Sharp polynomial decay rates for the damped wave equation with Hölder-like damping
aa r X i v : . [ m a t h . A P ] M a r SHARP POLYNOMIAL DECAY RATES FOR THE DAMPED WAVEEQUATION WITH H ¨OLDER-LIKE DAMPING
KIRIL DATCHEV AND PERRY KLEINHENZ
Abstract.
We study decay rates for the energy of solutions of the damped wave equation on thetorus. We consider dampings invariant in one direction and bounded above and below by multiplesof x β near the boundary of the support and show decay at rate 1 /t β +2 β +3 . In the case where W vanishes exactly like x β this result is optimal by [Kle19]. The proof uses a version of the Morawetzmultiplier method. Introduction
Let W be a bounded, nonnegative damping function on a compact Riemannian manifold M , andlet v solve ( ∂ t v + W ∂ t v − ∆ v = 0 , t > , ( v, ∂ t v ) = ( v , v ) ∈ C ∞ ( M ) × C ∞ ( M ) , t = 0 . (1)We are interested in decay rates as t → ∞ for the energy E ( t ) = Z M | ∂ t v ( t ) | + |∇ v ( t ) | . When W is continuous, it is classical that uniform stabilization , namely a uniform decay rate E ( t ) ≤ Cr ( t ) E (0) with r ( t ) → t → ∞ , is equivalent to geometric control , namely the existenceof a length L such that all geodesics of length at least L intersect the set where W >
0. Moreover,in this case the optimal r ( t ) is exponentially decaying in t .When uniform stabilization fails, we look instead for r ( t ) such that E ( t ) / ≤ Cr ( t ) (cid:0) k v k H ( M ) + k ∂ t v k H ( M ) (cid:1) . (2)Then the optimal r ( t ) depends on the geometry of M and of the set where W >
0, and also onthe rate of vanishing of W . In this note we explore this dependence in precise detail for translationinvariant damping functions on the torus, where we prove decay of the form E ( t ) / ≤ Ct − α (cid:0) k v k H ( M ) + k ∂ t v k H ( M ) (cid:1) . (3) Theorem.
Let M be the torus ( R / π Z ) x × ( R / π Z ) y . Let C > , σ ∈ (0 , π ) , and β ≥ be given.Suppose W = W ( x ) obeys C V ( x ) ≤ W ( x ) ≤ C V ( x ) , V ( x ) = ( , | x | ∈ [0 , σ ] , ( | x | − σ ) β , | x | ∈ ( σ, π ] , (4) for all x ∈ [ − π, π ] . Then there is C , depending only on C , σ , and β , such that (3) holds with α = β + 2 β + 3 . (5) Date : March 27, 2020.
Remarks. (1) Our result is especially interesting when W = V near the set where | x | = σ . Then, byTheorem 1.1 of [Kle19], the value of α in (5) is the best possible. More specifically, in thissetting the second author proves that (3) is false for any α > β +2 β +3 by constructing a suitablesequence of quasimodes of the stationary operator − ∆ + iqW − q .(2) As is clear from the reduction to (8) at the beginning of the proof below, the same proofgives the same result (with the same constant C ) if the torus T = ( R / π Z ) x × ( R / π Z ) y isreplaced by another product ( R / π Z ) x × Σ y , where Σ is any compact Riemannian manifold.The equivalence of uniform stabilization and geometric control for continuous damping functionswas proved by Ralston [Ral69], Rauch and Taylor [RT75] (see also [BLR92] and [BG97], where M is also allowed to have a boundary). For some more recent finer results concerning discontinuousdamping functions, see Burq and G´erard [BG18].Decay rates of the form (2) go back to Lebeau [Leb96]. If we assume only that W ∈ C ( M ) isnonnegative and not identically 0, then the best general result is that r ( t ) in (2) is 1 / log(2 + t )[Bur98] and this is optimal on spheres and some other surfaces of revolution [Leb96]. At the otherextreme, if M is a negatively curved (or Anosov) surface, W ∈ C ∞ ( M ), W ≥ W
0, then r ( t )may be chosen exponentially decaying [DJN19].When M is a torus, these extremes are avoided and the best bounds are polynomially decaying asin (3). Anantharaman and L´eautaud [AL14] show (3) holds with α = 1 / W ∈ L ∞ , W ≥ W >
W > W
0. Anantharaman and L´eautaud [AL14] further show that ifsupp W does not satisfy the geometric control condition then (3) cannot hold for any α >
1. Theyalso show if W satisfies |∇ W | ≤ W − ε for ε > W ∈ W k , ∞ for some k then(3) holds with α = 1 / (1 + 4 ε ). For earlier work on the square and partially rectangular domainssee [LR05] and [BH07] respectively, and for polynomial decay rates in the setting of a degeneratelyhyperbolic undamped set, see [CSVW14].In [Kle19], the second author shows that, if W = V near σ , then (3) holds with α = ( β +2) / ( β +4).In the case of constant damping on a strip ( W = V and β = 0) the result that (3) holds with α = 2 / α > / σ ∈ (0 , π ), but one can also look at the behavior as σ approaches theendpoints of the interval. As σ → π , the constants in our estimates blow up. This makes sensebecause the problem becomes undamped, and no decay is possible in the limit ((3) holds only with α = 0). More interesting is to let σ →
0. In that case the constants in our estimates remainbounded, but better results are known by other methods.In [LL17], L´eautaud and Lerner show that if σ = 0, then (3) holds with α = ( β + 2) /β . They alsoconsider more general manifolds and damping functions. Note that, intriguingly, the decay ratedecreases as β increases when σ = 0, while the decay rate increases as β increases when σ ∈ (0 , π ).A key difference in the geometry is that, when σ = 0 the support of W is the whole torus (so allgeodesics interesect it) whereas when σ > − σ, σ )which do not intersect the support of W .Our result may be interpreted microlocally in the following way. The decay rates in (2) and (3)are related to time averages of W along geodesics [Non11]. When W has conormal singularities, asin the case that W = V near the set where | x | = σ , one must consider both transmitted and reflectedgeodesics. In our setting reflected geodesics originating in the undamped region remain undamped,which slows decay. Stronger singularities in W correspond to more reflection [dHUV15, GW18b],so we expect smaller values of β to lead to slower decay. (See also [DKK15, GW18a] for examplesof such phenomena for scattering resonances) By contrast, in the setting of [LL17], where σ = 0, HARP POLYNOMIAL DECAY RATES 3 reflected and transmitted geodesics are both equally damped. In that case, smaller values of β correspond to larger time averages of W along geodesics just because W is then larger, so we expectfaster decay. In terms of our estimates below, the effect of geodesics which remain undamped for along time is reflected in the fact that our bounds are weakest for angular momentum modes closeto the undamped (vertical) ones; in the notation of Section 2, this corresponds to E positive butnot too large. 2. Proof of Theorem
By a Fourier transform in time, we may study the associated stationary problem. More precisely,by Theorem 2.4 of [BT10], as formulated in Proposition 2.4 of [AL14], the decay (3) with α givenby (5) follows from showing that that there are constants C and q such that, for any q ≥ q , k ( − ∆ + iqW − q ) − k L ( T ) → L ( T ) ≤ Cq / ( β +2) . (6)Expanding in a Fourier series in the y variable we see that it is enough to show that there are C and q such that for any f ∈ L ( R / π Z ), any real E ≤ q and any q ≥ q , if u ∈ H ( R / π Z ) solves − u ′′ + iqW u − Eu = f, (7)then Z | u | ≤ Cq / ( β +2) Z | f | . (8)Here, and below, all integrals are over R / π Z . We will actually obtain a more precise dependenceon E , namely we will show that there is E > Z | u | ≤ C Z | f | , when E ≤ E , (9)and Z | u | ≤ CE − q / ( β +2) Z | f | , when E ≥ E . (10)The second of these, (10), is our main estimate.In our proofs we use a version of the Morawetz multiplier method, which we arrange using theenergy functional F ( x ) = | u ′ ( x ) | + E | u ( x ) | . (11)This method was introduced to prove wave decay for star-shaped obstacle scattering [Mor61], andour approach is inspired by that of [CV02], as adapted to cylindrical geometry in [CD17].We begin with some easier and essentially well-known estimates. We will often use the elementaryfact that if a, b, c, d, e ≥ θ ∈ [0 , a + b ≤ cb − θ d θ + e = ⇒ a + θb ≤ θc /θ d + e. (12) Lemma 1.
For any E ∈ R , q > and u, f solving (7) we have Z W | u | ≤ q − Z | f u | . (13) Also, for any ψ ∈ C ∞ ( R / π Z ) which vanishes near [ − σ, σ ] , there is C > such that for any q > , E ∈ R and u, f solving (7) we have Z ψ | u ′ | ≤ C (1 + max(0 , E ) q − ) Z | f u | . (14) Finally, there are positive constants E and C such that for any q > , E ≤ E , and u, f solving (7) we have (9) . KIRIL DATCHEV AND PERRY KLEINHENZ
Proof.
To prove (13) we multiply (7) by ¯ u and take the imaginary part, integrating by parts to seethat the first term is real.To prove (14), we integrate by parts twice and use (7) to write Z ψ | u ′ | = − Re Z ψ ′ u ′ ¯ u − Re Z ψu ′′ ¯ u = 12 Z ψ ′′ | u | + E Z ψ | u | + Re Z ψf ¯ u. (15)Now use | ψ ′′ | + | ψ | ≤ CW and (13) to conclude.To prove (9), we multiply (7) by ¯ u and by a positive function b ∈ C ∞ ( R / π Z ) to be determinedlater, integrate, and take the real part to obtain − Re Z bu ′′ ¯ u − E Z b | u | = Re Z bf ¯ u. Integrating by parts twice (as in (15)), gives Z b | u ′ | + Z (cid:18) − b ′′ − Eb (cid:19) | u | = Re Z bf ¯ u. Now choose b such that b ′′ < − σ, σ ]. Then, as long as E ≤ E for some E sufficientlysmall, adding a multiple of (13) gives Z (cid:0) | u ′ | + | u | (cid:1) . Z | f u | ≤ (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / , which implies (9) by (12). (cid:3) It remains to show (10). We proceed by proving two lemmas:
Lemma 2.
Let δ > be given, and let µ = µ ( x ) = ( q δ , | x | ∈ [ σ, σ + q − δ ] , , | x | ∈ [0 , σ ) ∪ ( σ + q − δ , π ] . Then we have Z µ | u ′ | + Eµ | u | . Z | f | + q Z W | uu ′ | . (16) Lemma 3.
Let δ = β +2 and let χ ( x ) = , | x | ∈ [0 , σ ] ,q δ ( | x | − σ ) , | x | ∈ [ σ, σ + q − δ ] , , | x | ∈ [ σ + q − δ , π ] . Then we have Z µ | u ′ | + Eµ | u | . (1 + E − q δ ) Z | f | + q / (cid:18)Z | f u | (cid:19) / (cid:18)Z | W χf u | (cid:19) / . (17)We then prove (10). Proof of Lemma 2.
Fix τ ∈ ( σ, π ) and a continuous and piecewise linear b such that b ′ ( x ) = , | x | ∈ [0 , σ ) ,q δ , | x | ∈ ( σ, σ + q − δ ) , , | x | ∈ ( σ + q − δ , τ ) , − M, | x | ∈ ( τ, π ) , with M > b is 2 π periodic. We assume q is large enough that σ + q − δ < τ when q ≥ q . HARP POLYNOMIAL DECAY RATES 5
With F as in (11), we have( bF ) ′ = b ′ | u ′ | + Eb ′ | u | + 2 b Re u ′′ ¯ u ′ + 2 E Re u ¯ u ′ = b ′ | u ′ | + Eb ′ | u | − b Re f ¯ u ′ + 2 qb Re iW u ¯ u ′ . Using Z ( bF ) ′ = 0 , gives Z b ′ | u ′ | + Eb ′ | u | ≤ Z b | f u ′ | + 2 q Z bW | uu ′ | . Add a multiple of (13) and (14) to both sides, and apply (12), to get the desired statement. (cid:3)
Proof of Lemma 3.
To estimate the last term of Lemma 2 we use (13): (cid:18)Z W | uu ′ | (cid:19) ≤ (cid:18)Z W | u | (cid:19) (cid:18)Z W | u ′ | (cid:19) . q − (cid:18)Z | f u | (cid:19) (cid:18)Z V | u ′ | (cid:19) . (18)We write Z V | u ′ | = Z V (1 − χ ) | u ′ | + Z V χ | u ′ | . For the first term use the fact that V (1 − χ ) is supported on [ σ, σ + q − δ ] where it obeys V (1 − χ ) ≤ q − δβ = q − δβ − δ µ. To handle the
V χ term we integrate by parts. Z V χ | u ′ | = − Re Z ( V χ ) ′ u ′ ¯ u − Re Z V χu ′′ ¯ u. For the first resulting term we use | ( V χ ) ′ | . q δ W, and for the other (7) and (13) give − Re Z V χu ′′ ¯ u = E Z V χ | u | + Re Z V χf ¯ u . Eq − Z | f u | + Z | W χf u | . Putting everything into (18) gives (cid:18)Z W | uu ′ | (cid:19) . q − − δβ − δ (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / Z µ | u ′ | + q − δ (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / Z W | uu ′ | + Eq − (cid:18)Z | f | (cid:19) (cid:18)Z | u | (cid:19) + q − (cid:18)Z | f u | (cid:19) (cid:18)Z | W χf u | (cid:19) , which, by (12), implies (cid:18)Z W | uu ′ | (cid:19) . q − − δβ − δ (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / Z µ | u ′ | + ( Eq − + q − δ ) (cid:18)Z | f | (cid:19) (cid:18)Z | u | (cid:19) + q − (cid:18)Z | f u | (cid:19) (cid:18)Z | W χf u | (cid:19) . KIRIL DATCHEV AND PERRY KLEINHENZ
Inserting into Lemma 2 gives Z µ | u ′ | + Eµ | u | . Z | f | + q (1 − δβ − δ ) / (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / (cid:18)Z µ | u ′ | (cid:19) / + ( E / + q δ ) (cid:18)Z | f | (cid:19) / (cid:18)Z | u | (cid:19) / + q / (cid:18)Z | f u | (cid:19) / (cid:18)Z | W χf u | (cid:19) / , and using again (12) we obtain Z µ | u ′ | + Eµ | u | . (1 + E − q − δβ − δ + E − q δ ) Z | f | + q / (cid:18)Z | f u | (cid:19) / (cid:18)Z | W χf u | (cid:19) / . We choose δ = 1 / ( β + 2) to optimize the dependence on q , giving Lemma 3. (cid:3) Proof of (10) . Let η = δ and let N ∈ N to be chosen later and η j ∈ (0 , δ ) with η j − > η j for j = 1 , . . . N also to be chosen later. By linearity, we may consider separately the N + 3 cases(1) | x | ≤ σ on supp f ,(2) | x | ∈ [ σ, σ + q − δ ] on supp f ,(3) | x | ∈ [ σ + q − η j , σ + q − η j +1 ] on supp f , for j = 0 , . . . , N − | x | ≥ σ + q − η N on supp f .1. In the case that | x | ≤ σ on supp f , the last term in Lemma 3 vanishes and we have (10).2. In the case that | x | ∈ [ σ, σ + q − δ ] on supp f , we use the fact that µ = q δ there to write Z | f u | ≤ q − δ/ (cid:18)Z | f | (cid:19) / (cid:18)Z µ | u | (cid:19) / , and, moreover, since W ≤ q − βδ there, by (13) we have Z | W χf u | ≤ q − βδ/ Z | f W / u | . q − / − βδ/ (cid:18)Z | f | (cid:19) / (cid:18)Z | f u | (cid:19) / ≤ q − / − βδ/ − δ/ (cid:18)Z | f | (cid:19) / (cid:18)Z µ | u | (cid:19) / . (19)Inserted into Lemma 3, these give Z µ | u ′ | + Eµ | u | . (1 + E − q δ ) Z | f | + q / − βδ/ − δ/ (cid:18)Z | f | (cid:19) / (cid:18)Z µ | u | (cid:19) / , which, by (12), implies Z µ | u ′ | + Eµ | u | . (1 + E − q δ + E − / q / − βδ/ − δ/ ) Z | f | , which implies (10).3. In the case that | x | ∈ [ σ + q − η j , σ + q − η j +1 ], since W ≥ q − η j β there and by (13) we have, Z | f u | . q η j β/ Z | f W / u | ≤ q η j β/ − / (cid:18)Z | f | (cid:19) / (cid:18)Z | f u | (cid:19) / , or Z | f u | . q − η j β Z | f | , which also gives, as in (19), Z | W χf u | . q − / − βη j +1 / (cid:18)Z | f | (cid:19) / (cid:18)Z | f u | (cid:19) / . q − − βη j +1 / η j β/ Z | f | . HARP POLYNOMIAL DECAY RATES 7
Inserting these into Lemma 3 gives Z µ | u ′ | + Eµ | u | . (1 + E − q δ + q − / η j β/ − βη j +1 / ) Z | f | .
4. In the case that | x | ≥ σ + q − η N on supp f we estimate similarly: Z | f u | . q βη N / Z | f W / u | ≤ q βη N / − / (cid:18)Z | f | (cid:19) / (cid:18)Z | f u | (cid:19) / , or Z | f u | . q − βη N Z | f | , which gives Z | χW f u | . q − / (cid:18)Z | f | (cid:19) / (cid:18)Z | f u | (cid:19) / . q − βη N / Z | f | . Inserted into Lemma 3, these give Z µ | u ′ | + Eµ | u | . (1 + E − q δ + q − / βη N / ) Z | f | . These are optimized when ( η j = 4 η j +1 − η j +2 , j = 0 , , . . . , N − η N − = 4 η N . Recalling that η = δ this is solved by η k = δ N +1 − k N +1 − , this gives (10) in all cases 3 and 4 as long as N is chosen large enough that β ≤ N +1 − (cid:3) Acknowledgments.
The authors are grateful to Jared Wunsch and Matthieu L´eautaud for help-ful comments and suggestions. KD was partially supported by NSF Grant DMS-1708511. PKwas supported in part by the National Science Foundation grant RTG: Analysis on manifolds atNorthwestern University.
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Department of Mathematics, Purdue University, West Lafayette, IN, USA
E-mail address : [email protected] Department of Mathematics, Northwestern University, Evanston, IL, USA
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