On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates

Transcription

1 On a class of nonlinear viscoelastic Kirchhoff plates: well-poseness an general ecay rates M. A. Jorge Silva Department of Mathematics, State University of Lonrina, Lonrina, PR, Brazil. J. E. Muñoz Rivera National Laboratory of Scientific Computation, Petrópolis, RJ, Brazil, an Institute of Mathematics, Feeral University of Rio e Janeiro, Rio e Janeiro, PR, Brazil. R. Racke Department of Mathematics an Statistics, University of Konstanz, Konstanz, Germany. Abstract This paper is concerne with well-poseness an energy ecay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem correspons to a class of fourth orer viscoelastic equations with a non-locally Lipschitz perturbation of p- Laplacian type. The only amping effect is given by the memory component. We show that no aitional amping is neee to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy ecay are similar to ones given by the memory kernel, but generally not with the same spee, mainly when we consier the nonlinear problem with large initial ata. Keywors: Kirchhoff plates, well-poseness, p-laplacian, general rates ecay MSC: 35B35, 35B4, 35L75, 74D99. 1 Introuction This paper is motivate by moels of Kirchhoff plates subject to a weak viscoelastic amping u tt σ u tt + µ u + marcioajs@uel.br, corresponing author rivera@lncc.br reinhar.racke@uni-konstanz.e µ t s us s = F, 1.1 1

2 where σ > is the uniform plate thickness, the kernel µ > correspons to the viscoelastic flexural rigiity, an F = Fx, t, u, u t,... represents aitional amping an forcing terms. The unknown function u = ux, t represents the transverse isplacement of a plate filament with prescribe history u x, t, t. The erivation of the linear mathematical moel 1.1 with F = is given in Lagnese [16] an Lagnese an Lions [17], by assuming viscoelastic stress-strain laws on an isotropic material occupying a region of R 3 an constant Poisson s ratio. Lagnese [16, Chapter 6] stuie the behavior of the energy associate to the linear moel 1.1 in a boune omain R, by introucing bounary feeback laws which inuce further issipation in the system, geometrical escriptions of, an also µ C [,, µt >, µ t <, µ t, µ >, see also Lagnese [15]. Muñoz Rivera an Naso [6] consiere an abstract moel which encompasses equation 1.1 in the cases F = u t or else F = u t. They showe that the associate semigroup is not exponential stable in the first case weak amping whereas in the secon case strong amping the corresponing semigroup is exponential stable. We note that in both cases F introuces an aitional issipation to the system. More recently, Jorge Silva an Ma [13, 14] investigate the asymptotic behavior of a N-imensional system like 1.1 with σ = without rotational inertia, by consiering F = p u fu + hx + u t, where p u := iv u p u, p. Then 1.1 becomes to u tt + µ u p u + fu + µ t s us s u t = hx, In such case the strong amping plays an important role to obtain global well-poseness mainly uniqueness in higher imensions N 3 ue to the presence of the p-laplacian term p u. If we take u = for t, µ = 1 an gt = µ t, then 1.1 can be rewritten as follows, u tt σ u tt + u gt s us s = F. 1. Barreto et al. [3] investigate problem 1. in a boune omain R with mixe bounary conition, suitable geometrical hypotheses on, an F =. They establishe that the energy ecays to zero with the same rate of the kernel g such as exponential an polynomial ecay. To o so in the secon case they mae assumptions on g, g an g which means that g 1+t p for p >. Then they obtaine the same ecay rate for the energy. However, their approach can not be applie to prove similar results for 1 < p. Concerning N-imensional systems which cover the system 1. with σ =, both Cavalcanti et al. [5] an Anrae et al. [1] investigate the global existence, uniqueness an stabilization of energy. By taking a boune or unboune open set an F = M u x u t as a kin of non-egenerate weak amping, where Ms > m > for all s, the authors showe in [5] that the energy goes to zero exponentially provie that g goes to zero at the same form. In [1] the authors stuie the same concepts by

3 consiering a boune omain an F = p u fu + u t, but replacing the fourth orer memory term in 1. by a weaker memory of the form gt s us s. It is worth noting that in both cases, respectively, the weak or strong amping constitutes an important role to obtain uniqueness an energy ecay. If we consier 1. with the Laplace operator instea of the bi-harmonic one we get the moel u tt σ u tt u + gt s us s = F, 1.3 which correspons to a viscoelastic wave equation of secon orer. Equation 1.3 an relate quasilinear problems with u t ρ u tt instea, ρ >, have been extensively stuie by many researches with possible external forces F like source f 1 u an amping f u t. See for instance [4, 6, 1, 11, 1, 19,,, 3, 4, 3, 31, 3] an the references therein. In 8 Messaoui [, 3] establishe a general ecay of the energy solution to a viscoelastic equation corresponing to 1.3 with σ =, by taking F = an F = u γ u, γ >. More precisely, he consiere the following ecay conition on the memory kernel g t ξtgt, t >, 1.4 uner proper conitions on ξt >, an prove general ecay of energy such as Et c e c 1 ξs s, t, 1.5 for some c, c 1 > epening on the weak initial ata. Ever since several authors have use this conition to obtain arbitrary ecay of energy for problems relate to 1.3. See for instance the papers by Han an Wang [1, 11], Liu [19], Liu an Sun [1], Park an Park [3]. It is worth pointing that in all papers mentione above when authors eal with nonlinear systems then c 1 is a proportional constant to E enote here by c 1 E, but it is not specifie how this occurs. More recently, in [1], the author has illustrate that 1.5 provies ecay rates which are faster than exponential one in the linear case. See also Messaoui et al. [18, 5], Tatar [31] an Wu [3] for other kins of interesting arbitrary ecay rates in viscoelastic wave moels relate to 1.3. There are also previous an recent works which encompass viscoelastic wave equations in a history framework an only employ memory issipation to treat asymptotic behavior of solutions. We refer, for instance, the papers by Dafermos [7], Giorgi et al. [9], Muñoz Rivera an Salvatierra [7], an Pata [8, 9], Fabrizio et al. [8] an Araújo []. Our main goal in the present paper is to iscuss the well-poseness an the asymptotic behavior of energy to the following nonlinear viscoelastic Kirchhoff plate equation u tt σ u tt + u ivf u with simply supporte bounary conition gt s us s = in R +, 1.6 u = u = on R +, 1.7 3

4 an initial conitions u, = u an u t, = u 1 in, 1.8 where is a boune omain of R N with smooth bounary, σ, F : R N R N is a vector fiel an g : [, R + is a real function. The hypotheses are given later. Essentially, we consier the moel 1. with F = ivf u which constitutes a nonlinear an non-locally Lipschitz perturbation. Our main results are Theorems.3,.8 an.1. Making a comparison with the above relate papers on the viscoelastic plate moel 1. our main results yiel the following improvements an contributions: 1. The only amping effect is cause by the memory term. Besies, our conition on the kernel g like 1.4 is less restrictive than those use in [3, 6, 5, 1]. Nevertheless, our general ecay of energy see.14 an.17 generalizes all results on stability obtaine in [1, 3, 6, 5]. We also specify how the ecay rate epens on the initial ata.. The p-laplacian term p u is consiere as a particular case. We show that in the presence of the rotational inertia term σ > the well-poseness of is achieve without strong amping term. Moreover, for σ all results on stability are shown by exploiting only the memory issipation. No aitional weak or strong issipation is necessary. Therefore our results improve those ones given in [1, 5, 13]. 3. No further C -smoothness is impose on the relaxation function g as regare e.g. in [3, 5]. Moreover, the ecay rate 1 + t κ hols for every κ > when g has a polynomial behavior. We note that the case < κ was not approache in [3]. 4. The parameter σ is relate to the uniform plate thickness an the results on stability hol by moving σ [, uniformly. Further, there is no result by now which treats the asymptotic behavior for plates with perturbation of p-laplacian type just using issipation from the memory. We also exemplify other interesting types of ecay rates of energy beyon exponential an polynomial ones. Remark 1.1. It is worth pointing out that the parameter σ changes the character of the system 1.6 epening whether it is null or not. In the case of choosing σ >, the term σ u tt acts as a regularizing term by allowing us to consier stronger solutions an uniqueness see Theorem.3. This is possible because the rotational inertia term gives a way to control the nonlinear perturbation ivf u. On the other han, if we consier σ = in 1.6 then we can also check the existence of weak solutions see Theorem.1 i but uniqueness an stronger solutions are not provie once the term ivf u spoils the estimates along with lack of regularity for u t. In spite of having two ifferent systems accoring to parameter σ all results on stability hol in both cases see Theorems.8 an.1 ii. In the secon case when σ = the stability is obtaine first for approximate solutions an then for weak solutions by taking lim inf on the approximate energy. The rest of the paper is organize as follows. In Section we fix some notations an present our assumptions an main results. Section 3 is evote to show that problem 4

5 is well pose. Section 4 is eicate to the proof of the energy ecay. Finally, section 5 consists in an appenix where we first give some examples for ifferent rates of ecay. Then we provie some properties an examples for the vector fiel F. Assumptions an main results We begin by introucing the following Hilbert spaces V = L, V 1 = H 1, V = H H 1, an with norms V 3 = {u H 3 H 1 ; u H 1 }, u V = u, u V1 = u, u V = u, an u V3 = u, respectively. As usual, p means the L p -norm as well as, enotes either the L - inner prouct or else a uality pairing between a Banach space V an its ual V. The constants λ, λ 1, λ, λ > represent the embeing constants λ u u, λ 1 u u, λ u u, λ = 1 λ λ, for u V 1. We also consier the following phase spaces with their respective norms H = V V 1 with u, v H = u + v, H 1 = V 3 V with u, v H 1 = u + v, W = V V with u, v W = u + v..1 The problem with rotational inertia Let us first consier 1.6 with σ >. Without loss of generality we can take σ = 1. Setting I = [, T ] with T > arbitrary, weak solutions are efine as follows. Given initial ata u, u 1 H, we call a function U := u, u t CI, H a weak solution of the problem on I if U = u, u 1 an, for every ω V, [ ] u t t, ω + u t t, ω t + F ut, ω + ut, ω gt s us, ω s = a.e. in I. The energy corresponing to the problem with rotational inertia is efine as Et = 1 u tt + 1 u tt + ht ut + 1 g ut + f ut x,.1 5

6 where ht is given below in.6 an g wt := Now let us precise the hypotheses on g an F. gt s wt ws s. Assumption A1. The C 1 -function g : [, R + satisfies l := 1 gs s > an g t, t.. Assumption A. F : R N R N is a C 1 -vector fiel given by F = F 1,..., F N such that F j z k j 1 + z p j 1/, z R N,.3 where, for every j = 1,..., N, we consier k j > an p j satisfying p j 1 if N = 1, an 1 p j N + N if N 3..4 Moreover, F is a conservative vector fiel with F = f, where f : R N value function satisfying R is a real with α an α [, λ. α αl z fz F z z + αl z, z R N,.5 Remark.1. From the choice of l an α we have ht := 1 gs s l, t, an β := l 1 αλ >..6 Also, applying.4 it follows from Sobolev embeing that V W 1,p j+1, j = 1,..., N..7 Thereby, the constants µ p1,..., µ pn > represent the embeing constants for u pj +1 µ pj u, j = 1,..., N. Remark.. Without loss of generality we can consier F =. Inee, if F = F, then we efine Gz = F z F so that G satisfies G =, G j z = F j z, j = 1,..., N, an Gz = fz, where fz = fz F z. Also, it is easy to check that f an Gz satisfy.5 for some constants α, α [, λ. Therefore, G is a C 1 -conservative vector fiel satisfying In the Section 5 we give some examples of vector fiels satisfying such properties. 6

7 Our first two main results establish the Haamar well-poseness of with respect to weak solutions, an a general ecay rate of the energy. Theorem.3 Well-Poseness. Uner Assumptions A1 an A we have: i If u, u 1 H 1, then problem has a stronger weak solution satisfying u L locr +, V 3, u t L locr +, V, I u tt L locr +, V 1..8 ii If u, u 1 H, then problem has a weak solution satisfying u L locr +, V, u t L locr +, V 1, I u tt L locr +, V..9 iii In both cases we have continuous epenence on initial ata in H, that is, given U = u, u 1, V = v, v 1 H, let us consier the corresponing weak solutions U = u, u t, V = v, v t of the problem Then Ut V t H C T U V H, t I,.1 for some constant C T = C U H, V H, T >. In particular, problem has a unique weak solution. Remark.4. The proof of existence is given by the Faeo-Galerkin metho. We first prove the existence of stronger weak solutions an then the existence of a weak solution is given by ensity arguments. The uniqueness follows as a consequence of the continuous epenence of stronger an weak solutions. The proofs are given in Section 3. Lemma.5. Uner the assumptions of Theorem.3 the energy Et satisfies t Et = 1 g ut gt ut, t >..11 Remark.6. From conitions. an.11 it follows that t Et is nonincreasing. Since gt g, for each t, if we take g = then g, g an Lemma.5 implies that Et is constant That is, the system is conservative. This motivates us to efine the following ecay conition on the memory kernel gt. Assumption A3. g >, an there exist a constant ξ an a C 1 -function ξ : [, R + such that g t ξtgt, t >,.1 an ξt >, ξ t, ξ t ξt ξ, t..13 Remark.7. The first two conitions in.13 allow us to conclue that ξt ξ := ξ 1 >, t. Also, conition.1 implies that the memory kernel has the uniform ecay gt ge ξs s, t. 7

8 Then our secon main result is given by the following Theorem.8. Uner the assumptions of Theorem.3, let u, u t be the weak solution of problem with given initial ata u, u 1 H. If we aitionally assume Assumption A3 an α = in.5, then where c = 3E e γ 1 ξs s >, an γ k 1 + [E] p 1 Et c e γ ξs s, t,.14 with k > an p = max j},...,n if E 1, min j},...,n if E < 1. Remark.9. Theorem.8 is prove in Section 4. Concerning to estimate.14 it is worth point out two issues: i when every component of F = F 1,..., F N is linear, namely, when p 1,..., p N = 1 in.3 an so p = 1, then the general estimate.14 is similar to the ecay of the memory kernel g an provies us several kins of ecay accoring to the feature of ξt inepenently of the size of the initial ata; ii otherwise, in the presence of the nonlinear perturbation F, then estimate.14 can be very slow for large initial ata even if the memory kernel g ecays quickly. In the Section 5 we consier some concrete examples for function ξt.. The problem without rotational inertia Let us now consier 1.6 with σ =. Let us also take I = [, T ] with T >. Given initial ata u, u 1 W, we say a function U := u, u t CI, W is a weak solution of on I if U = u, u 1 an, for every ω V, t u tt, ω + ut, ω + F ut, ω gt s us, ω s = a.e. in I. Now the energy associate to the problem without rotational inertia is given by Et = 1 u tt + ht ut + 1 g ut + f ut x..15 Our thir main result is the following theorem. Theorem.1. Uner Assumptions A1 an A, we have: i If u, u 1 W, then problem has a weak solution in the class u L locr +, V, u t L locr +, V, u tt L locr +, V..16 ii Besies, if Assumption A3 hols an α = in.5, then Et also satisfies Et c e γ ξs s, t,.17 where c > an γ > are given in terms of E as in Theorem.8. 8

9 Remark.11. To prove Theorem.1 i one also uses the Faeo-Galerkin metho. Only one a priori estimate is necessary to get a weak solution satisfying.16. Since we o not have regularity for u t the estimate.17 is shown first for the approximate energy. Then the proof of Theorem.1 ii will hol true by passing the approximate energy to the limit. The etails of the proof of Theorem.1 are very similar to those ones use in the proofs of Theorem.3 an Theorem.8. Thus we omit them here. 3 Well-poseness In this section we prove Theorem.3. We start with the following approximate problem u n ttt, ω j + u n ttt, ω j + u n t, ω j F u n t, ω j gt s u n s, ω j s =, u n = u n an u n t = u n 1, 3. for j = 1,..., n, which has a local solution u n t = n y jn tω j [ω 1,..., ω n ], on [, t n, n N, given by ODE theory, where ω j j N is an orthonormal basis in V given by eigenfunctions of with bounary conition 1.7. The a priori estimates below imply that the local solution can be extene to the interval [, T ] an allow us to conclue the existence of a weak solution. Proof of Theorem.3 i. Let us take regular initial ata u, u 1 H 1 := V 3 V. Then we consier the approximate problem with u n u in V 3 an u n 1 u 1 in V. 3.3 A Priori Estimate I. Replacing w j by u n t t in 3.1 an since it hol gt s u n s, u n t t s = 1 { t } g u n t gs s u n t t + 1 g u n t 1 gt un t, 3.4 an it follows that F u n t u n t t x = f u n t u n t t x = f u n t x, 3.5 t t En t = 1 g u n t 1 gt un t, t >, 3.6 9

10 where E n t = 1 un t t + 1 un t t + ht un t + 1 g un t + f u n t x. Assumption. an 3.6 imply that E n t E n. From.6 an the first conition in.5 we get ht un t + f u n t x β un t α, an consequently, 1 un t t + β un t E n t + α E n + α. From 3.3, secon conition in.5, 5.3 an Höler s inequality we conclue u n t t + u n t M 1, t [, T ], n N, 3.7 where M 1 = M 1 u 1, u, >. A Priori Estimate II. Replacing w j by u n t t in 3.1, since 3.4 hols with in the place of, an also F u n t u n t t x = F u n t u n t x + J F, t with J F given by [ = F1 u n t u n t t,..., F N u n t u n t t ] u n t x, J F then we infer t F n t = 1 g u n t 1 gt un t + J F J F, t >, 3.8 where F n t = 1 un t t + 1 un t t + ht un t + 1 g un t I F with I F = F u n t u n t x. Let us estimate the right han sie of 3.8. Using assumption.3, generalize Höler inequality, an.7 we get J F F j u n t u n t t u n t x k j 1 + u n t p j 1/ u n t t u n t x p k j µ pj j +1 + u n t 1 p j +1 u n t t u n t.

11 From estimate 3.7 an using again.7 we obtain p k j µ pj j +1 + u n t p j +1 C <. From this an Young s inequality there exists a constant C 1 = C 1 u 1, u > such that J F C 1 u n t t + u n t. 3.9 Inserting 3.9 in 3.8 an integrating from to t T, yiels F n t F n + C 1 u n t s + u n s s, t. 3.1 On the other han, from 5.3 in the appenix with F = an Höler s inequality, we have I F F u n t u n t x K K u n t + u n t p j+1/ u n t x N u n t + u n t Moreover, the estimates 3.7 an.7 imply K N u n t + u n t p j +1 p j +1 p j +1 p j +1 u n t. C <. Using Young s inequality there exists a constant C = C u 1, u > such that Since ht l, then an, consequently, Combining 3.1 an 3.11 we arrive at I F C + l 4 un t. ht un t I F l 4 un t C, l 4 un t t + l 4 un t F n t + C u n t t + u n t 4 l C + F n + 4C 1 l 11 u n t s + u n s s.

12 Taking into account 3.3 an applying Gronwall s inequality, we finally conclue u n t t + u n t M, t [, T ], n N, 3.1 where M = M u 1, u,, T >. The estimates 3.7 an 3.1 are sufficient to pass to the limit in the approximate problem an to obtain a stronger weak solution satisfying I u tt = u + ivf u + u, u t C [, T ], H L, T ; H 1, T >, 3.13 gt s us s in L, T ; V This finishes the proof of the existence of regular weak solutions. Remark 3.1. Unless for the term which involves ivf u, the limit on the other terms in the approximate system can be one in a usual way. With respect to this term we only nee to apply estimates 3.7 an 5. along with the Aubin-Lions Lemma. Then it will hol later in the case of weak solutions, see for instance [1, 13]. Proof of Theorem.3 iii stronger weak solutions. We first show that solution in 3.13 satisfies the continuous epenence property.1. Let us consier two stronger weak solutions U = u, u t, V = v, v t of the problem corresponing to initial ata U = u, u 1, V = v, v 1 H 1, respectively. By setting w = u v, then function w, w t = U V solves the equation w tt w tt + w in L, T ; V 1, with initial ata w, w t = U V. gt s ws s = ivf u ivf v 3.15 Since w t t V V 1, then multiplying equation 3.15 by w t t an integrating over, we get where t W t = 1 g w t 1 gt wt + L F L F, t >, 3.16 W t = 1 w tt + 1 w tt + ht wt + 1 g wt, t, an L F = [F ut F vt] w t t x. 1

13 Estimate 5. from the appenix, the generalize Höler inequality, an.7 imply L F F ut F vt w t t x K K 1 + ut / + vt p j 1/ wt w t t x p µ pj j +1 + ut p j +1 + vt From.7 an 3.13 we obtain K p µ pj j +1 + ut p j +1 + vt p j +1 p j +1 wt w t t. C <, an making use of Young inequality there exists a constant C 3 = C 3 u 1, u > such that L F C 3 wt + w t t Inserting 3.17 into 3.16 an integrating from to t T, one has W t W + C 3 On the other han it is easy to check that ws + w t s s, t wt + w t t W t, t, 3.19 l an W λ w + w t. 3. Combining an applying Gronwall s inequality we conclue wt + w t t C T w + w t, t [, T ], 3.1 for some constant C T = C U H, V H, T >. This shows that the estimate.1 is guarantee for regular solutions since we have w, w t = U V. Proof of Theorem.3 ii. Let us take initial ata u, u 1 H. Then there exists a sequence u n, u n 1 H 1 such that u n u in V an u n 1 u 1 in V For each regular initial ata u n, u n 1, n N, there exists a regular solution u n, u n t satisfying Taking the multiplier u n t t in 3.14 an proceeing analogously as in then estimate 3.7 hols true. This implies u n, u n t u, u t in L, T ; H

14 Besies, if we consier m, n N, m n, an w = u m u n, then function w, w t satisfies w tt w tt + w gt s ws s = ivf u m ivf u n in L, T ; V 1, with initial ata w, w t = u m u n, u m 1 u n 1. Taking the multiplier w t t an using analogous arguments as given in , the estimate 3.1 also hols. This means that u m t u n t + u m t t u n t t C u m u n + u m 1 u n 1, for any t [, T ], an some constant C = C u, u 1 H, T >. From 3.13 an 3., an since C is a constant epening only on the initial ata in H, we infer u n, u n t u, u t in C[, T ], H. 3.4 Finally, we note that the limits 3.3 an 3.4 are enough to pass to the limit in the approximate problem an to obtain a weak solution u, u t C [, T ], H satisfying.9 an I u tt = u + ivf u + This conclues the proof on existence of weak solutions. gt s us s in L, T ; V. Remark 3.. In a similar proceure we can check that conition 3.3 also hols in the problem without rotational inertia, namely, u n, u n t u, u t in L, T ; W. This is sufficient to pass the limit on the corresponing approximate problem to obtain u tt = u + ivf u + gt s us s in L, T ; V. Therefore, we can also conclue the proof of Theorem.1 i. Proof of Theorem.3 iii weak solutions. Given initial ata U = u, u 1, V = v, v 1 H, let us consier the corresponing initial regular ata U n = u n, u n 1, V n = v n, v1 n H 1 such that U n, V n U, V in H H, 3.5 an the respective regular solutions U n = u n, u n t, V n = v n, v n t converging to the weak solutions U = u, u t, V = v, v t as in 3.4, namely U n, V n U, V in C[, T ], H H. 3.6 Since.1 hols for stronger weak solutions we have U n t V n t H C T U n V n H, t [, T ], n N, 3.7 for some constant C T = C U H, V H, T >. Therefore,.1 is given for weak solutions after passing 3.7 to the limit when n an applying In particular, we have uniqueness of solution in both cases. This completes the proof of Theorem.3. 14

15 4 Uniform ecay of the energy Our methos on stability are similar to but not equal to those for viscoelastic wave equations, see for instance [11, 19,, 3, 3, 31]. The proofs of Lemma.5 an Theorem.8 are given first for regular solutions. Then by stanar ensity arguments the conclusion of Theorem.8 also hols for weak solutions. Proof of Lemma.5. By taking the multiplier u t with 1.6 an using the ientities for the solution, then the energy efine in.1 satisfies.11. Therefore, the proof of Lemma.5 follows reaily. Before proving Theorem.8 we nee to state some technical lemmas. Lemma 4.1. Uner the assumptions of Theorem.3 we have t Et 1 g ut, t >. 4.1 Proof. Inequality 4.1 is an immeiate consequence of Lemma.5. Let us first efine the functionals φ χ t = gt s χt χs s x, ψ χ t = ζ χ t = g t s χt χs s x, gt s χs s x. Lemma 4.. Uner the assumptions of Theorem.3 we have: a φ u t b φ u t 1 l λ 1 g ut, t. 1 l λ g ut, t. c φ u t 1 lg ut, t. e ψ u t g λ 1 g ut, t. ψ u t g λ g ut, t. f ζ u t 1 lg ut + 1 l ut, t. 15

16 Proof. To prove the items a-e it is enough to apply Höler s inequality along with the embeings V V 1 V an the first conition in.6. Moreover, from.6 an item c of Lemma 4., we prove the item f as follows. ζ u t gt s ut us + ut s x gt s ut us s + φ u t + gs s ut 1 lg ut + 1 l ut. gt s s ut x Let us now efine the functionals Gt = Et + ɛ 1 Φt + ɛ Ψt, t, 4. where ɛ 1, ɛ > will be fixe later an Φt = ξt ut t u t t ut x, 4.3 Ψt = ξt ut t u t t gt sut uss x. 4.4 Lemma 4.3. Uner the assumptions of Theorem.8 there exists a constant c > such that Φt given in 4.3 satisfies where β 1 = β/ >. [ ] t Φt c ξt u t t + u t t + g ut [ ] β1 ξt ut + Et, t >, 4.5 Proof. Differentiating t Φt, using equation 1.6 an integrating by parts we get where [ ] t Φt = ξt u t t + u t t + ξ tj 1 + ξtj [ ] ξt ut + F ut ut x, 4.6 J 1 = u t t, ut + u t t, ut, J = gt s us, ut s. 16

17 Now, applying Young s inequality with η 1 > an η >, it is easy to check that J 1 η 1 λ ut + 1 ut t + u t t, 4η 1 J gs s ut + η ut + 1 g ut. 4η Inserting these two last estimates into 4.6, using the thir conition in.13, Aing an subtracting ξtet, we obtain 3 Φt ξt t + ξ [ ] 1 u t t + u t t + ξt 4η g ut 4η ξt ht [ ] ut + ξt f ut F ut ut x + ξtλξ η 1 + η ut ξtet. Now applying assumption.5 an conition.6, we have 3 Φt ξt t + ξ [ ] 1 u t t + u t t + ξt 4η g ut 4η β ξt λξ η 1 η ut ξtet. 4.7 Since β 1 = β >, so choosing η = η 1 β 1 1+λξ, an setting c = max { 3 + ξ 4η 1, η 1 } in 4.7, we conclue that 4.5 hols true. This completes the proof of Lemma 4.3. Lemma 4.4. Uner the assumptions of Theorem.8, an given any δ >, then there exists a constant c δ > such that Ψ efine in 4.4 satisfies t Ψt δ1 + ξ + c δ 1 + [E] p 1 gs s [ ] ξt u t t + u t t + 4δξt ut ξtg ut + c δ g ut, t >, 4.8 where p = { max{p1,..., p N } if E 1, min{p 1,..., p N } if E < 1. Proof. Differentiating Ψ, using equation 1.6 an integrating by parts we get t [ ] t Ψt = gs s ξt u t t + u t t + 6 I j + I F,

18 where I 1 = ξ t u t t gt sut us s x, I = ξ t u t t gt s ut us s x, I 3 = ξt ut gt s ut us s x, I 4 = ξt gt s us s gt s ut us s x, I 5 = ξt u t t g t sut us s x, I 6 = ξt u t t g t s ut us s x, I F = ξt F ut gt s ut us s x, Now let us estimate I j, j = 1,..., 6, an I F. From Young s inequality with δ >, item a of Lemma 4. an assumption.13 we obtain I 1 ξ t ξt δ u ξt t t + 1 4δ φ ut δξ ξt u t t + ξ 4δλ 1 1 lξtg ut 4.1 Analogously, but using items b an c of Lemma 4. instea of a, we have I δξ ξt u t t + ξ 4δλ 1 lξtg ut, 4.11 I 3 δξt ut lξtg ut δ Again from Young s inequality with δ >, items c, f an of Lemma 4., we euce an I 4 δξtζ u t + 1 4δ ξtφ ut δ1 l ξt ut + δ lξtg ut, δ I 5 δξt u t t + 1 4δ ξtψ ut δξt u t t + g ξ 1 4δλ 1 g ut

19 Similarly with item e in the place of in Lemma 4., we also have I 6 δξt u t t + g ξ 1 4δλ g ut Now with respect to I F we have I F ξt gt s F ut ut us x s. } {{ } :=IF 1 Applying 5.3 from the appenix with F =, Höler s inequality, Young s inequality with δ >, an since V W 1,p j+1 V 1, j = 1,..., N, we infer N IF 1 K 1 + ut / ut ut us x K p j +1 + ut ut [ K λ δ ut + 1 4δ p j +1 ut pj +1 ut us p µ pj j +1 + ut p j +1 ] ut us [ K λ p µ pj j +1 + ut p j +1 ] } {{ } :=IF ut us. Since ut pj +1 µ pj ut an β ut Et E for any t >, then N IF K p µ λ pj j +1 + K p j +1 4 µ λ p j [E] 4 β µ 1 + µ [E] p 1, where we consier { max{p1,..., p p := N } if E 1, min{p 1,..., p N } if E < 1, N µ 1 := K p µ λ pj j +1 an µ := K λ Thus, I 1 F δ ut + 1 4δ from where it follows that I F δξt ut + 1 4δ p j +1 µ p j β µ 1 + µ [E] p 1 ut us 4 µ 1 + µ [E] p 1 ξtg ut

20 Inserting into 4.9, an since 1 l < 1, yiels t Ψt δ1 + ξ [ ] gs s ξt u t t + u t t + 4δξt ut + g ξ 1λ 4δ g ut ξ λ + 8δ + µ 1 + µ [E] p 1 ξtg ut. 4δ Therefore, inequality 4.8 follows by taking c δ = 1 4δ max{ + ξ λ + 8δ + µ 1, µ, gξ 1 λ}. This conclues the proof of Lemma 4.4. Lemma 4.5. Uner the assumptions of Theorem.8 an fixing any t >, then for some positive constant ɛ 1 t Gt ɛ 1ξtEt, t t, 4.17 c 1, with c 1+[E] p 1 1 > inepenent of the initial ata. Proof. From efinition of Gt in 4., an Lemmas 4.1, 4.3 an 4.4, we get ɛ t Gt 1 c + ɛ δb 1 gs s [ ] ξt u t t + u t t β 1 ɛ 1 4δɛ ξt ut ɛ 1 ξtet ɛ 1 c + ɛ c δ 1 + [E] p 1 1 ξtg ut + ɛ c δ g ut, for any δ, ɛ 1, ɛ >, where we enote b 1 = 1 + ξ >. By fixing any t > we note that gs s gs s := g >, t t. From this an conition.1 we can rewrite 4.18 as follows t Gt [ ] ɛ g δb 1 ɛ 1 c ξt u t t + u t t β 1 ɛ 1 4δɛ ξt ut ɛ 1 ξtet ɛ 1c ɛ c δ 1 + [E] p 1 g ut, for every t t. Now we first choose < δ min{ g b 1, g β 1 3c }. Thus g δb 1 g an 4δ g β 1 8c. 4.

21 Once fixe δ > we pick out ɛ 1 > an ɛ > small enough such that 1 ɛ < c 1 1 ɛ 1 < ɛ < min, g g 8c δ 1 + [E] p Therewith imply that ɛ g δb 1 ɛ 1 c >, ɛ 1 β 1 4δɛ >, 1 ɛ 1c ɛ c δ 1 + [E] p 1 >. Therefore, since g, we obtain from 4.19 that estimate 4.17 hols true for some positive constant ɛ 1 c 1 / 1 + [E] p 1, where c 1 > is inepenent of the initial ata. This completes the proof of Lemma 4.5. With the lemmas above we have obtaine the main ingreients to prove Theorem.8. Proof of Theorem.8. First of all we note that where we take ɛ 1 an ɛ such that 1 Et Gt 3 Et, t, 4. < ɛ 1, ɛ < β 1 min { 1 ξ 1, 1 λξ 1 }. 4.3 Inee, to prove 4. we use Höler an Young inequalities, Lemma 4. a-b, an.5 with α =. Now let us fix t = 1 an choose ɛ 1 > an ɛ > so that 4.1 an 4.3 are satisfie. Then from Lemma 4.5 an 4. we have t Gt ɛ 1ξtEt ɛ 1 ξtgt, t 1. 3 A straightforwar computation implies that Gt G1e γ 1 ξs s, t 1, where γ = ɛ 1 k/ 1 + [E] p 1, for some k >. Applying again 4. an since 3 Et is nonincreasing we obtain from where it follows that Et 3E1 e γ 1 ξs s where c = 3E e γ 1 ξs s, an γ k/ 3E e γ 1 ξs s e γ ξs s, Et c e γ ξs s, t 1, [E] p 1 1 for some positive constant k.

22 On the other han, since < ξt ξ 1 > for any t, we see that 1 < e γ 1 t ξs s = e γ 1 ξs s e γ ξs s < e γξ 1 <, t [, 1]. Then, since E e γ 1 ξs s < c, we get Et E c e γ ξs s, t [, 1]. 4.5 Therefore, the uniform ecay.14 is ensure by estimates The proof of Theorem.8 is now complete. Remark 4.6. The proof of Theorem.1 ii can be one with minor changes on the above calculations. In fact, we can formally check that the energy Et satisfies.17 by excluing the terms appearing ue to the presence of rotational inertia term. 5 Appenix 5.1 Rates of energy ecay We emphasize below that.14 provies several rates of energy ecay accoring to the function ξt. In fact, the Examples are motivate by [, 3, 19, 31, 3, 5]. Besies, the Examples 5.5 an 5.6 were first shown in [1] an illustrate the wie variety of ecay rates provie by.14 which are faster than pure exponential ecay when we consier small initial ata or else the linear case. Example 5.1. Let us consier ξt = κ >. Then conition.13 is fulfille an from.14 we obtain the following exponential ecay Et c e κγt, t. Example 5.. Let us also consier ξt = κ lna + 1, where a > an κ >. Then, ξt satisfies.13 an applying.14 results Et c a + 1 κγt, t. Example 5.3. Let us take a rational function ξt = κ, with κ >. It is also easy to t+1 check that conition.13 hols true. From.14 we have the following polynomial-type ecay Et c, t. t + 1 κγ The case < κγ is contemplate if we consier κ small enough. However, the interesting case consists in consiering polynomial ecay of higher orer which is possible by taking large values for κ.

23 Example 5.4. Let us now take ξt = ξ lnt + e + 1 t = κ [t + e lnt + e] < κ, with κ >, then t+e lnt+e an ξ t ξt = 1 t + e lnt + e + 1 t + e < e, for every t. From.14 we get the logarithmic ecay type Et c, t. [lnt + e] κγ Example 5.5. If ξt = κ cotht + θ, t, where κ > an θ = ln1 +, then ξ t = κ[cosht + θ] < an ξ t ξt = 1 cosht + θ sinht + θ < 1, for every t. From.14 we obtain the following hyperbolic ecay Et c, t. [sinht + θ] κγ Example 5.6. We also consier ξt = κ1+ lnt+e1/, t, where κ >. Thus ξt >, t+e 1/ an ξ t = κ1 lnt + e1/ an ξ t t + e 1/ ξt 1 t + e < 1 1/ e, 1/ for every t, see for instance [1]. So ξt also fulfills conition.13. From.14 we conclue that the energy has a transcenental ecay like Et for all t, where c κ = e 3κ/4 c >. 5. The vector fiel F c κ t + e 1/ κγ1+lnt+e1/, In this section we show an interesting property to vector fiels which satisfy a conition like.3. This is given as an application of the Mean Value Inequality. Lemma 5.1. Let F : R N R N be a C 1 -vector fiel given by F = F 1,..., F N. a If there exist positive constants k 1,..., k N an q 1,..., q N such that F j z k j 1 + z q j, z R N, j = 1,..., N. 5.1 Then, there exists a constant K = Kk j, q j, N >, j = 1,..., N, such that F x F y K 1 + x q j + y q j x y, x, y R N. 5. 3

24 b In particular, we have F x F + K 1 + x q j x, x R N. 5.3 Proof. a From conition 5.1 it is not so ifficult to check that F : R N LR N satisfies F z LR N N k j 1 + z q j, z R N. Given x, y R N, we consier z [x, y] R N written as Thus, from where it follows that z = 1 θy + θx, θ = θx, y [, 1]. z q j q j x q j + y q j, j = 1,..., N, F z LR N K 1 + x q j + y q j, z [x, y], where K = N max 1 j N {q j k j }. Applying the Mean Value Inequality we obtain 5.. b It is suffices to efine Gz = F z F. 5.3 Examples for F We finally give examples of vector fiels satisfying conitions like.3 an.5. More generally, we show below some applications of conservative C 1 -vector fiels F = f such that 5.1 hols an also a a 1 z fz F z z + a z, z R N, 5.4 for some nonnegative constants a, a 1, a 3. Example 5.7. Let us first consier F : R N R N z F z = z q z, q. Denoting by F z = F 1 z,..., F N z an z = z 1,..., z N R N, then F j z = z q z j, j = 1,..., N. If we consier z R N an i, j = 1,..., N, we have z i F j z = q z q z i z j for i j, z j F j z = q z q z j + z q for i = j

25 It is also easy to check by efinition that F j = = lim F j z, i, j = 1,..., N. z i z z i Thus, the components F 1,..., F N are C 1 -functions in R N. Besies, from 5.5 we get F j z z i 1 + q z q, i, j = 1,..., N. This is suffices to ensure that 5.1 hols true. Moreover, we note that F = f, where f : R N R z fz = 1 q + z q+, an then conition 5.4 is reaily verifie for any a, a 1, a. Therefore, this vector fiel generates the following operator ivf u = iv u q u, which consists a p-laplacian one by taking p = q + 1 satisfying.4. Example 5.8. The above argument can be applie to the vector fiel F = f, where fz = k q + z q+ + τ z, z = z 1,..., z N R N, with q, k >, an τ = τ 1,..., τ N R N. Thus, conition 5.1 follows analogously as above whereas 5.4 is fulfille with a = τ a 1 = 1, an any a. Example 5.9. Another case of p-laplacian operator arises when we consier the vector fiel F = F 1,..., F N whose components F j, j = 1,..., N, are given by where p. In this case F j z = z j p z j, z = z 1,..., z N R N, ivf u = u x j x j p u Example 5.1. To illustrate another vector fiel, ifferent one of p-laplacian type, we consier F = f, where the potential function is given by fz = ln z + 1, z = z 1,..., z N R N. In such case we have z F z = z + 1, z RN, which vanishes when z. It is easy to check that F an f satisfy 5.1 an 5.4. Acknowlegement The authors have been supporte by the Brazilian Agency CNPq within the project Ciências sem Fronteiras, grant #4689/1-7. x j. 5

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