Some L 1 - L 1 estimates for solutions to visco-elastic damped σ -evolution models
aa r X i v : . [ m a t h . A P ] N ov Some L - L estimates for solutions to visco-elastic damped σ -evolution models Tuan Anh Dao
Abstract.
This note is to conclude L ´ L estimates for solutions to thefollowing Cauchy problem for visco-elastic damped σ -evolution models: u tt ` p´ ∆ q σ u ` p´ ∆ q σ u t “ , x P R n , t ě ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n , (1)where σ ą
1, in all space dimensions n ě Mathematics Subject Classification (2010).
Keywords. L estimates ‚ σ -evolution models ‚ Visco-elastic damping.
1. Introduction
At prsent, there has been very little work on the question of getting L ´ L estimates for solutions to (1). Back in 2000, let us first recall the pioneeringpaper of Shibata [3] devoting to the study of one of the most well-knownequations of (1) in the case σ “
1, the so-called strongly damped wave equa-tion. In the cited paper, relying on the very special structure of fundamentalsolutions to the wave equation he succeeded in obtaining the following L ´ L estimates: } u p t, ¨q} L À p ` t q n } u } L ` p ` t q n ` } u } L if n ě , p ` t q n ´ } u } L ` p ` t q n ` } u } L if n ě , by taking into considerations the connection to Fourier multipliers appearingfor wave models. Quite recently, regarding a different interesting model of (1),namely that with σ “
2, D’Abbicco and his collaborators [1] have employeda different technique in comparison with that used in [3] to derive L ´ L estimates for solutions to the strongly damped plate equation as follows: } u p t, ¨q} L À p ` t q n } u } L ` p ` t q n ` } u } L for any n ě . Here this limitation of the space dimensions comes from the technical dif-ficulty. More precisely, the authors used Bernstein inequality to estimate Tuan Anh DaoFourier multipliers for small frequencies. To apply this technique, it is neces-sary to require the above restriction to space dimensions. Independently fromthe above mentioned results, Dao-Reissig in the recent paper [2] have consid-ered the more general cases of (1) for any σ ą L ´ L estimatesfor solutions localized to small frequencies by applying the theory of modifiedBessel functions linked to Fa`a di Bruno’s formula. Unfortunately, this strat-egy fails in the treatment of large frequencies. For this reason, the presentpaper is to fill this lack and report some L ´ L estimates for solutions to(1) as well.In order to state our main result, we introduce the following notationsused in this paper: ‚ We write f À g when there exists a constant C ą f ď Cg , and f « g when g À f À g . ‚ The spaces H a with a ě L spaces, where (cid:10) D (cid:11) a denote the pseudo-differential operators with symbols (cid:10) ξ (cid:11) a . ‚ For a given number s P R , we denote r s s : “ max k P Z : k ď s ( and r s s ` : “ max t s, u as its integer part and its positive part, respectively.The main purpose of this note is to prove the following result. Theorem 1.1 (Main result).
Let σ ą . Then, the Sobolev solutions to (1)satisfy the following L ´ L estimates: ›› | D | a u p t, ¨q ›› L À p ` t q p `r n sq´ a σ } u } H a ` p ` t q ` p `r n sq´ a σ } u } H r a ´ σ s` , ›› | D | a u t p t, ¨q ›› L À p ` t q p `r n sq´ a σ } u } H σ `r a ´ σ s` ` p ` t q p `r n sq´ a σ } u } H σ `r a ´ σ s` , where a ě and for all space dimensions n ě . Remark 1.1.
Here we want to underline that at the first glance the decayestimates for solutions produced from the results of [3] or [1] are somehowbetter than those of Theorem 1.1 when we choose σ “ σ “ σ “ σ “ L ´ L estimates for solutions to (1) in gerenalcases of σ ą n ě
2. Proof of main result
At first, using partial Fourier transformation to (1) we obtain the Cauchyproblem for p u p t, ξ q : “ F p u p t, x qq , p u p ξ q : “ F p u p x qq and p u p ξ q : “ F p u p x qq as - L estimates for solutions to visco-elastic damped σ -evolution models 3 follows: p u tt ` | ξ | σ p u t ` | ξ | σ p u “ , p u p , ξ q “ p u p ξ q , p u t p , ξ q “ p u p ξ q . (2)The characteristic roots are λ , “ λ , p ξ q “ ´ ´ | ξ | σ ˘ a | ξ | σ ´ | ξ | σ ¯ . The solutions to (2) are presented by the following formula (here we assume λ ‰ λ ): p u p t, ξ q “ λ e λ t ´ λ e λ t λ ´ λ p u p ξ q ` e λ t ´ e λ t λ ´ λ p u p ξ q“ : p K p t, ξ q p u p ξ q ` p K p t, ξ q p u p ξ q . Taking account of the cases of small and large frequencies separately we have1. λ , “ λ , p ξ q “ ´ ` | ξ | σ ¯ i a | ξ | σ ´ | ξ | σ ˘ and λ , „ ´| ξ | σ ˘ i | ξ | σ , λ ´ λ „ i | ξ | σ for | ξ | P p , ´ σ q ,2. λ , “ λ , p ξ q “ ´ ` | ξ | σ ¯ a | ξ | σ ´ | ξ | σ ˘ and λ „ ´ , λ „ ´| ξ | σ , λ ´ λ „ | ξ | σ for | ξ | P p σ , .Let χ k “ χ k p| ξ |q with k “ , , χ p| ξ |q “ | ξ | ď ´ σ , | ξ | ě ´ σ , χ p| ξ |q “ | ξ | ě σ , | ξ | ď σ , and χ p| ξ |q “ ´ χ p| ξ |q ´ χ p| ξ |q . We note that χ p| ξ |q “ ´ σ ď | ξ | ď σ and χ p| ξ |q “ | ξ | ď ´ σ or | ξ | ě σ . Let us now decompose the solutions to (1) into three parts localizedindividually to small, middle and large frequencies, that is, u p t, x q “ u χ p t, x q ` u χ p t, x q ` u χ p t, x q , where u χ k p t, x q “ F ´ ` χ k p| ξ |q p u p t, ξ q ˘ with k “ , , . For this reason, we shall divide our considerations into three cases as follows.
We follow the staments from Corollary 3 . Proposition 2.1.
Let σ ą and n ě . The Sobolev solutions to (1) satisfythe L ´ L estimates ›› | D | a u χ p t, ¨q ›› L À p ` t q p `r n sq´ a σ } u } L ` p ` t q ` p `r n sq´ a σ } u } L , ›› | D | a B t u χ p t, ¨q ›› L À p ` t q p `r n sq´ a σ } u } L ` p ` t q p `r n sq´ a σ } u } L , for any a ě . Tuan Anh Dao
At first, let us represent the characteristic roots in the form λ p ξ q “ ´ ´ φ p ξ q and λ p ξ q “ ´| ξ | σ ` ` φ p ξ q , (3)where φ p ξ q “ ´ ` ż ´ ´ | ξ | σ s ¯ ´ ds. (4)For the sake of transparent representation for large frequencies, we introducethe following notations: K u p t, x q : “ F ´ ´ λ p ξ q e λ p ξ q t λ p ξ q ´ λ p ξ q p u p ξ q χ p| ξ |q ¯ p t, x q , K u p t, x q : “ F ´ ´ λ p ξ q e λ p ξ q t λ p ξ q ´ λ p ξ q p u p ξ q χ p| ξ |q ¯ p t, x q , K u p t, x q : “ F ´ ´ e λ p ξ q t λ p ξ q ´ λ p ξ q p u p ξ q χ p| ξ |q ¯ p t, x q , K u p t, x q : “ F ´ ´ e λ p ξ q t λ p ξ q ´ λ p ξ q p u p ξ q χ p| ξ |q ¯ p t, x q . Then, our main goal of this section is to show the following assertions.
Proposition 2.2.
Let σ ą and n ě . The following estimates hold: ›› B jt | D | a K u p t, ¨q ›› L À e ´ ct } u } H a , ›› B jt | D | a K u p t, ¨q ›› L À e ´ ct } u } H σj `r a ´ σ s` , ›› B jt | D | a K u p t, ¨q ›› L À e ´ ct } u } H r a ´ σ s` , ›› B jt | D | a K u p t, ¨q ›› L À e ´ ct } u } H σj `r a ´ σ s` , where c is a suitable positive constant, for any t ą , a ě and for all integernumber j ě . In order to prove Proposition 2.2 let us recall the following auxiliary estimatesfrom Lemma 3.5 in [2].
Lemma 2.1.
The following estimates hold in R n for sufficiently large | ξ | : ˇˇ B αξ | ξ | pσ ˇˇ À | ξ | pσ ´| α | for all α and p P R , (5) ˇˇ B αξ φ p ξ q ˇˇ À | ξ | ´ σ ´| α | for all α, (6) ˇˇˇ B αξ ´ λ p ξ q e λ p ξ q t λ j p ξ q| ξ | b λ p ξ q ´ λ p ξ q ¯ˇˇˇ À e ´ ct | ξ | σj ` b ´ σ ´| α | for all α, (7) for any b P R , j ě and t ą , where c is a suitable positive constant , - L estimates for solutions to visco-elastic damped σ -evolution models 5 ˇˇˇ B αξ ´ e λ p ξ q t λ j p ξ q| ξ | b λ p ξ q ´ λ p ξ q ¯ˇˇˇ À e ´ ct | ξ | σj ` b ´ σ ´| α | for all α, (8) for any b P R , j ě and t ą , where c is a suitable positive constant , ˇˇˇ B αξ ´ λ p ξ q e λ p ξ q t λ j p ξ q| ξ | b λ p ξ q ´ λ p ξ q ¯ˇˇˇ À e ´ ct | ξ | b ´| α | for all α, (9) for any b P R , j ě and t ą , where c is a suitable positive constant , ˇˇˇ B αξ ´ e λ p ξ q t λ j p ξ q| ξ | b λ p ξ q ´ λ p ξ q ¯ˇˇˇ À e ´ ct | ξ | b ´ σ ´| α | for all α, (10) for any b P R , j ě and t ą , where c is a suitable positive constant . Proof of Proposition 2.2.
In order to indicate some estimates for K u p t, x q ,we may wirte B jt | D | a K u p t, x q“ F ´ ´ λ p ξ q e λ p ξ q t λ j p ξ q| ξ | min t a,σ u´ σj λ p ξ q ´ λ p ξ q χ p| ξ |q| ξ | σj `r a ´ σ s ` p u p ξ q ¯ p t, x q“ F ´ ´ λ p ξ q e λ p ξ q t λ j p ξ q| ξ | min t a,σ u´ σj λ p ξ q ´ λ p ξ q χ p| ξ |q ¯ p t, x q ˚ | D | σj `r a ´ σ s ` u p x q“ : F ´ ` p L p t, ξ q ˘ p t, x q ˚ | D | σj `r a ´ σ s ` u p x q . By choosing b “ min t a, σ u ´ σj in (7), we get ˇˇˇ B αξ ` p L p t, ξ q ˘ˇˇˇ À e ´ ct | ξ | min t a,σ u´ σ ´| α | À e ´ ct | ξ | ´ σ ´| α | , where c is a suitable positive constant. Since e ixξ “ n ÿ k “ x k i | x | B ξ k e ixξ , (11)carrying out m steps of partial integration we derive F ´ ` p L p t, ξ q ˘ p t, x q “ C ÿ | α |“ m p ix q α | x | | α | F ´ ´ B αξ ` p L p t, ξ q ˘¯ p t, x q . For this reason, we obtain the following estimates: ˇˇ F ´ ` p L p t, ξ q ˘ p t, x q ˇˇ À | x | ´ m ››› F ´ ´ B αξ ` p L p t, ξ q ˘¯ p t, ¨q ››› L À | x | ´ m ›› B αξ ` L p t, ξ q ˘ p t, ¨q ›› L À | x | ´ m e ´ ct ż | ξ | ´ σ ´ m ` n ´ d | ξ |À e ´ ct | x | ´p n ´ q if 0 ă | x | ď m “ n ´ , | x | ´p n ` q if | x | ě m “ n ` , Tuan Anh Daowhere the assumption σ ą ›› F ´ ` p L p t, ξ q ˘ p t, ¨q ›› L À ż | x |ď ˇˇ F ´ ` p L p t, ξ q ˘ p t, x q ˇˇ dx ` ż | x |ě ˇˇ F ´ ` p L p t, ξ q ˘ p t, x q ˇˇ dx À e ´ ct ż d | x | ` e ´ ct ż | x | ´ d | x | À e ´ ct . Then, employing Young’s convolution inequality we have proved the secondstatement in Proposition 2.2. In the same way, we may also conclude the laststatement and the third statement in Proposition 2.2, respectively, by using(8) and (10). Let us come back to estimate the first statement. Indeed, wecan see that B jt | D | a K u p t, x q “ B jt | D | a F ´ ´ e λ p ξ q t χ p| ξ |q p u p ξ q ¯ p t, x q` B j ` t | D | a F ´ ´ e λ p ξ q t λ p ξ q ´ λ p ξ q χ p| ξ |q p u p ξ q ¯ p t, x q , (12)by using the relation λ p ξ q e λ p ξ q t λ p ξ q ´ λ p ξ q “ e λ p ξ q t ` B t ´ e λ p ξ q t λ p ξ q ´ λ p ξ q ¯ . In an analogous treatment to get the third statement, we derive the followingestimate for the second term: ››› B j ` t | D | a F ´ ´ e λ p ξ q t λ p ξ q ´ λ p ξ q χ p| ξ |q p u p ξ q ¯ p t, ¨q ››› L À e ´ ct } u } H r a ´ σ s` . (13)In order to control the first term, using the relation λ p ξ q “ ´ ´ φ p ξ q wewrite e λ p ξ q t “ e ´ t e ´ φ p ξ q t “ e ´ t ´ te ´ t φ p ξ q ż e ´ φ p ξ q tr dr. Hence, we obtain F ´ ´ e λ p ξ q t χ p| ξ |q p u p ξ q ¯ p t, x q (14) “ e ´ t F ´ `p u p ξ q ˘ p x q ´ e ´ t F ´ ` p ´ χ p| ξ |qq p u p ξ q ˘ p x q´ te ´ t F ´ ´ φ p ξ q χ p| ξ |q p u p ξ q ż e ´ φ p ξ q tr dr ¯ p t, x q . (15)Obviously, we have ››› B jt | D | a ´ e ´ t F ´ `p u p ξ q ˘¯ p t, ¨q ››› L “ e ´ t ›› | D | a u ›› L À e ´ t } u } H a . (16) - L estimates for solutions to visco-elastic damped σ -evolution models 7 Now, we re-write B jt | D | a ´ te ´ t F ´ ´ φ p ξ q χ p| ξ |q p u p ξ q ż e ´ φ p ξ q tr dr ¯¯ p t, x q“ j ÿ ℓ “ B j ´ ℓt p te ´ t q B ℓt | D | a F ´ ´ φ p ξ q χ p| ξ |q p u p ξ q ż e ´ φ p ξ q tr dr ¯ p t, x q“ j ÿ ℓ “ B j ´ ℓt p te ´ t q F ´ ´ φ ℓ ` p ξ q| ξ | min t a,σ u χ p| ξ |q ż e ´ φ p ξ q tr p´ r q ℓ dr ¯ p t, x q˚ | D | r a ´ σ s ` u p x q“ : j ÿ ℓ “ B j ´ ℓt p te ´ t q F ´ ` p L p t, ξ q ˘ p t, x q ˚ | D | r a ´ σ s ` u p x q . Thanks to (5) and (6), by using the Leibniz rule we have ˇˇ B αξ ` p L p t, ξ q ˘ˇˇ À e t | ξ | ´ σℓ ´ σ ` min t a,σ u´| α | À e t | ξ | ´ σ ´| α | . Using again (11), and carrying out n ´ n ` ˇˇ F ´ ` p L p t, ξ q ˘ p t, x q ˇˇ À e t | x | ´p n ´ q if 0 ă | x | ď , | x | ´p n ` q if | x | ě . It follows ˇˇˇ j ÿ ℓ “ B j ´ ℓt p te ´ t q F ´ ` p L p t, ξ q ˘ p t, x q ˇˇˇ À e ´ ct | x | ´p n ´ q if 0 ă | x | ď , | x | ´p n ` q if | x | ě , where c is a suitable positive constant. Therefore, we derive ››› j ÿ ℓ “ B j ´ ℓt p te ´ t q F ´ ` p L p t, ξ q ˘ p t, ¨q ››› L À e ´ ct . Applying Young’s convolution inequality gives ››› B jt | D | a ´ te ´ t F ´ ´ φ p ξ q χ p| ξ |q x u p ξ q ż e ´ φ p ξ q tr dr ¯¯ p t, ¨q ››› L À e ´ ct } u } H r a ´ σ s` . (17)Moreover, due to 1 ´ χ P C , we derive ››› B jt | D | a ´ e ´ t F ´ ` ´ χ p| ξ |q ˘¯ p t, ¨q ››› L À e ´ t . By using again Young’s convolution inequality we obtain ››› B jt | D | a ´ e ´ t F ´ ` p ´ χ p| ξ |qq p u p ξ q ˘¯ p t, ¨q ››› L À e ´ t } u } L . (18)Combining from (12) to (18) we may conclude the first statement in Propo-sition 2.2. Summarizing, the proof of Proposition 2.2 is completed. ˝ Tuan Anh DaoFrom the statements in Proposition 2.2 we obtain immediately the followingresult.
Proposition 2.3.
Let σ ą and n ě . The Sobolev solutions to (1) satisfythe L ´ L estimates ›› B jt | D | a u χ p t, ¨q ›› L À e ´ ct ´ }p u , u q} H σj `r a ´ σ s` ` } u } H a ` } u } H r a ´ σ s` ¯ , where c is a suitable positive constant, for any t ą , a ě and for all integernumber j ě . Now let us turn to consider some estimates for middle frequencies, where3 ´ σ ď | ξ | ď σ . Our goal is to clarify the exponential decay for solutionsand some of their derivatives to (1) localized to middle frequencies, whichwere neglected or not well-studied in the references. Proposition 2.4.
Let σ ą and n ě . The Sobolev solutions to (1) satisfythe L ´ L estimates ›› B jt | D | a u χ p t, ¨q ›› L À e ´ ct }p u , u q} L , where c is a suitable positive constant, for any t ą , a ě and j “ , .Proof. At first, with 3 ´ σ ď | ξ | ď σ we use Cauchy’s integral formula tore-write the above Fourier multipliers in the following form: p K p t, ξ q χ p| ξ |q “ πi ´ ż Γ p z ` | ξ | σ q e zt z ` | ξ | σ z ` | ξ | σ dz ¯ χ p| ξ |q , (19) p K p t, ξ q χ p| ξ |q “ πi ´ ż Γ e zt z ` | ξ | σ z ` | ξ | σ dz ¯ χ p| ξ |q , (20)where Γ is a closed curve containing the two characteristic roots λ , . We cansee that λ “ λ when | ξ | “ σ and ξ P R n : | ξ | “ σ ( is not a singularset because we may give equivalent formulas as follows: p K p t, ξ q “ e λ t ´ λ e λ t ż t e p λ ´ λ q s ds, x K p t, ξ q “ e λ t ż t e p λ ´ λ q s ds. Therefore, it is reasonable to assume λ ‰ λ . Since 3 ´ σ ă | ξ | ă σ , thiscurve additionally is contained in t z P C : Re z ď ´ c u , where c is apositive constant. In order to verify (19) we express p z ` | ξ | σ q e zt z ` | ξ | σ z ` | ξ | σ “ p z ` | ξ | σ q e zt p z ´ λ qp z ´ λ q “ ´ λ λ ´ λ e zt z ´ λ ` λ λ ´ λ e zt z ´ λ . - L estimates for solutions to visco-elastic damped σ -evolution models 9 For this reason, applying Cauchy’s integral formula we obtain12 πi ż Γ p z ` | ξ | σ q e zt z ` | ξ | σ z ` | ξ | σ dz “ ´ λ λ ´ λ ´ πi ż Γ e zt z ´ λ dz ¯ ` λ λ ´ λ ´ πi ż Γ e zt z ´ λ dz ¯ “ ´ λ λ ´ λ e λ t ` λ λ ´ λ e λ t χ p| ξ |q “ p K p t, ξ q for middle frequencies. Here we split the curve Γ into two closed sub-curvesseparately Γ and Γ containing λ and λ , respectively. In the same way wemay conclude the relation (20). Now, taking account of estimates for p K p t, ξ q we get F ´ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘ “ ż R n e ixξ | ξ | a p K p t, ξ q χ p| ξ |q dξ “ n ÿ k “ x k i | x | ż R n B ξ k ` e ixξ ˘ | ξ | a p K p t, ξ q χ p| ξ |q dξ, where we used (11). By induction argument, carrying out m steps of partialintegration we derive F ´ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘ “ C ÿ | α |“ m p ix q α | x | | α | F ´ ´ B αξ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘¯ , for any non-negative integer m and C is a suitable constant. Hence, we arriveat the following estimates: ˇˇ F ´ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘ˇˇ À | x | ´ m ››› F ´ ´ B αξ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘¯››› L À | x | ´ m ›› B αξ ` | ξ | a p K p t, ξ q χ p| ξ |q ˘›› L À | x | ´ m e ´ ct , where c is a suitable positive constant, since 3 ´ σ ă | ξ | ă σ . This estimateimmediately implies ›› F ´ ` | ξ | a p K χ p| ξ |q ˘ p t, ¨q ›› L À e ´ ct . In an analogous way we may also conclude ›› F ´ ` | ξ | a p K χ p| ξ |q ˘ p t, ¨q ›› L À e ´ ct . Similarly, we may arrive at the exponential decay for the following estimates: ›› F ´ ` | ξ | a B t p K χ p| ξ |q ˘ p t, ¨q ›› L À e ´ ct , ›› F ´ ` | ξ | a B t p K χ p| ξ |q ˘ p t, ¨q ›› L À e ´ ct , where we notice that B t p K p t, ξ q “ ´| ξ | σ p K p t, ξ q and B t p K p t, ξ q “ p K p t, ξ q ´ | ξ | σ p K p t, ξ q . ›› B jt | D | a u χ p t, ¨q ›› L À ›› F ´ ` | ξ | a B jt p K χ p| ξ |q ˘ p t, ¨q ›› L } u } L ` ›› F ´ ` | ξ | a B jt p K χ p| ξ |q ˘ p t, ¨q ›› L } u } L À e ´ ct }p u , u q} L . Summarizing, the proof to Proposition 2.4 is completed. ˝ Proof of Theorem 1.1.
We combine the statements from Propositions 2.1, 2.3and 2.4 to conclude immediately all the desired estimates. This completes ourproof. ˝ References [1] M. D’Abbicco, G. Girardi, J. Liang, L ´ L estimates for the strongly dampedplate equation, J. Math. Anal. Appl. , (2019), 476–498.[2] T.A. Dao, M. Reissig, L estimates for oscillating integrals and their applica-tions to semi-linear models with σ -evolution like structural damping, DiscreteContin. Dyn. Syst. A , (2019), 5431–5463.[3] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci. , (2000), 203–226.Tuan Anh DaoSchool of Applied Mathematics and Informatics,Hanoi University of Science and Technology,No.1 Dai Co Viet road, Hanoi, Vietnam.Faculty for Mathematics and Computer Science,TU Bergakademie Freiberg,Pr¨uferstr. 9, 09596, Freiberg, Germany.e-mail:(2000), 203–226.Tuan Anh DaoSchool of Applied Mathematics and Informatics,Hanoi University of Science and Technology,No.1 Dai Co Viet road, Hanoi, Vietnam.Faculty for Mathematics and Computer Science,TU Bergakademie Freiberg,Pr¨uferstr. 9, 09596, Freiberg, Germany.e-mail: