A SECOND ORDER FRACTIONAL DIFFERENTIAL EQUATION UNDER EFFECTS OF A SUPER DAMPING model to treat dynamics of nonlinear networks in elastic crystals.

. In this work we study asymptotic properties of global solutions for an initial value problem of a second order fractional diﬀerential equation with structural damping. The evolution equation considered includes plate equation problems. We show asymptotic proﬁles depending on the exponents of the Laplace operators involved in the equation and optimality of the decay rates for the associated energy and the L 2 norm of solutions.


1.
Introduction. In this work we consider the Cauchy problem for a generalized second order evolution equation given by      u tt + (−∆) δ u tt + (−∆) θ u t + (−∆) α u = 0, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), (1.1) with u = u(t,x), (t,x) ∈ (0,∞)×R n (n ≥ 1) and the exponents of the Laplace operators α, δ and θ satisfying α > 0 and 0 < δ ≤ α, α+δ 2 ≤ θ < α+δ or α < θ if δ = 0. The Cauchy problem we consider in (1.1) models several physical phenomena, especially hydrodynamic problems such as propagation of waves in shallow waters. Problems of plate vibrations, wave propagation in general are also modeled by this type of equation as it appears in (1.1), depending on the choice of the fractional exponents of the Laplace operators. Several other applications can be studied by this model, for example in [19] (see also [1]) Maugin proposed a type of Boussinesq model to treat dynamics of nonlinear networks in elastic crystals. In Sander-Hutter [26] a brief account of the history of the evolution of these problems in hydrodynamics can be found, especially in the theory of solitary waves. The first attempt of mathematical modeling of waves in shallow water was made by Joseph Valentin Boussinesq  in 1871, who took into account the vertical accelerations of the fluid particles and admitted a solitary wave as a solution. However, he neglected a number of derivative product terms in his modeling, considering only nonlinear first-order effects. These equations are called classical Boussinesq equations. In order to obtain better models, some of these hydrodynamic phenomena and other equations were later developed, originating from the classical equations, considering higher order terms that Boussinesq ignored. These equations are called the Boussinesq class. Among them these are equations of Korteweg-de Vries studied by Green-Naghdi [8], Peregrine [23] and Serre [27]. Other equations related to those of Boussinesq, are the so-called IBq (improved Boussinesq) and IMBq (improved modified Boussinesq). More details on Boussinesq equations and their characteristics can be found in the articles due to Esfahani-Farah-Wang [5], Wang-Chen [33], Wang-Xue [36], Wang-Xu [34] and [35].
For the model we consider in this work, when δ = α = 2 we have a sixth order linear Boussinesq equation and when, in addition, θ = 1, we have a Boussinesq equation under the effect of a hydrodynamic dissipation. We note that when δ = 1, α = 2, θ = 0, we have in (1.1) a linear equation for plate vibrations under rotational inertial effects (see [6]) and frictional dissipation. Particular results on some problems associated with equations together with these properties can be found in Charão-Luz-Ikehata [2], Luz-Charão [17], Sugitani-Kawashima [29] and the references therein. Moreover, when α = 1, δ = θ = 0 we have a linear wave equation with frictional dissipation.
The results in this paper include regularity-loss type of estimates, which have their origin in [10,11,21], and seem to be new for general equations considered here. For particular cases of our problem we can cite several works [2,3,14,15,16,17,20,28,29,30,33], and especially Horbach-Ikehata-Charão [9]. Some of our mathematical tools and ideas that are used to develop part of this work follow several ideas from those works.
We also mention that our work generalizes the works by Ikehata [13,12] which gave us the idea to study the system with strong damping considered in this work. We note that in [13] Ikehata studies only the case θ = 2, α = 1 and δ = 0, so even in the case of (1.1) with θ = 2, α = 1, and δ = 1 (rotational inertia effect exists): the corresponding study seems to be quite new. Important works that are deserved to be mentioned are special cases of the equation (1.1), and are reported in the references of the classic works due to Matsumura [20] and Ponce [25]. Several results on the IMBq and the generalized Boussinesq equations can be seen in Wang-Chen [32,33] and in Wang-Xue [36], respectively. This paper is organized as follows. In Section 2 we collect several facts and notation, which will be used in the next sections. In Section 3 we study the special case of (1.1) with δ = 0 and θ > α, that is, and mainly prove the following result, which shows the optimality of the decay rates of the solution in L 2 -sense to problem (1.2). In this connection, one can cite recent results due to [4] and [24] on the precise L 1 ∩ L m -L m (m ∈ (1, 2]) and L p -L q estimates of solutions to the equation (1.2) with θ ∈ (0, α] and α ≥ 1. The case of [4] and [24] does not include such regularity-loss structure.
for all t 1, where C 1 and C 2 are positive constants, and that is, the regularity of the initial data u 1 depend on the choice of κ.
In Section 4 we consider the general case 0 < δ ≤ α, and the condition to strong damping we consider is α+δ 2 ≤ θ < α + δ. In fact, we can report the following general result, in the case of δ > 0 and α+δ 2 ≤ θ < α + δ, concerning the optimality of the L 2 -norm of solutions to problem (1.1).
Then there exist constants C 1 > 0, C 2 > 0 and t 0 1 such that holds for t ≥ t 0 , where u(t, x) is the solution to problem (1.1), and Remark 2. The case 0 ≤ θ ≤ α+δ 2 was essentially studied in [9]. The case θ ≥ α+δ is still open. It should be emphasized that the results in Theorem 1.2 are essentially new under the regularity-loss type structure of the equations with rotational inertia effects. The new result just obtained above includes the case of δ = 1, α = 1, and θ ∈ (1, 2) in (1.1) as an example.
In order to obtain the decay rates, we have employed the method of energy in the Fourier space (see [31]) combined with the explicit solution of the associated problem in the Fourier space, and an asymptotic profile obtained from the explicit solution. Our aim is mainly concentrated on proving the optimality of decay rates for the L 2 norm of solutions as shown in above theorems, although we can also prove the optimality for decay rates of the L 2 norm of the derivatives of solutions by using the same method.
2. Notation and basic estimates. We consider the following spaces 4436   RUY COIMBRA CHARÃO, JUAN TORRES ESPINOZA AND RYO IKEHATA for m, p ∈ R n , with p ≥ 1 and its norm given by Note that by definition u ∈Ẇ m,p (R n ) implies v = (−∆) m 2 u ∈ L p (R n ). We also consider the usual Sobolev spaces H s (R n ), s ∈ R, with norms given by where Fu =û represents the usual Fourier transform of u .
The following lemma justifies the equivalence between the norms given above and the usual norm in H s . Lemma 2.1. For all ξ ∈ R n and s ≥ 0 it holds that For u ∈ H α (R n ) the operator (−∆) α is defined via the Fourier transform by With this definition we can characterize the Fourier transform of (−∆) α u , u ∈ H α (R n ), as The next two lemmas can be proved in a standard way (see Horbach-Ikehata-Charão [9]).
ii) If 0 < κ ≤ 1 and f ∈ L 1,κ (R n ) then for all ξ ∈ R n it is true that with L, N , K and M positive constants depending only on the dimension n and κ.
The next lemma is well known. The proofs of next two lemmas appear in [9] for the case α = 1. The case α > 0 is quite similar.
Lemma 2.5. Let n > 2α and θ > α 2 . Then there exists t 0 > 0 such that for t ≥ t 0 with C a positive constant depending only on n, θ and α.
Lemma 2.6. Let n ≥ 1 and θ > α 2 . Then there exists t 0 > 0 such that for t ≥ t 0 , where C is a positive constant depending only on n, θ and α.
Our goal in the next sections is to obtain decay rates, an asymptotic expansion (profile) and to prove the optimality of the decay rates to the Cauchy problem (1.1) with the exponent θ of the damping term to be large. Now we are going to study optimal decay rates for the L 2 norm of solutions. Decay rates for L 2 norms of the derivatives are also obtained in this work, but we do not prove the optimality. Indeed, by using the same method and similar estimates to the ones for L 2 -norm of solutions the optimality can be proved more easily. We begin with considering two cases on the exponents in (1.1): • Case δ = 0, 0 < α < θ.
• Case 0 < δ ≤ α, α+δ 2 ≤ θ < α + δ. We note that in case δ > 0 the restriction θ < α + δ is considered to get the optimality of the decay rates. The optimality for the case α + δ ≤ θ we leave it open, however, to get the decay rates such additional restriction is not necessary.
3. Case δ = 0 and θ > α. In this section we prove L 2 decay rates to the linear problem (1.1) when δ = 0, that is to a Cauchy problem of type with (t, x) ∈ (0, ∞) × R n and the assumption θ > α > 0. The results of this section generalize the work by Ikehata-Iyota [13] that considered the particular case α = 1 and θ = 2.
Remark 3 (Existence of solutions). On the existence of solutions to the Cauchy problem (3.1) we note that we can write it in an abstract form as follows From the definition of the operator A we may see that it is self-adjoint and nonnegative. In addition, for initial data in the energy space This regularity says that u t ∈ C (0, ∞), D((−∆) θ ) . Note that the item (b) is the regularizing effect of the strong damping.
3.1. Decay Estimates. We apply an improvement (cf. [9]) of the energy method in the Fourier space (cf. [31]) to obtain the decay estimates. For this purpose, set which defines the associated density of energy in the Fourier space, defined for t ≥ 0 and ξ ∈ R n . Furthermore, one defines the important function ρ : A way to choose the best function ρ(ξ) can be seen in [18]. Then as in the similar argument to [13] and/or [18] one can arrive at the following significant energy estimate in the Fourier space. and holds for all ξ ∈ R n and t ≥ 0.
Applying the above Lemma 3.1 we can prove the next result on the total energy. The result below is a generalization to α > 0 of that already obtained for the case of α = 1 in [13]. We state it without proof because the proof is almost the same as that of [13].
On the asymptotic behavior of the L 2 -norm of solutions one has the following result by basing on Lemma 3.1. One also mentions without proof (see [13]).
In order to impose less regularity on the initial data we may only consider u 1 ∈Ẇ κ−α,2 (R n ) with κ > 0. In this case we get the following estimate on the zone of high frequency Asymptotic expansion: estimates for low frequencies. In this subsection we shall consider the asymptotic expansion of the solution in the L 2 -norm. The result in this subsection is a generalization of [13], which dealt with the case of δ = 0, α = 1, and θ = 2. For the sake of reader's convenience we shall state its full proof based on Lemma 3.1.
We first observe that the characteristic equation associated to the Cauchy problem (3.2) in the Fourier space is given by We are interested in the case θ > α > 0. The associated characteristics roots are We fix 0 < δ 0 < 1. In this case we have |ξ| 4θ−2α − 4 ≤ δ 4θ−2α 0 − 4 ≤ −3 < 0 for ξ ∈ R n such that |ξ| ≤ δ 0 < 1. Thus, the the characteristics roots are complex and we can write them as The explicit solution of (3.2) is given bŷ Now, using Lemma 2.2 forû 0 andû 1 we define the functions whereû j = A j (ξ) − iB j (ξ) + P j , i = 0, 1, and Applying the mean value theorem on the functions sin(b(ξ)t) and cos(b(ξ)t) we get that To obtain an asymptotic expansion forû(t, ξ) we need to deal with b −1 (ξ) in the first term of the right hand side ofû(t, ξ). Using the mean value theorem we can see that 2 That is 2 Therefore, we have for |ξ| ≤ δ 0 < 1, t > 0 To simplify the notation we define the functions We need L 2 estimates for K j (t, ξ), (i = 1, . . . , 6) on the zone of low frequency |ξ| ≤ δ 0 < 1 and t > 0. We use Lemmas 2.3 and 2.4 and the fact that a(ξ) = |ξ| 2θ 2 . From the estimate In fact, .
On the other hand, it is easy to see that , with constants C > 0 and d > 0, and t ≥ 1.
Next step is to prove Theorem 1.1 by relying on Proposition 1. For this we need the upper bound for the following quantities.
• For n > 2α Proof of Theorem 1.1. By using the Plancherel theorem, Lemmas 2.5, 2.6 and Proposition 1 we first obtain for t ≥ max{1, t 0 }, where t 0 is fixed by the above mentioned lemmas. Now, for κ > (n−2α)(θ−α) 2θ we can see from (3.9) that there exists a constant C 1 > 0 such that the estimate from below u(t, ·) ≥ C 1 |P 1 |t − n−2α 4θ holds for t 1. On the other side, using again the Plancherel theorem, (3.7), (3.8) and Proposition 1 we can get the following estimate from above provided that κ > (n−2α)(θ−α) 2θ and t ≥ 1, where I 0 is a constant defined in the statement of Theorem 1.1. Hence, we have just proved Theorem 1.1, which shows the optimality of the decay rates of the solution to problem (1.2). 4. Case δ > 0 and α+δ 2 ≤ θ < α + δ. These conditions under consideration on the exponents of the Laplace operator include the case we consider as a strong damping θ > α. An equivalent problem for the case α+δ 2 ≥ θ with α = 2 was studied in [9]. The aim of this section is to get optimal decay rates to the Cauchy problem (1.1) under the assumption 0 < δ ≤ α, α+δ 2 ≤ θ < α + δ. The results of this section essentially new under the effect of rotational inertia terms. where υ = θ α > 0. The equation (4.1) is written in an abstract form described in Theorem 2.1 of [7], except for the linear and continuous operator P . In fact, in [7] P = Id the identity operator. But reviewing the proof of existence and regularity in the article by [7], it can be proved that the bounded linear operator P does not affect the existence of solution to the Cauchy problem associated with the equation (4.1) and thus to problem (1.1).
To get several estimates to (1.1), for the case in consideration, we again consider the associated Cauchy problem in the Fourier space given by 4.1. Decay estimates. In this subsection we also use the energy method in the Fourier space. This framework is standard nowadays and we develop this section in a similar way to the previous sections. To do that, we define the functionals of energy which satisfy the following identities of energy.
with ε > 0 to be chosen. This function ρ(ξ) is quite similar to the one defined in (3.4). The time derivative of e(t, ξ) combined with (4.4) says that the next identity follows.
H α+ , t > 0, with C > 0 a constant that is possible to be calculated explicitly.
Proof. By the Plancherel theorem and Lemma 4.2, for t > 0 it holds that We now estimate the integral on the right hand side of the above inequality on the zones of low and high frequency.
To the L 2 norm of the solution itself we prepare the next result.
Proof. By Lemma 4.2 we have for ξ = 0, • On the zone of low frequency |ξ| ≤ 1 one can see as in the proof of Theorem 4.3 that ρ(ξ) ≥ c 1 |ξ| 2θ . Thus • On the region of high frequency |ξ| ≥ 1 we can also observe as in the proof of Theorem 4.3 that the estimate So, in a similar way to the previous theorem we can conclude that Remark 6. Note that in the proof, the assumption u 1 ∈ H δ+ −α (R n ) can be changed by u 1 ∈Ẇ δ+ −α,2 (R n ) in the case > 0.
Therefore, applying Lemma 2.2, we can rewrite the explicit solution forû aŝ u(t, ξ) = e −a(ξ)t P 0 cos(|ξ| α t) + e −a(ξ)t P 1 sin(|ξ| α t) |ξ| α + K 1 (t, ξ) (4.17) to the case θ ≥ α+δ 2 , where P 0 and P 1 are again defined as (see Lemma 2.2) The first two terms in the right side of (4.17) give the leading term of the solutions. In fact, in the next step we are going to prove that the functions K i , i = 1, · · · , 6 decay faster than such leading term.
Similarly to the previous section we get L 2 decay estimates for low frequencies to the functions K i (t, ξ), i = 1, ..., 6, using the assumptions on the exponents and Lemma 2.4.
The above estimates imply that the next result is true.

4.4.
Optimal decay rates. Finally, combining the estimates for low and high frequency in previous subsections we get the next lemma.