qΓ(1α)
Γ(1+α)z2Γ(α)Eα,α(z)1
1 + qΓ(1+α)
Γ(1+2α)z2(z0 ; 0 < α 1);
1
1 + Γ(βα)
Γ(β)zΓ(β)Eα,β (z)1
1 + Γ(β)
Γ(β+α)z(z0; 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 zCbe given.
(i) If α, β, γ Care such that (α)>0,(β)>0,(βα)>0, then
zEγ
α,β(z) = Eγ
α,βα(z)Eγ1
α,βα(z).
8 N. FAUSTINO AND J. MARQUES
(ii) If α, β, γ Care such that (α)>0,(β)>0,αβ /N0, then
zEα,β (z) = Eα,β α(z)1
Γ(βα).
(iii) If α, β Care such that (α)>0,(β)>1, then
αE2
α,β (z) = Eα,β1(z)(1 + αβ)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 ,·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)eix·ξdx
and the corresponding inverse Fourier transform as
(2.10) ϕ(x) := (F1bϕ)(x) = 1
(2π)nZRnbϕ(ξ)eix·ξ ,
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 Lpspaces, 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
(ϕψ)(x) = ZRn
ϕ(xy)ψ(y)dy,(2.12)
intertwined by the Fourier convolution formula (cf. [31, Theorem 5.8])
F(ϕψ)(ξ) = bϕ(ξ)b
ψ(ξ)(2.13)
share many interesting features of the Lpspaces. Of special interest so far is the
Young’s inequality (cf. [31, Theorem 4.2])
kψϕkqκpκr
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 σEVOLUTION EQUATIONS WITH POWER-LAW MEMORY 9
Despite the Caputo-Djrbashian derivative (1.2), the fractional differential oper-
ator (∆)γ
2(γ0), defined viz the spectral formula
F(∆)γ
2ϕ(ξ) = |ξ|γbϕ(ξ),(2.15)
or equivalenty, as (∆) γ
2=F1|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 γ0
˙
Wγ,p (Rn) := nϕ S(Rn) : (∆) γ
2ϕLp(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) = (Rγϕ)(x),with Rγ(x) = 2γπn
2Γnγ
2
Γγ
21
|x|nγ
(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
n2γthe sharp fractional Sobolev inequality
(cf. [31, cf. Theorem 5.7]))
kϕkp(Sn,γ )1
2k(∆)γ
2ϕk2, Sn,γ =Γn2γ
2
22γπγΓn+2γ
2 Γ(n)
Γn
2!2γ
n
(2.17)
yields from the sharp HLS inequality
ZZRn×Rn
ϕ(x)ϕ(y)
|xy|λdxdyπλ
2Γnλ
2
Γnλ
2 Γ(n)
Γn
2!1λ
n
kϕk2
p,(2.18)
where 0 < λ < n and p=2n
2nλ. 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=n2γin the inequality (2.18) gives rise to
ZRn
ϕ(x).(∆)γϕ(x)dxSn,γkϕk2
p,
10 N. FAUSTINO AND J. MARQUES
˙
H0(Rn) = L2(Rn) (γ= 0)
˙
H0(Rn)
Figure 1. Function space equality involving the L2space and
the homogeneous Sobolev space ˙
H0.
p=2n
n2γ(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 σEVOLUTION 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, λCand ηR.
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 ϕLp(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 (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·ξ =(2π)n
2
|x|n
21Z
0
φ(ρ)Jn
21(ρ|x|)ρn
2
it immediately follows that eq. (3.3) simplifies to
Kη
σ,n(t, x|α, β, λ) = 1
(2π)n
2|x|n
21Z
0
ρηEα,β (λρσtα)Jn
21(ρ|x|)ρn
2
=1
(2π)n
2e
Hn
21φ(|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
21on the weighted Lebesgue spaces Lr((0,), ρn1), 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,), ρn1).
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) ωn1=2πn
2
Γn
2the (n1)dimensional measure of the sphere Sn1;
(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. LpLqand 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
ρnrs1
(1 + σ)a =Γnrs
σΓ+rsn
σ
σΓ(a)bnrs
σ
(3.5)
0<nrs
σ<(a),
Z
0
ρn+r(σs)1
(1 + σ)2r =
Γn+r(σs)
σΓr(σ+s)n
σ
σΓ(2r)bn+r(σs)
σ
(3.6)
0<nrs
σ<(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 s0be given.
In case that rsatisfies the following condition:
r < n
s(s6= 0) max 1,n
σ+s< r <
one has
Z
0ρsEα,β (λρσtα)rρn11
r
C(s)
r,σ,n(α, β, λ)1
r
|Γ(β)|tα
σ(n
rs),(3.7)
STRUCTURALLY DAMPED σEVOLUTION EQUATIONS WITH POWER-LAW MEMORY 13
with C(s)
r,σ,n(α, β, λ)equals to
Γnrs
σΓr(2σ+s)n
σ
σΓ(2r) sΓ(1 + α)
Γ(1 + 2α)λ!nrs
σ
, λ 0, β =α
Γnrs
σΓr(σ+s)n
σ
σΓ(r)Γ(β)
Γ(β+α)λnrs
σ
, λ 0, β {1} (α, +)
CrΓnrs
σΓr(σ+s)n
σ
σΓ(r)|λ|nrs
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 s0be given. In case
that rsatisfies the following condition:
r < n
s(s6= 0) max 1,n
σ+s< r <
one has
Z
0ρσsEα,α(λρσtα)rρn11
r
D(s)
r,σ,n(α, λ)1
r
Γ(α)tα
σ(n
r+σs),(3.8)
with D(s)
r,σ,n(α, λ)equals to
Γn+r(σs)
σΓr(σ+s)n
σ
σΓ(2r) sΓ(1 + α)
Γ(1 + 2α)λ!n+r(σs)
σ
, λ 0
CrΓn+r(σs)
σΓr(σ+s)n
σ
σΓ(2r)|λ|n+r(σs)
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 λ0. The strategy considered to compute the best upper
bound, in terms of the norm of weighted Lebesgue space Lr((0,), ρn1), 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
κqnωn1C(0)
r,σ,n(α, β, λ)1
r
(2π)n
2|Γ(β)|tα
σ·n
rkϕkp,(3.9)
(∆)σ
2Eα,α λ(∆)σ
2tαϕ
q
κpκr
κqnωn1D(0)
r,σ,n(α, λ)1
r
(2π)n
2Γ(α)tαα
σ·n
rkϕkp,(3.10)
whereby ωn1, 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
qκpκr
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
21gives rise to the norm equality
Kη
σ,n(t, ·|α, β, λ)
r=1
(2π)n
2ZRne
Hn
21[|x|ηEα,β (λ|x|σtα) ]
r
dx1
r
Also, from eq. (3.4) we infer that x7→ e
Hn
21[|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=(ωn1)1
r
(2π)n
2Z
0e
Hn
21[ργEα,β (λρσtα) ]
r
ρn11
r
,
where ωn1denotes the (n1)dimensional measure of the sphere Sn1(see
Notation 3.1).
Then, for every max n1,n
σo< r 2, the optimal inequality
Kη
σ,n(t, ·|α, β, λ)
r(ωn1)1
r
(2π)n
2Z
0|ρηEα,β (λρσtα)|rρn11
r
.
yields from the fact that ke
Hn
21k= 1 is the operator norm associated to the
following mapping property (cf. [15, Proposition 2.1.]), in case of 1 < r 2:
e
Hn
21:Lr(0,), ρn1 Lr(0,), ρn1.
STRUCTURALLY DAMPED σEVOLUTION 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,
1max n1,n
σo<2max n1,n
σon
σ
ensures that rn
σ>0 is fulfilled for every max n1,n
σo< r 2 so that the
boundedness of e
Hn
21yield 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 LpLqdecay
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 Lpdecay
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. ˙
HγLqand ˙
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 2p=2n
n2γ
in case of 0 < γ < n
2σ > n
2+γ.
(ii) Sobolev embedding cannot be applied in general
max 1,n
σ+γ< r 2p= 2
in case of 0 < γ < n
2σ > n
2γ.
(iii) No Sobolev embedding
max 1,n
σ+γ< r < n
γp= 2
in case of n
2γ < 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
κqnωn1C(0)
r,σ,n(α, β, λ)1
r(Sn,γ )1
2
(2π)n
2|Γ(β)|tαn
σr
(∆)γ
2ϕ
2,(3.11)
(∆)σ
2Eα,α λ(∆)σ
2tαϕ
q
κpκr
κqnωn1D(0)
r,σ,n(α, λ)1
r(Sn,γ )1
2
(2π)n
2Γ(α)tααn
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
κqnωn1C(γ)
r,σ,n(α, β, λ)1
r
(2π)n
2|Γ(β)|tα
σ(n
rγ)
(∆)γ
2ϕ
2,(3.13)
(∆)σ
2Eα,α λ(∆)σ
2tαϕ
q
κpκr
κqnωn1D(γ)
r,σ,n(α, λ)1
r
(2π)n
2Γ(α)tαα
σ(n
rγ)
(∆)γ
2ϕ
2.(3.14)
Hereby ωn1, 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
n2γpermits 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) = Kηγ
σ,n (t, ·|α, β, λ)(∆) γ
2ϕ(x).
STRUCTURALLY DAMPED σEVOLUTION 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
Kηγ
σ,n (t, ·|α, β, λ)
rk(∆)γ
2ϕk2.(3.15)
On the other hand, to prove the boundedness of
Kηγ
σ,n (t, ·|α, β, λ)
rwe ob-
serve that in the view of lemmata 3.2 and 3.3, the optimal inequality
Kηγ
σ,n (t, ·|α, β, λ)
r(ωn1)1
r
(2π)n
2Z
0ρηγEα,β (λρσtα)rρn11
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
2σ+γ > n
2;
(iii) max 1,n
σ+γ< r < n
γin case of n
2γ < 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 λ0), 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].
18 N. FAUSTINO AND J. MARQUES
0< α 1, β > 0
σ > n
2Proposition 3.5
1p <
n
2γ < n
σ > 0
Proposition 3.8 (iii)
p= 2
0< γ < n
2
σ > n
2γ
Proposition 3.8 (ii)
p= 2
σ > n
2+γ
Proposition 3.8 (i)
p=2n
n2γ
Figure 3. Schematic represention of the parameter constraints
considered on Proposition 3.5 and Proposition 3.8.
STRUCTURALLY DAMPED σEVOLUTION 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