Global existence results for semi-linear structurally damped wave equations with nonlinear convection
aa r X i v : . [ m a t h . A P ] F e b GLOBAL EXISTENCE RESULTS FOR SEMI-LINEAR STRUCTURALLYDAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION
TUAN ANH DAO AND HIROSHI TAKEDA
Abstract.
In this paper, we consider the Cauchy problem for semi-linear wave equations withstructural damping term ν p´ ∆ q u t , where ν ą Introduction
In this paper, let us study the following Cauchy problem for the semi-linear wave equation withstructural damping term: u tt ´ ∆ u ` ν p´ ∆ q u t “ a ˝ ∇ f p u, u t q , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n , (1)where ν is a positive constant, a ‰ R n and ˝ represents the inner productin R n . Here we mainly deal with two kinds of the nonlinear terms: f p u, u t q “ | u | p , | u t | p , where p ą
1. We are interested in finding out the conditions for the growth order of the nonlin-earities, the so-called admissible exponents p , which ensure the global solutions to (1) for the smalldata.The crux of our proof ideas of the main results is to apply the derived decay estimates for solutionsto the corresponding linear Cauchy problem in the treatment of the nonlinear convection terms.For this reason, here we would like to give the brief historical survey for the linear equation andsome previous results related to nonlinear convection. Namely, the corresponding linear equationof (1) is given by u tt ´ ∆ u ` ν p´ ∆ q u t “ , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n . (2)As we can see, this equation was proposed by Ghisi-Gobbino-Haraux [8] as one of the modeling caseof the second order abstract evolution equation. In the cited paper, the authors claimed that theregularity loss type estimates appear not only for solutions u p t, x q itself but also for their derivativesin time u t p t, x q to (2). After that, Ikehata-Iyota [10] obtained the asymptotic profile of solutions to(2) with the weighted L , initial data in suitable space. Moreover, they also proved that the lowfrequency part is dominant and the solution is approximated by the diffusion wave as t Ñ 8 . Quiterecently, Fukushima-Ikehata-Michihisa [7] have studied the higher order asymptotic expansion ofsolutions to (2), which corresponds to the additional regularity assumptions on the initial data.
Mathematics Subject Classification.
Key words and phrases.
Wave equations; Structural damping; Nonlinear convection; Global existence.
The other point worthy of mentioning is that the authors have determined a threshold of theregularity condition for the initial data to classify whether the leading factor of the asymptoticprofiles of solutions as t Ñ 8 is given by the low frequency parts or not. Concerning the studyof nonlinear convection, we want to refer the interested readers to a series of previous work, forexamples, [5, 13, 1, 6] in term of the convection-diffusion equation and references therein. Oneshould recognize that in the cited papers the large time behavior of solutions has been explored bysupposing the initial data with some kind of different regularities.To the best of the authors’ knowledge, there seem not so many research papers regarding theinvestigation of the semi-linear equation (1) so far. For this reason, generally speaking, previouspapers suggest that it is difficult to construct sharp time decay estimates for solutions to the dissi-pative hyperbolic equations with derivative loss structure. Especially, the same situation happenswhen the growth order of nonlinearities is nearby the critical case even if the nonlinear functionbelongs to C (cf. [2]). We also remark that this is observed when the nonlinearities are smooth(see [12]). Notations:
Throughout this paper, we use the following notations. We denote f À g if there existsa constant C ą f ď Cg , and f „ g if g À f À g . We write p h p t, ξ q : “ F x Ñ ξ ` h p t, x q ˘ asthe Fourier transform with respect to the spatial variable of a function h p t, x q . As usual, H s and H s , with s ě
0, stand for Bessel and Riesz potential spaces based on L spaces. For any σ ě L ,σ are defined by L ,σ “ ! f P L : } f } L ,σ : “ ż R n p ` | x |q σ | f p x q| dx ă `8 ) . Moreover, if we introduce the space D : “ ` L ,σ X H s ˘ ˆ ` L ,σ X H s ˘ with σ, s , s ě
0, then thenorm is defined by }p v , v q} D “ } v } L ,σ ` } v } H s ` } v } L ,σ ` } v } H s . Now we consider the nonlinear function f p u, u t q “ | u | p in (1), i.e. the following semi-linearequation: u tt ´ ∆ u ` ν p´ ∆ q u t “ a ˝ ∇ | u | p , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n . (3)Our first result states the global (in time) existence of solutions to (3) with their sharp decayproperties. Theorem 1.1.
Let n “ , , , and ε is a sufficiently small positive constant. Suppose that thefollowing condition: p ą if n “ ,p ě ` n ´ if n “ , , . (4) Then, there exists a constant ε ą such that for any small data p u , u q P A : “ ` L X H n ` ε ˘ ˆ ` L X L ˘ satisfying the assumption }p u , u q} A ď ε , the Cauchy problem (3) admits a unique global in timesolution in the class u P C ` r , , H n ` ε ˘ X C ` r , , L ˘ . EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 3
Moreover, the following estimates hold: } u p t, ¨q} L À log p t ` e q}p u , u q} A if n “ , p ` t q ´ n ` }p u , u q} A if n “ , , , (5) ›› ∇ n ` ε u p t, ¨q ›› L À p ` t q ´ n ´ ` ε }p u , u q} A if n “ , , , p ` t q ´ ` ε }p u , u q} A if n “ , (6) } u t p t, ¨q} L À p ` t q ´ n }p u , u q} A . (7) Remark 1.1.
Here we note that the estimate (6) for n “ n “ , , n “
5. Then, we need to modify the solution space to obtain the sharp decay property of the L norm (5) for n “ f p u, u t q “ | u t | p in (1), i.e. the following semi-linearequation: u tt ´ ∆ u ` ν p´ ∆ q u t “ a ˝ ∇ | u t | p , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n . (8)In this case, we can construct global solutions with their sharp decay properties because of thegained regularity of derivative in time of solutions from the linear principal part as mentioned in[8]. Theorem 1.2.
Let n “ , , and ε is a sufficiently small positive constant. We assume thefollowing condition: p ě ` max ! n , ) . (9) Then, there exists a constant ε ą such that for any small data p u , u q P B : “ ` L X H n ` ε ˘ ˆ ` L X H n ` ε ˘ satisfying the assumption }p u , u q} B ď ε , we have a uniquely determined global (in time) solution u P C ` r , , H n ` ε ˘ X C ` r , , H n ` ε ˘ to (8) . Moreover, the following estimates hold: } u p t, ¨q} L À log p t ` e q}p u , u q} B if n “ , p ` t q ´ n ` }p u , u q} B if n “ , , (10) ›› ∇ n ` ε u p t, ¨q ›› L À p ` t q ´ n ´ ` ε }p u , u q} B , (11) } u t p t, ¨q} L À p ` t q ´ n }p u , u q} B , (12) ›› ∇ n ` ε u t p t, ¨q ›› L À p ` t q ´ n ` ε }p u , u q} B . (13) Remark 1.2.
It is clear that we have assumed the higher regularity for the second data in Theorem1.2 in comparison with Theorem 1.1. This comes from the treatment of the nonlinear function f p u, u t q “ | u t | p appearing in (8) in the place of the nonlinear function f p u, u t q “ | u | p appearing in(3).This paper is organized as follows: Section 2 is devoted to the preparation of decay estimates forsolutions to the corresponding linear equation (2), which play an essential role to prove the global(in time) existence of small data solutions in Theorems 1.1 and 1.2. We will give the proofs ofTheorem 1.1 and Theorem 1.2 in Section 3 and Section 4, respectively. Some further discussionsincluding the large time behavior of the obtained global solutions and several results for (1) with themixture of nonlinearities will be provided in Section 5. Finally, to end this paper, we summarize thewell-known estimates, which are useful to show the decay properties of solutions in the appendix. TUAN ANH DAO AND HIROSHI TAKEDA The treatment of the linear equation
In this section, at first, we are going to prove the decay estimates for the fundamental solutions to(2). Some of them are already derived in the previous papers [10, 7]. However, for the convenienceof the readers, we rephrase them to our notation.2.1.
Representation of solutions.
At first, using partial Fourier transformation to (2) we obtainthe following Cauchy problem: p u tt ` ν | ξ | p u t ` | ξ | p u “ , p u p , ξ q “ x u p ξ q , p u t p , ξ q “ x u p ξ q . (14)The characteristic roots are λ ˘ “ λ ˘ p ξ q “ ´ ´ ν | ξ | ˘ a ν | ξ | ´ | ξ | ¯ . The solutions to (14) 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 ´ x u p ξ q ` e λ ` t ´ e λ ´ t λ ` ´ λ ´ x u p ξ q“ : x K p t, ξ q x u p ξ q ` x K p t, ξ q x u p ξ q , which leads to the representation formula of solutions to (2) as follows: u p t, x q “ K p t, x q ˚ x u p x q ` K p t, x q ˚ x u p x q . (15)Here K k p t, x q with k “ , x K k p t, ξ q with respect to spatialvariables.2.2. Pointwise estimates in Fourier space.
Taking account of the cases of small and largefrequencies separately we have the asymptotic behavior of the characteristic roots as follows: λ ˘ „ ´ ν | ξ | ˘ i | ξ | , λ ` ´ λ ´ „ i | ξ | for | ξ | ! ,λ ` „ ´ ν | ξ | ´ , λ ´ „ ´ ν | ξ | , λ ` ´ λ ´ „ ν | ξ | for | ξ | " . To make further discussion rigorously, we follow the notation of [11] to introduce the radial, smoothcut-off functions χ L , χ M and χ H defined by χ L p| ξ |q “ | ξ | ď ρ { , | ξ | ě ρ, χ H p| ξ |q “ | ξ | ě ρ, | ξ | ď ρ,χ M p| ξ |q “ ´ χ L p| ξ |q ´ χ H p| ξ |q , where ρ ą ρ ă ˆ ν ˙ . As an easy consequence, we have the following point-wise estimates for the fundamental solutionsin the Fourier space.
Lemma 2.1.
Let j “ , and s ě . Then, the following estimates hold: | ξ | s χ L p| ξ |q ˇˇ B jt x K p t, ξ q ˇˇ À e ´ ct | ξ | | ξ | s ` j , | ξ | s χ L p| ξ |q ˇˇ B jt x K p t, ξ q ˇˇ À e ´ ct | ξ | | ξ | s ` j ´ , and | ξ | s ` χ M p| ξ |q ` χ H p| ξ |q ˘ˇˇ B jt x K p t, ξ q ˇˇ À | ξ | s ` | ξ | ´ j e ´ ct | ξ | ´ ` | ξ | ´ ` j e ´ ct | ξ | ˘ , (16) | ξ | s ` χ M p| ξ |q ` χ H p| ξ |q ˘ˇˇ B jt x K p t, ξ q ˇˇ À | ξ | s ´ ` | ξ | ´ j e ´ ct | ξ | ´ ` | ξ | j e ´ ct | ξ | ˘ , (17) EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 5 where c is a suitable positive constant.Proof. Since the desired estimates for the low frequency part are shown in [10], we only show theestimates for the high frequency part, i.e. (16) and (17). Observing that χ H p| ξ |qB jt x K p t, ξ q “ λ ` λ j ´ e λ ´ t ´ λ ´ λ j ` e λ ` t λ ` ´ λ ´ χ H p| ξ |q„ | ξ | ´ ` | ξ | ´ j e ´ ct | ξ | ´ ` | ξ | ´ ` j e ´ ct | ξ | ˘ , we have the estimate (16). Here we notice that the characteristic roots λ ˘ are negative in themiddle frequency part ξ P R n : | ξ | P “ ρ { , ρ ‰( . So, the corresponding estimates for this partpossess an exponential decay. Similarly, we can obtain the estimate (17) by χ H p| ξ |qB jt x K p t, ξ q “ λ j ` e λ ` t ´ λ j ´ e λ ´ t λ ` ´ λ ´ χ H p| ξ |q„ | ξ | ´ ` | ξ | ´ j e ´ ct | ξ | ´ ` | ξ | j e ´ ct | ξ | ˘ . This completes the proof of Lemma 2.1. (cid:3)
Decay estimates.
Now we define the functions by K kℓ p t, x q : “ F ´ ` x K k p t, ξ q χ ℓ p| ξ |q ˘ (18)for k “ , ℓ “ L, M, H . Once we have Lemma 2.1, we can easily conclude the decay propertiesof the frequency-wise evolution operators K kℓ p t, x q˚ x for k “ , ℓ “ L, M, H as follows.
Lemma 2.2.
Let α ě , j “ , and r P r , s . Then, the following estimates hold: ›› B jt ∇ α K L p t, x q ˚ x g ›› L À p ` t q ´ n p r ´ q´ α ` j } g } L r , (19) ›› B jt ∇ α K L p t, x q ˚ x g ›› L À $’’’&’’’% t ´ α } g } L if p n, j, α q P t u ˆ t u ˆ r , q , a log p t ` e q} g } L if p n, j, α q P p , , q , p , , q ( , p ` t q ´ p r ´ q´ α ` j ´ } g } L r if p n, j, α q R t u ˆ t u ˆ r , s and p n, j, α q ‰ p , , q , (20) and ›› B jt ∇ α ` K M p t, x q ` K H p t, x q ˘ ˚ x g ›› L À e ´ ct t ´ n p r ´ q´ α ´ ` j ´ β ›› ∇ β g ›› L r ` p ` t q ´ β ›› ∇ p α ´ j ` β q ` g ›› L , (21) ›› B jt ∇ α ` K M p t, x q ` K H p t, x q ˘ ˚ x g ›› L À e ´ ct t ´ n p r ´ q´ α ´ ` j ´ β ›› ∇ β g ›› L r ` p ` t q ´ β ›› ∇ p α ´ ´ j ` β q ` g ›› L , (22) for any space dimensions n ě and β , β ě , where c is a suitable positive constant and p γ q ` : “ max t γ, u for γ P R .Proof. One can find the proof of the estimates (19) and (20) in [10, 7] except for the case (20) with p n, j, α q P t u ˆ t u ˆ p , s . On the other hand, applying the same arguments as in [10, 7] we caneasily obtain the following estimates for n “ ›››› ∇ α F ´ ˆ e ´ ν | ξ | t sin p t | ξ |q| ξ | χ L p| ξ |q ˙ ˚ x g ›››› L ď t ´ α } g } L if α P r , q , a log p t ` e q} g } L if α “ , TUAN ANH DAO AND HIROSHI TAKEDA which are to conclude the remainder case. Then, it suffices to only prove the estimates (21) and(22). Firstly, let us prove the estimate (21). Indeed, using the formula of Parseval-Plancherel andthe estimate (16) one derives ›› B jt ∇ α ` K M p t, x q ` K H p t, x q ˘ ˚ x g ›› L À ›› | ξ | α ´ j e ´ ct | ξ | ´ ` χ M p| ξ |q ` χ H p| ξ |q ˘p g ›› L ` ›› | ξ | α ´ ` j e ´ ct | ξ | ` χ M p| ξ |q ` χ H p| ξ |q ˘p g ›› L . Now, we may see easily that ›› | ξ | α ´ j e ´ ct | ξ | ´ ` χ M p| ξ |q ` χ H p| ξ |q ˘p g ›› L “ ›› | ξ | ´ β e ´ ct | ξ | ´ ` χ M p| ξ |q ` χ H p| ξ |q ˘ | ξ | α ´ j ` β p g ›› L ď ∇ p α ´ j ` β q ` g ›› L if 0 ď t ď ,Ct ´ β ›› ∇ p α ´ j ` β q ` g ›› L if t ě , where C is a suitable positive constant, which gives ›› | ξ | α ´ j e ´ ct | ξ | ´ ` χ M p| ξ |q ` χ H p| ξ |q ˘p g ›› L À p ` t q ´ β ›› ∇ p α ´ j ` β q ` g ›› L . In addition, we denote by r , the conjugate number of r , i.e. r ` r “
1. The application of H¨older’sinequality and the Hausdorff-Young inequality entails ›› | ξ | α ´ ` j e ´ ct | ξ | ` χ M p| ξ |q ` χ H p| ξ |q ˘p g ›› L À ›› | ξ | α ´ ` j ´ β e ´ ct | ξ | ` χ M p| ξ |q ` χ H p| ξ |q ˘›› L r ´ r ›› | ξ | β p g ›› L r À e ´ ct t ´ n p r ´ q´ α ´ ` j ´ β ›› ∇ β g ›› L r . From the two above estimates, we can conclude the desired estimate (21). The remaining estimateis shown in a similar way. Hence, our proof is completed. (cid:3)
From Lemma 2.2, we may conclude immediately the following proposition.
Proposition 2.1.
Let ℓ , ℓ ě and r P r , s . Then, the solutions to (2) satisfy the followingestimates: } u p t, ¨q} L À $’’’’’’’’’’&’’’’’’’’’’% p ` t q ´ p r ´ q } u } L r ` p ` t q ´ ℓ } u } H ℓ ` t } u } L ` p ` t q ´ ℓ ´ } u } H ℓ if n “ , p ` t q ´ p r ´ q } u } L r ` p ` t q ´ ℓ } u } H ℓ ` a log p t ` e q} u } L ` p ` t q ´ ℓ ´ } u } H ℓ if n “ , p ` t q ´ n p r ´ q } u } L r ` p ` t q ´ ℓ } u } H ℓ `p ` t q ´ n p r ´ q` } u } L r ` p ` t q ´ ℓ ´ } u } H ℓ if n ě , } u p t, ¨q} H s À p ` t q ´ n p r ´ q´ s } u } L r ` p ` t q ´ ℓ ´ s } u } H ℓ ` p ` t q ´ n p r ´ q´ s ´ } u } L r ` p ` t q ´ ℓ ´ s ´ } u } H ℓ if ď s ď min t ℓ ` , ℓ ` u , } u t p t, ¨q} H s À p ` t q ´ n p r ´ q´ s ` } u } L r ` p ` t q ´ ℓ ´ s ´ } u } H ℓ ` p ` t q ´ n p r ´ q´ s } u } L r ` p ` t q ´ ℓ ´ s ´ } u } H ℓ if ď s ď min t ℓ ` , ℓ u , for any space dimensions n ě . EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 7
Remark 2.1.
We want to point out that the obtained decay estimates for solutions to (2) andseveral their derivatives in Proposition 2.1 are regularity loss type estimates, which bring somedifficulties to treat the associated semi-linear equations like (1). However, to overcome such kindof difficulties, we can use appropriate regularities for the initial data by the suitable choice of ℓ and ℓ appearing in Proposition 2.1. Additionally, some technical steps of our proofs to deal withthe nonlinear convection terms come into play in the next sections.3. Proof of main result (I)
In this section, we give the proof of Theorem 1.1.
Proof of Theorem 1.1 with n “ , , . We introduce the solution space X p t q : “ C ` r , , H n ` ε ˘ X C ` r , , L ˘ with the norm } u } X p t q : “ sup ď τ ď t ´ ℓ p τ q} u p τ, ¨q} L ` p ` τ q n ´ ` ε ›› ∇ n ` ε u p τ, ¨q ›› L ` p ` τ q n } u t p τ, ¨q} L ¯ , where ℓ p τ q “ log ´ p τ ` e q if n “ , p ` τ q n ´ if n “ , . As mentioned in (15), we can write the solutions of the corresponding linear Cauchy problem withvanishing right-hand side to (3) as follows: u ln p t, x q “ K p t, x q ˚ x u p x q ` K p t, x q ˚ x u p x q . Using Duhamel’s principle we get the formal implicit representation of solutions to (3) in thefollowing form: u p t, x q “ u ln p t, x q ` ż t K p t ´ τ, x q ˚ x ` a ˝ ∇ | u p τ, x q| p ˘ dτ “ : u ln p t, x q ` u nl p t, x q . We define a mapping Φ : X p t q ÝÑ X p t q byΦ r u sp t, x q “ u ln p t, x q ` u nl p t, x q . In order to conclude the uniqueness and the global (in time) existence of small data solutions to(3) as well, we have to prove the following pair of inequalities: } Φ r u s} X p t q À }p u , u q} A ` } u } pX p t q , (23) } Φ r u s ´ Φ r v s} X p t q À } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ . (24)We firstly compute the norms of the nonlinear term in X p t q . Applying Proposition 5.3, from thedefinition of the norm in X p t q we have the following useful estimate: ›› u p τ, ¨q ›› L À log p τ ` e q ˘ ε p ` ε q p ` τ q ´ } u } X p t q if n “ , p ` τ q ´ n ´ } u } X p t q if n “ , . This leads to the following estimates for the nonlinear terms for p ě ›› | u p τ, ¨q| p ›› L r ď } u p τ, ¨q} p ´p ´ r q L } u p τ, ¨q} ´ rL À p ` τ q ´ p p ´ r q` p ` ε } u } pX p t q if n “ , p ` τ q ´ n p p ´ r q` p } u } pX p t q if n “ , , with the choice of a sufficiently small constant ε satisfying 0 ă ε ă p ´ when n “
2. Thus, itfollows that ›› | u p τ, ¨q| p ›› L r À p ` τ q ´ n p p ´ r q` p ` ǫ } u } pX p t q TUAN ANH DAO AND HIROSHI TAKEDA with n “ , , r “ ,
2, where ǫ is given by ǫ “ ε P ` , p ´ ˘ if n “ , n “ , . (25)In addition, one also derives ›› ∇ | u p τ, ¨q| p ›› L ď } u p τ, ¨q} p ´ L } ∇ u p τ, ¨q} L . The application of Proposition 5.2 gives } ∇ u p τ, ¨q} L À } u p τ, ¨q} ´ θL } u p τ, ¨q} θ H n ` ε with θ “ n ` ε À log p τ ` e q ˘ ε p ` ε q p ` τ q ´ } u } X p t q if n “ , p ` τ q ´ n } u } X p t q if n “ , . Thus, it follows immediately ›› ∇ | u p τ, ¨q| p ›› L ď } u p τ, ¨q} p ´ L } ∇ u p τ q} À p ` τ q ´ n p p ´ q` p ´ ` ǫ } u } pX p τ q . First let us prove the inequality (23) . From the definition of the data space, it is obvious thatwe need to indicate the following inequality instead of (23): } u nl } X p t q À } u } pX p t q . (26)Our proof is divided into two steps.Step 1: We may control the norm } u nl p t, ¨q} L by using the estimates from Lemma 2.2 asfollows: } u nl p t, ¨q} L À ż t ›› ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ›› | u p τ, ¨q| p ›› L dτ ` ż tt { ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż tt { p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ ¯ } u } pX p t q . Noting the assumptions (4) and (25), we easily see ´ n p p ´ q ` p ` ǫ ď ´ . The direct calculationshows that the first two integrals are estimated as follows: ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n log p t ` e q if p ě ` ` ǫ n ´ , p ` t q ´ n ´ n p p ´ q` p ` ǫ ` if p ă ` ` ǫ n ´ , À p ` t q ´ n ` with p satisfying (4), and ż tt { p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n p p ´ q` p ` ` ǫ À p ` t q ´ n ` . EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 9
After applying Lemma 5.1, we can obtain the estimate for the third integral: ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` τ q ´ min , n p p ´ q´ p ´ ǫ ( À p ` t q ´ n ´ ď p ` t q ´ n ` since ´ ă ´ n ´ for n ď ´ n p p ´ q ` p ` ǫ ď ´ n ´ under the assumptions (4) and(25). Therefore, we can conclude that } u nl p t, ¨q} L À p ` t q ´ n ` } u } pX p t q for n “ , , . Step 2: By using the same ideas, we may deal with the remaining norms ›› ∇ n ` ε u nl p t, ¨q ›› L and } u nl t p t, ¨q} L as follows: ›› ∇ n ` ε u nl p t, ¨q ›› L À ż t ›› ∇ n ` ` ε K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ n ` ε ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x ∇ | u p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ` ε ›› | u p τ, ¨q| p ›› L dτ ` ż tt { p ` t ´ τ q ´ p n ` ε q ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ` n ` ε ›› ∇ | u p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n ` ε p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż tt { p ` t ´ τ q ´ p n ` ε q p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż t p ` t ´ τ q ´ ` n ` ε p ` τ q ´ n p p ´ q` p ´ dτ ¯ } u } pX p t q . Then, repeating some arguments as we did in Step 1 we may conclude ›› ∇ n ` ε u nl p t, ¨q ›› L À p ` t q ´ n ´ ` ε } u } pX p t q . By an analogous manner, we can proceed as follows: } u nl t p t, ¨q} L À ż t ›› B t ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› B t ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ´ ›› | u p τ, ¨q| p ›› L dτ ` ż tt { p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ ` ż t e ´ c p t ´ τ q ›› ∇ | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż tt { p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż t e ´ c p t ´ τ q p ` τ q ´ n p p ´ q` p ´ ` ǫ dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ ¯ } u } pX p t q . For this reason, repeating again some arguments as we did in Step 1 one also obtains ż t { p ` t ´ τ q ´ n ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n ż tt { p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n . To estimate the remaining integral, we shall employ Lemma 5.2 to achieve ż t e ´ c p t ´ τ q p ` τ q ´ n p p ´ q` p ´ ` ǫ dτ À p ` t q ´ n p p ´ q` p ´ ` ǫ À p ` t q ´ n ´ ď p ` t q ´ n , where we have used ´ n p p ´ q ` p ´ ` ǫ ď ´ n ´ } u nl t p t, ¨q} L À p ` t q ´ n } u } pX p t q . Therefore, from the definition of the norm in X p t q we obtain immediately the inequality (26). Next let us prove the inequality (24) . We shall follow the strategy used in the proof of theinequality (26). The new difficulty is to require the estimates for the term | u p τ, ¨q| p ´ | v p τ, ¨q| p in L , L and H . Then, repeating an analogous treatment as in the proof of the inequality (26)we may conlcude the inequality (24). Indeed, by using H¨older’s inequality we get ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À } u p τ, ¨q ´ v p τ, ¨q} L p ` } u p τ, ¨q} p ´ L p ` } v p τ, ¨q} p ´ L p ˘ , ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À } u p τ, ¨q ´ v p τ, ¨q} L p ` } u p τ, ¨q} p ´ L p ` } v p τ, ¨q} p ´ L p ˘ . Analogously to the proof of (26), employing Proposition 5.2 to the norms } u p τ, ¨q ´ v p τ, ¨q} L η , } u p τ, ¨q} L η , } v p τ, ¨q} L η with η “ p and η “ p we may arrive at the following estimates: ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q` p ` ǫ } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ , ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q` p ` ǫ } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ . Let us now turn to estimate the norm ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› H “ ›› ∇ ` | u p τ, ¨q| p ´ | v p τ, ¨q| p ˘›› L . At first, observing that dd u | u | p ´ u “ p p ´ q| u | p ´ for p ě | u | p ´ u ´ | v | p ´ v “ p p ´ q ` ω | u | p ´ ` p ´ ω q| v | p ´ ˘ p u ´ v q for some ω P r , s . On the other hand, we see ∇ ` | u p τ, x q| p ´ | v p τ, x q| p ˘ “ p | u p τ, x q| p ´ u p τ, x q ∇ u p τ, x q ´ p | v p τ, x q| p ´ v p τ, x q ∇ v p τ, x q“ p | u p τ, x q| p ´ u p τ, x q ` ∇ u p τ, x q ´ ∇ v p τ, x q ˘ ` p ∇ v p τ, x q ` | u p τ, x q| p ´ u p τ, x q ´ | v p τ, x q| p ´ v p τ, x q ˘ . Therefore, we have ˇˇ ∇ ` | u p τ, x q| p ´ | v p τ, x q| p ˘ˇˇ ď C | u p τ, x q| p ´ | ∇ u p τ, x q ´ ∇ v p τ, x q|` C | ∇ v p τ, x q| ` | u p τ, x q| p ´ ` | v p τ, x q| p ´ ˘ | u p τ, x q ´ v p τ, x q| . Then, we can conclude the estimate ›› ∇ ` | u p τ, ¨q| p ´ | v p τ, ¨q| p ˘›› L ď C } u p τ, ¨q} p ´ L } ∇ u p τ, ¨q ´ ∇ v p τ, ¨q} L ` C } ∇ v p τ, ¨q} L ` } u p τ, ¨q} p ´ L ` } v p τ, ¨q} p ´ L ˘ } u p τ, ¨q ´ v p τ, ¨q} L , which implies the desired estimate ›› ∇ ` | u p τ, ¨q| p ´ | v p τ, ¨q| p ˘›› L À p ` τ q ´ n p p ´ q` p ´ } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ (27)by the aid of Proposition 5.2. This completes the proof of inequality (24). (cid:3) Proof of Theorem 1.1 with n “ . We need to modify the solution space as X p t q : “ C ` r , , H ` ε ˘ X C ` r , , L ˘ with the norm } u } X p t q : “ sup ď τ ď t ´ p ` τ q } u p τ, ¨q} L ` p ` τ q ´ ε ›› ∇ ` ε u p τ, ¨q ›› L ` p ` τ q } u t p τ, ¨q} L ¯ . At first, we note that } u p τ, ¨q} À } u p τ, ¨q} ´ θ ›› ∇ ` ε u p τ, ¨q ›› θ À p ` τ q ´ ` ε p ` ε q } u } X p t q with θ “ ` ε by Proposition 5.3, and } ∇ u p τ, ¨q} À } u p τ, ¨q} ´ θ ›› ∇ ` ε u p τ, ¨q ›› θ À p ` τ q ´ ` ε p ` ε q } u } X p t q with θ “ ` ε by Proposition 5.2. Now we choose a constant ε fulfilling ε ě ε p ` ε q , which solves ´ ` ε p ` ε q ď ´ ` ε and ´ ` ε p ` ε q ď ´ ` ε , so that one arrives at the following estimates: } u p τ, ¨q} À p ` τ q ´ ` ε } u } X p t q , } ∇ u p τ, ¨q} À p ` τ q ´ ` ε } u } X p t q . Then we define a mapping Φ : X p t q ÝÑ X p t q byΦ r u sp t, x q “ u ln p t, x q ` u nl p t, x q . As we see in the proof of Theorem 1.1 with n “ , ,
4, the proof of Theorem 1.1 with n “ } Φ r u s} X p t q À }p u , u q} A ` } u } pX p t q , (28) } Φ r u s ´ Φ r v s} X p t q À } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ . (29)Similar arguments to the proof of Theorem 1.1 with n “ , , ›› | u p τ, ¨q| p ›› L ď } u p τ, ¨q} p ´ L } u p τ, ¨q} L À p ` τ q ´ p p ´ q` ε p p ´ q } u } pX p t q , (30) ›› | u p τ, ¨q| p ›› L ď } u p τ, ¨q} p ´ L } u p τ, ¨q} L À p ` τ q ´ p p ´ q` ε p p ´ q } u } pX p t q , (31) ›› ∇ | u p τ, ¨q| p ›› L ď } u p τ, ¨q} p ´ L } ∇ u p τ, ¨q} L À p ` τ q ´ p p ´ q` ε p } u } pX p t q . (32)For the proof of Theorem 1.1 with n “
5, the following form of the estimates from (30) to (32) areuseful: ›› | u p τ, ¨q| p ›› L À p ` τ q ´ } u } pX p t q , ›› | u p τ, ¨q| p ›› L À p ` τ q ´ ` ε p } u } pX p t q , ›› ∇ | u p τ, ¨q| p ›› L À p ` τ q ´ ` ε p } u } pX p t q . They follow by the same method as in the previous section.
First let us prove the inequality (28) . As in the proof of the estimate (23), we only show theestimate } u nl } X p t q À } u } pX p t q . (33)Our proof is divided into two steps.Step 1: We may estimate the norm } u nl p t, ¨q} L as follows: } u nl p t, ¨q} L À ż t ›› ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ´ ż t p ` t ´ τ q ´ p ` τ q ´ dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ ` ε p dτ ¯ } u } pX p t q . The employment of Lemma 5.1 implies immediately that } u nl p t, ¨q} L À p ` t q ´ } u } pX p t q . Step 2: By using the same ideas, we may control the remaining norms ›› ∇ n ` ε u nl p τ, ¨q ›› L and } u nl t p t, ¨q} L EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 13 as follows: ›› ∇ ` ε u nl p t, ¨q ›› L À ż t ›› ∇ ` ` ε K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ ` ε ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x ∇ | u p τ, x q| p ›› L dτ À ż t p ` t ´ τ q ´ ` ε ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ` ` ε ›› ∇ | u p τ, ¨q| p ›› L dτ À ´ ż t p ` t ´ τ q ´ ` ε p ` τ q ´ dτ ` ż t p ` t ´ τ q ´ ` ε p ` τ q ´ ` ε p dτ ¯ } u } pX p t q . Then, after applying Lemma 5.1 again, we may conclude the following estimate: ›› ∇ ` ε u nl p t, ¨q ›› L À p ` t q ´ ` ε } u } pX p t q . Now we turn to deal with the norm } u nl t p t, ¨q} L by } u nl t p t, ¨q} L À ż t ›› B t ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› B t ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ ` ż t e ´ c p t ´ τ q ›› ∇ | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ´ ż t p ` t ´ τ q ´ p ` τ q ´ dτ ` ż t e ´ c p t ´ τ q p ` τ q ´ ` ε p dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ ` ε p dτ ¯ } u } pX p t q . Then, using some arguments as we did in Step 2 in the proof of the case n “ , , } u nl t p t, ¨q} L À p ` t q ´ } u } pX p t q . Therefore, from the definition of the norm in X p t q we obtain immediately the inequality (33). Next let us prove the inequality (29) . By the same way as in the proof of the estimate (24), wehave ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À p ` τ q ´ } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ , ›› | u p τ, ¨q| p ´ | v p τ, ¨q| p ›› L À p ` τ q ´ ` ε p } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ , ›› ∇ ` | u p τ, ¨q| p ´ | v p τ, ¨q| p ˘›› L À p ` τ q ´ ` ε p } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ˘ . This completes the proof of inequality (29). (cid:3) Proof of main result (II)
This section is devoted to the proof of Theorem 1.2.
Proof of Theorem 1.2.
We introduce the solution space Y p t q : “ C ` r , t s , H n ` ε ˘ X C ` r , t s , H n ` ε ˘ with the norm } u } Y p t q : “ sup ď τ ď t ´ ℓ p τ q} u p τ, ¨q} L ` p ` τ q n ´ ` ε ›› ∇ n ` ε u p τ, ¨q ›› L ` p ` τ q n } u t p τ, ¨q} L ` p ` τ q n ` ε ›› ∇ n ` ε u t p τ, ¨q ›› L ¯ , where ℓ p τ q is defined in the previous section. In the sequel, we follow the strategy in the previoussection. Then, we have the integral equation corresponding to (8) in the following form: u p t, x q “ u ln p t, x q ` ż t K p t ´ τ, x q ˚ x ` a ˝ ∇ | u t p τ, x q| p ˘ dτ “ : u ln p t, x q ` ˜ u nl p t, x q . We define a mapping Φ : Y p t q ÝÑ Y p t q in the following way:Φ r u sp t, x q “ u ln p t, x q ` ˜ u nl p t, x q . As we see in the proof of Theorem 1.1, the proof of Theorem 1.2 is reduced to prove the followingestimates: } Φ r u s} Y p t q À }p u , u q} B ` } u } pY p t q , (34) } Φ r u s ´ Φ r v s} Y p t q À } u ´ v } Y p t q ` } u } p ´ Y p t q ` } v } p ´ Y p t q ˘ . (35)Before indicating the both above inequalities, the application of Proposition 5.3 gives } u t p τ, ¨q} L À } u t p τ, ¨q} ´ θL } u t p τ, ¨q} θ H n ` ε with θ “ n n ` ε À p ` τ q ´ n } u } Y p t q . By the same fashion as in the proof of Theorem 1.1, we obtain the following auxiliary estimates forany p ě ›› | u t p τ, ¨q| p ›› L “ } u t p τ, ¨q} pL p À p ` τ q ´ n p p ´ q } u } pY p τ q , ›› | u t p τ, ¨q| p ›› L “ } u t p τ, ¨q} pL p À p ` τ q ´ n p p ´ q } u } pY p τ q , ›› ∇ | u t p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q´ } u } pY p τ q . First let us prove the inequality (34).
As in the proof of the estimate (23), we only show theestimate } ˜ u nl } Y p t q À } u } pY p t q . (36)We will follow by same method as in the previous section. Our proof is divided into two steps.Step 1: We may estimate the norm } ˜ u nl p t, ¨q} L as follows: } ˜ u nl p t, ¨q} L À ż t ›› ∇K L p t ´ τ, x q ˚ x | u t p τ, x q| p ›› L dτ ` ż t ›› ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u t p τ, x q| p ›› L dτ EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 15 À ż t { p ` t ´ τ q ´ n ›› | u t p τ, ¨q| p ›› L dτ ` ż tt { ›› | u t p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u t p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q dτ ` ż tt { p ` τ q ´ n p p ´ q dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ ¯ } u } pY p t q . When p ě ` n , it follows immediately ´ n p p ´ q ď ´
1. On the other hand, if 1 ` n ď p ă ` n ,we see ´ ă ´ n p p ´ q ď ´ . Hence, using the relations ` t ´ τ « ` t if τ P r , t { s ` τ « ` t if τ P r t { , t s to control the first two integrals we derive ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q dτ À p ` t q ´ n log p t ` e q if p ě ` n p ` t q ´ n ` if 1 ` n ď p ă ` n À p ` t q ´ n ` for any p ě ` n , and ż tt { p ` τ q ´ n p p ´ q dτ À p ` t q ´ n ` , where we used the fact that ´ n p p ´ q ` ď ´ n ` since the condition p ě ` n holds from (9).After applying Lemma 5.1, we arrive at the following estimate for the third integral: ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ À p ` t q ´ min , n p p ´ q ( À p ` t q ´ n ` . Therefore, we have proved that } ˜ u nl p t, ¨q} L À p ` t q ´ n ` } u } pY p t q À log p t ` e q} u } pY p t q if n “ , p ` t q ´ n ` } u } pY p t q if n “ , . Step 2: By using the same ideas, we may control the remaining norms ›› ∇ n ` ε ˜ u nl p t, ¨q ›› L , } ˜ u nl t p t, ¨q} L and ›› ∇ n ` ε ˜ u nl t p t, ¨q ›› L as follows: ›› ∇ n ` ε ˜ u nl p t, ¨q ›› L À ż t ›› ∇ n ` ` ε K L p t ´ τ, x q ˚ x | u t p τ, x q| p ›› L dτ ` ż t ›› ∇ n ` ε ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x ∇ | u t p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ` ε ›› | u t p τ, ¨q| p ›› L dτ ` ż tt { p ` t ´ τ q ´ p n ` ε q ›› | u t p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ` n ` ε ›› ∇ | u t p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n ` ε p ` τ q ´ n p p ´ q dτ ` ż tt { p ` t ´ τ q ´ p n ` ε q p ` τ q ´ n p p ´ q dτ ` ż t p ` t ´ τ q ´ ` n ` ε p ` τ q ´ n p p ´ q´ dτ ¯ } u } pY p t q . By the similar way to Step 1, we gain ›› ∇ n ` ε ˜ u nl p t, ¨q ›› L À p ` t q ´ n ´ ` ε } u } pY p t q . Furthermore, one gets } ˜ u nl t p t, ¨q} L À ż t ›› B t ∇K L p t ´ τ, x q ˚ x | u t p τ, x q| p ›› L dτ ` ż t ›› B t ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u t p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ´ ›› | u t p τ, ¨q| p ›› L dτ ` ż tt { p ` t ´ τ q ´ ›› | u t p τ, ¨q| p ›› L dτ ` ż t e ´ c p t ´ τ q ›› ∇ | u t p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u t p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n ´ p ` τ q ´ n p p ´ q dτ ` ż tt { p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ ` ż t e ´ c p t ´ τ q p ` τ q ´ n p p ´ q´ dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ ¯ } u } pY p t q Then, an analogous treatment as we estimated in Step 1 leads to ż t { p ` t ´ τ q ´ n ´ p ` τ q ´ n p p ´ q dτ À p ` t q ´ n ż tt { p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ À p ` t q ´ n ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q dτ À p ` t q ´ n . EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 17
After employing Lemma 5.2, one has ż t e ´ c p t ´ τ q p ` τ q ´ n p p ´ q´ dτ À p ` t q ´ n p p ´ q´ À p ` t q ´ n ´ À p ` t q ´ n because of the hypothesis (9). All the above estimates follow that } ˜ u nl t p t, ¨q} L À p ` t q ´ n } u } pY p t q . Now let us control the norm ›› ∇ n ` ε ˜ u nl t p t, ¨q ›› L in the following way: ›› ∇ n ` ε ˜ u nl t p t, ¨q ›› L À ż t ›› B t ∇ n ` ` ε K L p t ´ τ, x q ˚ x | u t p τ, x q| p ›› L dτ ` ż t ›› B t ∇ n ` ε ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x ∇ | u t p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ` ` ε ›› | u t p τ, ¨q| p ›› L dτ ` ż tt { p ` t ´ τ q ´ p n ` ` ε q ›› | u t p τ, ¨q| p ›› L dτ ` ż t e ´ c p t ´ τ q p t ´ τ q ´ p n ` ` ε q ›› | u t p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ` n ` ε ›› ∇ | u t p τ, ¨q| p ›› L dτ À ´ ż t { p ` t ´ τ q ´ n ` ` ε p ` τ q ´ n p p ´ q dτ ` ż tt { p ` t ´ τ q ´ p n ` ` ε q p ` τ q ´ n p p ´ q dτ ` ż t e ´ c p t ´ τ q p t ´ τ q ´ p n ` ` ε q p ` τ q ´ n p p ´ q dτ ` ż t p ` t ´ τ q ´ ` n ` ε p ` τ q ´ n p p ´ q´ dτ ¯ } u } pY p t q , where we used the estimate (22) from Lemma 2.2. Analogously to the estimation for } ˜ u nl t p t, ¨q} L ,we may derive ›› ∇ n ` ε ˜ u nl t p t, ¨q ›› L À p ` t q ´ n ` ε } u } pY p t q . Therefore, from the definition of the norm in Y p t q we obtain immediately the inequality (36). Next let us prove the inequality (35) . We shall follow the strategy used in the proof of theinequality (36). The new difficulty is to require the estimates for the term | u t p τ, ¨q| p ´ | v t p τ, ¨q| p in L , L and H . Then, repeating an analogous treatment as in the proof of the inequality (36)we may conlcude the inequality (35). Indeed, by using H¨older’s inequality we get ›› | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ›› L À } u t p τ, ¨q ´ v t p τ, ¨q} L p ` } u t p τ, ¨q} p ´ L p ` } v t p τ, ¨q} p ´ L p ˘›› | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ›› L À } u t p τ, ¨q ´ v t p τ, ¨q} L p ` } u t p τ, ¨q} p ´ L p ` } v t p τ, ¨q} p ´ L p ˘ . Analogously to the proof of (36), applying Proposition 5.2 to the norms } u t p τ, ¨q ´ v t p τ, ¨q} L η , } u t p τ, ¨q} L η , } v t p τ, ¨q} L η with η “ p and η “ p we may arrive at the following estimates: ›› | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q } u ´ v } Y p t q ` } u } p ´ Y p t q ` } v } p ´ Y p t q ˘ , ›› | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q } u ´ v } Y p t q ` } u } p ´ Y p t q ` } v } p ´ Y p t q ˘ . We also obtain the following estimates as (27): ›› ∇ ` | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ˘›› L ď C } u t p τ, ¨q} p ´ L } ∇ u t p τ, ¨q ´ ∇ v t p τ, ¨q} L ` C } ∇ v t p τ, ¨q} L ` } u t p τ, ¨q} p ´ L ` } v t p τ, ¨q} p ´ L ˘ } u t p τ, ¨q ´ v t p τ, ¨q} L , which leads the estimate ›› ∇ ` | u t p τ, ¨q| p ´ | v t p τ, ¨q| p ˘›› L À p ` τ q ´ n p p ´ q´ } u ´ v } Y p τ q ` } u } p ´ Y p τ q ` } v } p ´ Y p τ q ˘ . (37)This completes the proof of inequality (35). (cid:3) Further discussions
Large time behavior of global solutions.
This subsection is to discuss the large timebehavior of the derived global solutions to (3) and (8) in Theorems 1.1 and 1.2, respectively.Throughout this subsection, we denote some quantities and some kernels as follows: P j : “ ż R n u j p y q dy with j “ , , H p t, x q : “ F ´ ξ Ñ x ´ e ´ t | ξ | cos p t | ξ |q ¯ p t, x q , H p t, x q : “ F ´ ξ Ñ x ´ e ´ t | ξ | sin p t | ξ |q| ξ | ¯ p t, x q . Let us return to the Cauchy problems (3) and (8) in the following common form: u tt ´ ∆ u ` ν p´ ∆ q u t “ a ˝ ∇ ˇˇ B jt u ˇˇ p , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n , (38)where j “ ,
1. We intend to prove the large time behavior of global solutions to (38) in thefollowing result.
Theorem 5.1.
Let ε is a sufficiently small positive constant. We assume that the exponent p andthe space dimension n satisfy the following conditions: $’’’&’’’% p ą ` n ´ and n “ , , , if j “ ,p ą ` n and n “ , if j “ ,p ě and n “ if j “ . (39) Moreover, we choose the initial data p u , u q P ` L , X H n ` ε ˘ ˆ ` L , X H n ´ ` ε ˘ if j “ , p u , u q P ` L , X H n ` ` ε ˘ ˆ ` L , X H n ` ε ˘ if j “ . Then, the global (in time) small data solutions to (38) enjoy the following estimate for t " : ›› u p t, ¨q ´ P H p t, ¨q ´ P H p t, ¨q ›› L “ o ` t ´ n ` ˘ . (40)In order to prove our main result in this section, we need the following auxiliary estimates. EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 19
Proposition 5.1 (Theorem 1.3 in [10]) . Let ď n ď and ℓ ě . Let us assume p u , u q P ` L , X H ℓ ˘ ˆ ` L , X H ℓ ´ ˘ in (2). Then, the solutions to (2) satisfy the following estimate for t " : ›› u p t, ¨q ´ P H p t, ¨q ´ P H p t, ¨q ›› L À t ´ n ´ } u } L , ` t ´ n } u } L , ` e ´ ct }p u , u q} p L X L qˆp L X L q ` t ´ ℓ ` } u } H ℓ ` } u } H ℓ ´ ˘ , (41) where c is a suitable positive constant.Proof. In order to show the proof of Theorem 5.1, let us consider two cases including j “ j “ ‚ Case 1 : If j “
0, then we take ℓ “ n ` ε in Proposition 5.1. By virtue of the statement (41), toindicate the desired estimate (40), we need to show the following estimate instead: ››› ż t K p t ´ τ, x q ˚ x ∇ | u p τ, x q| p dτ ››› L “ o ` t ´ n ` ˘ (42)by using the representation of solutions u p t, x q “ u ln p t, x q ` u nl p t, x q to (3) as in the proof ofTheorem 1.1. Now we assume n “ , ,
4. First of all, recalling the proof of Theorem 1.1 we haveachieved the following estimates: ›› | u p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q` p ` ǫ , (43) ›› | u p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q` p ` ǫ , (44)where ǫ is given by (25). Similarly to the strategy which we have used in the proof of Theorem1.1, we separate the left-hand side term of (42) into several parts as follows: ››› ż t K p t ´ τ, x q ˚ x ∇ | u p τ, x q| p dτ ››› L À ż t ›› ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t { p ` t ´ τ q ´ n ›› | u p τ, ¨q| p ›› L dτ ` ż tt { ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż tt { p ` τ q ´ n p p ´ q` p ` ǫ dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ. In addition, the condition (39) follows immediately ´ n p p ´ q ` p ă ´ . Thus, it implies ´ n p p ´ q ` p ` ǫ ă ´
34 and ´ n ´ p ´ ¯ ` p ` ǫ ă ´ n ´ , which lead to ż t { p ` t ´ τ q ´ n p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n ` ` log p e ` t q ` p ` t q ´ n p p ´ q` p ` ǫ ` ˘ À p ` t q ´ n ` ´ ǫ and ż tt { p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n p p ´ q` p ` ` ǫ À p ` t q ´ n ` ´ ǫ for some sufficiently small constant ǫ ą
0. Moreover, as indicated in the proof of Theorem 1.1,one derives ż t p ` t ´ τ q ´ p ` τ q ´ n p p ´ q` p ` ǫ dτ À p ` t q ´ n ´ . Summing up all the above estimates gives the estimate (42) what we wanted to prove. Next weshow the estimate (42) for n “
5. Applying the estimates (30) and (31), instead of (43) and (44),we have ››› ż t K p t ´ τ, x q ˚ x ∇ | u p τ, x q| p dτ ››› L À ż t ›› ∇K L p t ´ τ, x q ˚ x | u p τ, x q| p ›› L dτ ` ż t ›› ∇ ` K M p t ´ τ, x q ` K H p t ´ τ, x q ˘ ˚ x | u p τ, x q| p ›› L dτ À ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ ` ż t p ` t ´ τ q ´ ›› | u p τ, ¨q| p ›› L dτ À ż t p ` t ´ τ q ´ p ` τ q ´ p p ´ q` ε p p ´ q dτ ` ż t p ` t ´ τ q ´ p ` τ q ´ p p ´ q` ε p p ´ q dτ. Therefore, by using Lemma 5.1 it holds ż t p ` t ´ τ q ´ p ` τ q ´ p p ´ q` ε p p ´ q dτ À $’&’% p ` t q ´ if p p ´ q ´ ε p p ´ q ą , p ` t q ´ log p e ` t q if p p ´ q ´ ε p p ´ q “ , p ` t q ´ p p ´ q` ε p p ´ q` if p p ´ q ´ ε p p ´ q ă “ o p t ´ q (45)as t Ñ 8 , because of the smallness of ε ą ´ p p ´ q ` ε p p ´ q ` ă ´ ż t p ` t ´ τ q ´ p ` τ q ´ p p ´ q` ε p p ´ q dτ À p ` t q ´ min , p p ´ q´ ε p p ´ q ( “ o p t ´ q (46) EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 21 as t Ñ 8 , where we have used the smallness of ε ą n “ ‚ Case 2 : If j “
1, then we take ℓ “ n ` ` ε in Proposition 5.1. Following the proof of Case 1,we may conclude the proof of Case 2 by the aid of the auxiliary estimates as follows: ›› | u t p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q , ›› | u t p τ, ¨q| p ›› L À p ` τ q ´ n p p ´ q , which we have obtained from the proof of Theorem 1.2.Conclusion, our proof is completed. (cid:3) Remark 5.1.
We want to point out that Theorem 5.1 is concerned with the large time behavior ofglobal derived solutions to (3) and (8) in the supercritical cases only, i.e. p ą ` n ´ n “ , , , p ą ` n for all space dimensions n “ ,
3, respectively. It remainsan open problem to explore such result in the critical cases, i.e. p “ ` n ´ n “ , , p “ ` n for n “ , ż s p ` ρ q ´ n p p ´ q` p dρ and ż s p ` ρ q ´ n p p ´ q dρ for s " , which are not infinitesimal quantities of s when p “ ` n ´ p “ ` n , respectively.5.2. Mixed nonlinearities.
In this subsection, relying on the proof of Theorems 1.1 and 1.2 onemay catch the global (in time) existence of small data solutions and their decay properties to (1)with the nonlinear function f p u, u t q “ ˇˇ B jt u ˇˇ p ` a ˝ ∇ ˇˇ B jt u ˇˇ q with j “ ,
1, i.e. the following semi-linearequations with mixing two different kinds of nonlinearities: u tt ´ ∆ u ` ν p´ ∆ q u t “ ˇˇ B jt u ˇˇ p ` a ˝ ∇ ˇˇ B jt u ˇˇ q , x P R n , t ą ,u p , x q “ u p x q , u t p , x q “ u p x q , x P R n , (47)where p, q ą
1. We obtain the following results.
Theorem 5.2 (Equation (47) with j “ . Let n “ , , , and ε is a sufficiently small positiveconstant. Assume that the following conditions for p and q hold: p ą , q ą if n “ ,p ą ` n ´ , q ě ` n ´ if n “ , , . (48) Then, we have the same conclusions as those in Theorem 1.1 and the estimates from (5) to (7) hold. Theorem 5.3 (Equation (47) with j “ . Let n “ , , and ε is a sufficiently small positiveconstant. Assume that the following conditions for p and q hold: p ą ` n and q ě ` max ! n , ) . (49) Then, we have the same conclusions as those in Theorem 1.2 and the estimates from (10) to (13) hold. Proof of Theorems 5.2 and 5.3.
We introduce the solution space X p t q ” $’&’% X p t q if n “ , , j “ ,X p t q if n “ j “ ,Y p t q if n “ , , j “ , where the spaces X p t q , X p t q and Y p t q appear as in the proof of Theorems 1.1 and 1.2. Thesolutions to (47) can be written by the following form: u p t, x q “ u ln p t, x q ` ż t K p t ´ τ, x q ˚ x ` |B jt u p τ, x q| p ` a ˝ ∇ |B jt u p τ, x q| q ˘ dτ “ : u ln p t, x q ` ¯ u nl p t, x q . We define a mapping Ψ : X p t q ÝÑ X p t q in the following way:Ψ r u sp t, x q “ u ln p t, x q ` ¯ u nl p t, x q . Our main is to indicate that the following pair of inequalities are fulfilled: } Ψ r u s} X p t q À }p u , u q} A ` } u } pX p t q ` } u } qX p t q , } Ψ r u s ´ Ψ r v s} X p t q À } u ´ v } X p t q ` } u } p ´ X p t q ` } v } p ´ X p t q ` } u } q ´ X p t q ` } v } q ´ X p t q ˘ . Then, repeating the similar approach to we did in the proof of Theorems 1.1 and 1.2 we may arriveat the desired inequalities above, which are to finish the proof of Theorems 5.2 and 5.3. (cid:3)
Remark 5.2.
In terms of the admissible exponents for p and q in (48), here one recognizes thatthe effect of the nonlinear convection a ˝ ∇ p|B jt u | q q is really remarkable in comparison with that ofthe usual power nonlinearities |B jt u | p , where j “ ,
1. More precisely, we can say that the formernonlinearities brings some more flexibility for lower bounds than those coming from the latternonlinearities.
Acknowledgments
This research of the first author (Tuan Anh Dao) is funded (or partially funded) by the Si-mons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science andTechnology. The work of the second author (H. TAKEDA) was supported in part by the Grant-in-Aid for Scientific Research (C) (No. 19K03596) from Japan Society for the Promotion of Science.The authors are grateful to the referee for his careful reading of the manuscript and for helpfulcomments.
Appendix
This section is to provide several useful inequalities, which play a significant role in the proofsof Sections 3 and 4.
Proposition 5.2 (Fractional Gagliardo-Nirenberg inequality) . Let ă r, r , r ă 8 , σ ą and s P r , σ q . Then, it holds } v } H sr À } v } ´ θL r } v } θ H σr , where θ “ θ s,σ p r, r , r q “ r ´ r ` sn r ´ r ` σn and sσ ď θ ď . For the proof one can see [9].
EMI-LINEAR STRUCTURALLY DAMPED WAVE EQUATIONS WITH NONLINEAR CONVECTION 23
Proposition 5.3 (Sobolev embedding) . Let ε ą . Then, it holds } v } L À } v } ´ θL } v } θ H n ` ε , where θ “ n n ` ε . The proof of Proposition 5.3 is well-known. However, for the convenience of the reader, we willshow it.
Proof. If } v } L “
0, we have v “ R n . Then, the statement is trivial. Now we assume } v } L ‰ R “ ˜ ›› ∇ n ` ε v ›› L } v } L ¸ n ` ε . It is easy to see } v } L “ } F ´ p p v q} L À } p v } L “ ż | ξ |ď R | p v p ξ q| dξ ` ż | ξ |ě R | p v p ξ q| dξ “ : A ` A . Now we apply H¨older inequality to have A ď ˜ż | ξ |ď R dξ ¸ } p v } L À R n } v } L and A ď ˜ż | ξ |ě R | ξ | ´p n ` ε q dξ ¸ ›› | ξ | n ` ε p v ›› L À R ´ ε ›› ∇ n ` ε v ›› L . Therefore, summing up the above we obtain } v } L À R n } v } L ` R ´ ε ›› ∇ n ` ε v ›› L À } v } L } ´ θ ›› ∇ n ` ε v ›› θL , where θ “ n n ` ε . This is the desired estimate. Hence, we have completed the proof of Proposition5.3. (cid:3) Moreover, the following lemmas comes into play.
Lemma 5.1.
Let α, β P R . Then, the following inequality holds: ż t p ` t ´ τ q ´ α p ` τ q ´ β dτ À $’&’% p ` t q ´ min t α,β u if max t α, β u ą , p ` t q ´ min t α,β u log p e ` t q if max t α, β u “ , p ` t q ´ α ´ β if max t α, β u ă . The proof of this lemma can be found in [3].
Lemma 5.2.
Let c ą , ď α ă and β P R . Then, the following inequality holds: ż t e ´ c p t ´ τ q p t ´ τ q ´ α p ` τ q ´ β dτ À p ` t q ´ β . For the ease of reading, we will prove this lemma even if it is standard and well-known.
Proof.
Let us distinguish our consideration into two cases as follows: ‚ If t P r , s , then it is obvious that ż t e ´ c p t ´ τ q p t ´ τ q ´ α p ` τ q ´ β dτ À ż t p t ´ τ q ´ α dτ “ ´ α t ´ α ď ´ α À p ` t q ´ β , where we have used the condition 0 ď α ă ‚ If t ě
1, then we split the integral of the left-hand side into the following two parts: I : “ ż t e ´ c p t ´ τ q p t ´ τ q ´ α p ` τ q ´ β dτ “ ż t { e ´ c p t ´ τ q p t ´ τ q ´ α p ` τ q ´ β dτ ` ż tt { e ´ c p t ´ τ q p t ´ τ q ´ α p ` τ q ´ β dτ “ : I ` I . Noticing that t ´ τ P r t { , t s for any τ P r , t { s in the first integral one derives I À e ´ ct { t ´ α ż t { p ` τ q ´ β dτ À $’&’% e ´ ct { p ` t q ´ α ´ β if β ă e ´ ct { p ` t q ´ α log p ` t q if β “ e ´ ct { p ` t q ´ α if β ą À p ` t q ´ β . Thanks to the relation 1 ` τ « ` t for any τ P r t { , t s , we may estimate the second integral inthe following way: I À p ` t q ´ β ż tt { e ´ c p t ´ τ q p t ´ τ q ´ α dτ “ p ` t q ´ β ż t { e ´ cρ ρ ´ α dρ p by the change of variables ρ “ t ´ τ q“ p ` t q ´ β ´ α ´ e ´ cρ ρ ´ α ˇˇˇ ρ “ t { ρ “ ` c ż t { e ´ cρ ρ ´ α dρ ¯ À p ` t q ´ β ´ α ´ e ´ ct { t ´ α ` c ż t { e ´ cρ { dρ ¯ p by 0 ď α ă qÀ p ` t q ´ β . Combining all the above estimates leads to what we wanted to prove. Therefore, our proof iscompleted. (cid:3)
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Tuan Anh DaoSchool of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1Dai Co Viet road, Hanoi, VietnamInstitute of Mathematics, Vietnam Academy of Science and Technology, No.18 Hoang Quoc Vietroad, Hanoi, Vietnam
Email address : [email protected] Hiroshi TakedaFaculty of Engineering, Fukuoka Institute of Technology3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295 Japan
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