MATHEMATICAL CONTROL AND doi:10 mcrf RELATED FIELDS Volume Number June pp
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MATHEMATICAL CONTROL AND doi:10 mcrf RELATED FIELDS Volume Number June pp

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MATHEMATICAL CONTROL AND doi:10.3934/mcrf.2011.1.251 RELATED FIELDS Volume 1, Number 2, June 2011 pp. 251–265 DECAY OF SOLUTIONS OF THE WAVE EQUATION WITH LOCALIZED NONLINEAR DAMPING AND TRAPPED RAYS Kim Dang Phung Yangtze Center of Mathematics, Sichuan University Chengdu 610064, China (Communicated by Sylvain Ervedoza) Abstract. We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data. 1. Introduction. We study a nonlinear wave equation in a bounded connected open set ? of Rn, n > 1 with a C2 boundary ∂?. Let M = ( ?ij ) 1≤i,j≤n ? C∞ ( ?;Rn?n ) be a symmetric and uniformly positive definite matrix. Therefore ( ?ij ) 1≤i,j≤n =M1/2 is well defined. Denote ? = (∑n j=1 ? 1j∂xj , . . . , ∑n j=1 ? nj∂xj ) and by ∆ = ∑n i,j=1 ∂xi(? ij∂xj ) the “Laplacian” associated to the matrix M. We deal with the following damped wave equation ? ?? ?? ∂2t u?∆u+ ag (∂tu) = 0 in ?? (0,+∞)

  • wave equation

  • holds when

  • well-known optimal

  • preliminary estimates

  • trapped rays

  • polynomial decay

  • optimal logarithmic

  • localized nonlinear


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MATHEMATICAL CONTROL AND RELATED FIELDS Volume 1 , Number 2 , June 2011
doi:10.3934/mcrf.2011.1.251 pp. 251–265
DECAY OF SOLUTIONS OF THE WAVE EQUATION WITH LOCALIZED NONLINEAR DAMPING AND TRAPPED RAYS
Kim Dang Phung Yangtze Center of Mathematics, Sichuan University Chengdu 610064, China (Communicated by Sylvain Ervedoza) Abstract. We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.
1. Introduction. We study a nonlinear wave equation in a bounded connected open set Ω of R n , n > 1 with a C 2 boundary Ω. Let M = α ij 1 i,j n C Ω; R n × n be a symmetric and uniformly positive definite matrix. Therefore β ij 1 i,j n = M 1 / 2 is well defined. Denote r = P jn =1 β 1 j x j , . . . , P jn =1 β nj x j and by Δ = P in,j =1 x i ( α ij x j ) the “Laplacian” associated to the matrix M . We deal with the following damped wave equation t 2 u Δ u + ag ( t u ) = 0 in Ω × (0 , + ) , ( u, ∂ t u ) ( , 0) = ( uu 0 , = u 1 0)ionnΩ Ω, × (0 , + ) , (1.1) where a = a ( x ) L (Ω) is a non-negative bounded function, a ( x ) 0 for all x Ω, and g : R R is a non-decreasing continuous function with g (0) = 0, sg ( s ) 0 and the additional conditions. ( i ) r [1 , ) , c 1 , c 2 > 0 , | s | ≤ 1 = c 1 | s | r ≤ | g ( s ) | ≤ c 2 | s | 1 /r . ( ii ) k [0 , 1] , p [1 , ) , c 3 , c 4 > 0 , | s | > 1 = c 3 | s | k ≤ | g ( s ) | ≤ c 4 | s | p . ( iii ) ( n 2) (1 k ) 4 r and ( n 2) ( p 1) 1 . For example, when g ( s ) = | | p 1 n take k = 1 and 1 r = p nn 12 . s s , we ca When g ( s ) = 1 s + s 2 , we can take k = 0 and r = p = 1. Standard arguments assure existence, uniqueness and regularity of the solution u of ( 1.1 ) (see [ 11 ], [ 10 ], [ 13 ]). For ( u 0 , u 1 ) H 01 (Ω) × L 2 (Ω), there exists a 2000 Mathematics Subject Classification. Primary: 35L05, 35L71; Secondary: 35B40, 35B35. Key words and phrases. Wave equation, decay estimates, localized nonlinear damping, trapped rays. This work was supported by the “Sichuan Youth Science & Technology Foundation” 2010JQ0013 and by the NSF of China under grants 10771149 and 60974035.
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252 KIM DANG PHUNG unique weak solution u C [0 , + ) ; H 01 (Ω) C 1 [0 , + ) ; L 2 (Ω) . Introduce the energy at instant t 0 E ( u, t )=12 Z Ω | t u ( x, t ) | 2 + |r u ( x, t ) | 2 dx . (1.2) E ( u, t ) is a non-increasing function of time and for t 2 > t 1 0 t 2 E ( u, t 2 ) E ( u, t 1 ) = Z Z Ω a ( x ) g ( t u ( x, t )) t u ( x, t ) dxdt 0 . (1.3) t 1 For more regular initial data ( u 0 , u 1 ) H 2 (Ω) H 01 (Ω) × H 01 (Ω), the solution u has the following regularity u L 0 , + ; H 2 (Ω) H 01 (Ω) W 1 , 0 , + ; H 01 (Ω) W 2 , 0 , + ; L 2 (Ω) . Moreover, there exists a constant c > 0 such that for a.e. t 0 Z Ω |r t u ( x, t ) | 2 + t 2 u ( x, t ) 2 dx cE 1 ( u 0 , u 1 ) , (1.4) where E 1 ( u 0 , u 1 ) = k u 0 ag ( u 1 ) , u 1 ) k 2 L 2 (Ω) × H 01 (Ω) . (1.5) Our main interest here is the decay rate of the energy when trapped ray occurs. The decay rate for solutions of the initial boundary value problem to the wave equation governed by a localized nonlinear dissipation is described in [ 17 , Theorem 1.2, p.506] and [ 2 , Theorem 2, p.307-308] (see also [ 5 ]). In [ 17 ] (see also [ 13 ]), a geometrical condition is added involving the sign of ( x x o ) ν ( x ) for x Ω ( ν ( x ) being the outward normal at x Ω). Here, no such condition is imposed and we consider the geometric case where the geometrical control condition (see [ 1 ]) may not be satisfied. When we deal with control of waves with localized damping, it is reasonable to choose the support of the damping large enough and to add a geometric assumption because of trapped rays. However, it is also natural to study what happen without such geometric assumption. We shall give two results in this context. Denote X ( u 0 , u 1 ) = E ( u, 0)+ E 1 ( u 0 , u 1 )+[ E 1 ( u 0 , u 1 )] (2 p 1) +[ E 1 ( u 0 , u 1 )]( 1+ rr + k 1 ).(1.6) Our main results are stated as follows. Theorem 1.1. Suppose that there exists ω Ω a non-empty subdomain such that a ( x ) a o > 0 a.e. in ω . Then there is C > 0 such that for any t > 0 and ( u 0 , u 1 ) H 2 (Ω) H 01 (Ω) × H 01 (Ω) , the solution u of ( 1.1 ) satisfies E ( u, t ) [ln (2 C + t )] 2 (( r 1) + X ( u 0 , u 1 )) . (1.7) Theorem 1.2. Suppose that there exist c > 0 and γ > 0 such that for any t > 0 and ( u 0 , u 1 ) H 2 (Ω) H 01 (Ω) × H 01 (Ω) , the solution u ` of t 2 u ` Δ u ` + a ( x ) t u ` = 0 in Ω × (0 , + ) , u ` = 0 on Ω × (0 , + ) , ( u ` , ∂ t u ` ) ( , 0) = ( u 0 , u 1 ) in Ω , satisfies E ( u ` , t ) tc γ k u 0 , u 1 ) k 2 L 2 (Ω) × H 01 (Ω) .
DECAY OF THE WAVE EQUATION WITH NONLINEAR DAMPING 253 Then there is C > 0 such that for any t > 0 and ( u 0 , u 1 ) H 2 (Ω) H 01 (Ω) × H 01 (Ω) , the solution u of ( 1.1 ) satisfies E ( u, t ) tC δ (( r 1) + X ( u 0 , u 1 )) , (1.8) γ with δ = 2 r ( γ +1)+ γ +2 . Remark 1. In Theorem 1.1 , the damping in our nonlinear mechanism is effective in an arbitrary non-empty subdomain. Theorem 1.1 improves the logarithmic decay in [ 2 ] (see also [ 5 ]). In [ 2 ] (see also [ 5 ]), the logarithmic decay is obtained from an inequality of observation type for the linear wave equation deduced by a Carleman estimate (see [ 8 ] or [ 6 ]) and a Fourier-Bros-Iagolnitzer transform (see [ 9 ]). Here, our logarithmic decay, that is ( 1.7 ), comes directly from the optimal decay rate for the linear damped wave equation established by [ 7 ] and [ 3 , Remarque 4.1, p.18]. In particular, we recover the optimal logarithmic decay when r = k = p = 1. Notice that following [ 6 ], the fact that a L (Ω) and Ω is only C 2 does not affect the getting of the resolvent estimate in [ 7 ]. Remark 2. Concerning Theorem 1.2 , we set an assumption linked with the linear damped wave equation which of course may be fulfilled. Indeed, the polynomial decay estimate for E ( u ` , t ) the energy of the damped linear wave, with respect to a stronger norm of the initial data, when trapped rays occur is described in different works (see [ 4 ], [ 12 ], [ 15 ], [ 14 ]). In particular, the conclusion of Theorem 1.2 holds when Δ = P i 2=1 2 x i , Ω R 2 is a partially rectangular domain and 0 a C Ω is such that a > 0 in the closure of the non-rectangular part of Ω (see [ 4 ] and more recently [ 14 ]). Also, the conclusion of Theorem 1.2 holds when Δ = P i 3=1 2 x i , Ω R 3 generates trapped rays bouncing up and down infinitely between two parallel parts Γ 1 , Γ 2 of the boundary Ω and a a o > 0 a.e. on a neighborhood of Ω \ 1 Γ 2 ) in R 3 (see [ 15 ]). The outline of the paper is the following. Next section gives technical estimates whoseproofisbasedonH¨olderinequalityandSobolevsimbeddingtheorem.In section 3, we recall some results for the linear damped wave equation and we es-tablish interpolation estimates for the nonlinear damped wave equation. In section 4, we complete the proof of Theorem 1.1 (the logarithmic decay) and the proof of Theorem 1.2 (the polynomial decay). Throughout the remainder of this paper, C denotes different positive constants independent of the initial data. 2. Preliminary estimates. In this section, we prove some technical estimates that will be needed in the sequel. Theses estimates are very similar to the ones in [ 2 ]. We have the following results. Proposition 1. If ( n 2) (1 k ) 4 r , then for a.e. t 0 , Z Ω a ( x ) | t u ( x, t ) | 2 dx r + k 1 ) Z |r t u ( x, t ) | 2 dx ( 1+ r (2.1) Ω 2 r +1 + C + ε 1  Z Ω a ( x ) g ( t u ( x, t )) t u ( x, t ) dx ε > 0 .
254 KIM DANG PHUNG Proposition 2. If ( n 2) (1 k ) 4 r and ( n 2) ( p 1) 1 , then for a.e. t 0 , Z Ω | g ( t u ( x, t )) | 2 dx a ( x ) k "Z Ω |r t u ( x, t ) | 2 dx ( 1+ rr +1 )+ Z |r t u ( x, t ) | 2 dx (2 p 1) # (2.2) Ω 2 r +1 + C + ε 1 3  Z Ω a ( x ) g ( t u ( x, t )) t u ( x, t ) dx ε > 0 . Corollary 1. If ( n 2) (1 k ) 4 r and ( n 2) ( p 1) 1 , then for a.e. t 0 , 2 k Δ u ( , t ) k 2 L 2 (Ω) + kr t u ( , t ) k L 2 (Ω) C [ E ( u, 0)] 1 r + C E 1 ( u 0 , u 1 ) + [ E 1 ( u 0 , u 1 )] (2 p 1) + [ E 1 ( u 0 , u 1 )]( 1+ rr + 1 k ) . (2.3) Proof of Corollary 1. First from the equation solved by u and ( 1.4 ), we have k Δ u ( , t ) k 2 L 2 (Ω) + kr t u ( , t ) k 2 L 2 (Ω) 2 ≤ kr t u ( , t ) k 2 L 2 (Ω) + 2 t 2 u ( , t ) L 2 (Ω) + 2 k ag ( t u ) ( , t ) k 2 L 2 (Ω) (2.4) 2 cE 1 ( u 0 , u 1 ) + 2 k ag ( t u ) ( , t ) k 2 L 2 (Ω) . Next, we check that , t )) | 2 dx Z Ω | a ( x ) g )(] r 1 t u ( x E 1 ( u 0 , u 1 )] (2 p 1) + [ E 1 ( u 0 , u 1 )]( 1+ rr + 1 k ) .(2.5) C [ E ( u, 0 + C [ Indeed, from Proposition 2 with ε = 1, we get Z | a ( x ) g ( t u ( x, t )) | 2 dx Ω ≤ k a k L (Ω) C "Z |r t u ( x, t ) | 2 dx ( 1+ rr + 1 k )+ Z Ω |r t u ( x, t ) | 2 dx (2 p 1) # Ω 2 + k a k L (Ω) ( C + 1) Z Ω a ( x ) g ( t u ( x, t )) r +1 t u ( x, t ) dx . (2.6) On the other hand, 2 Z Ω a ( x ) g ( t u ( x, t )) t u ( x, t ) dx r +1 2 2 ≤ k ag ( t u ) k r L +2 ( 1 Ω) k t u k r L +2 ( 1 Ω) by Cauchy-Schwarz, ε 0 k ag ( t u ) k L r 22+ ( 1 Ω) r +1 + r + r 1 ε 0 ( r 1+1) r 1 k t u k r L +22 ( 1 Ω r + r 1 ε ) 0 > 0 , (2.7)
DECAY OF THE WAVE EQUATION WITH NONLINEAR DAMPING 255 by Young’s inequality ( ab ε 0 a p + 1 q ε 1 0 p qp b q a, b, ε 0 > 0 , 1 p + q 1 = 1 , 1 < p, q < ). Choosing ε 0 > 0 small enough, we obtain the desired estimate from ( 2.6 ) thanks to ( 1.3 )-( 1.4 ). This concludes the proof of Corollary 1. The proof of Proposition 1 and Proposition 2 can easily be deduced from the four lemmas below. Let us introduce the following two sets. Ω 1 = { x Ω; | t u ( x, t ) | ≤ 1 } , Ω 2 = { x Ω; | t u ( x, t ) | > 1 } for a fixed t 0. We need to estimate the following four quantities. Z Ω 1 a ( x ) | t u ( x, t ) | 2 dx, Z Ω 2 a ( x ) | t u ( x, t ) | 2 dx, Z Ω 1 a ( x ) | g ( t u ( x, t )) | 2 dx, Z Ω 2 a ( x ) | t )) | 2 dx. g ( t u ( x, Lemma 2.1. There is C > 0 such that for a.e. t 0 , 2 Z Ω 1 dx C Z Ω a ( x, t )) t u ( x, t ) dx r +1 . (2.8) a ( x ) | t u ( x, t ) | 2 ( x ) g ( t u Proof. Indeed,using c 1 | s | r ≤ | g ( s ) | ≤ c 2 | s | 1 /r for | s | ≤ 1,we deduce that c 1 | s | r +1 | sg ( s ) | = sg ( s ) and 2 Z Ω 1 a | t u | 2 dx Z Ω 1 a 1 r 2+1 a r 2+1 c 1 1 g ( t u ) t u r +1 dx 2 c 1 1 r +1 k a k r L r + 11 (Ω) Z Ω ag ( t u ) t u ) r +21 dx (2.9) ( 2 C k a k L rr + 11 (Ω) Z Ω ag ( t u ) t udx r +1 because 0 < r +21 1andtheapplicationofH¨olderinequality, 2 2 Z Ω | f | r +21 dx ≤ | Ω | 1 r +1 Z Ω | f | dx r +1 f L 1 (Ω) .
Lemma 2.2. If ( n 2) (1 k ) 4 r , then there is C > 0 such that for a.e. t 0 , Z Ω 2 a ( x ) | t u ( x, t ) | 2 dx k Z |r t u ( x, t ) | 2 dx ( 2 rr +1+1 )(2.10) Ω 2 + Cε 1 Z a ( x ) g ( t u ( x, t )) t u ( x, t ) dx r +1 ε > 0 . Ω
256 KIM DANG PHUNG Proof. Indeed, using c 3 | s | k ≤ | g ( s ) | ≤ c 4 | s | p for | s | > 1, we deduce that c 3 | s | k +1 | sg ( s ) | = sg ( s ) and Z Ω 2 t u | 2 dx Z Ω 2 a | ∂ a | t u | 2 α | t u | α dx α 2 Z Ω 2 a 1 αk + α 1 a +1 | t u | α c 1 3 g ( t u ) t u k +1 dx (2.11) 1 α c 1 k + k a k 1 L k ( + Ω 1 ) Z Ω | t u | 2 α ( ag ( t u ) t u ) +1 dx 3 2 r k +1 C k a k L r r + 1 (Ω) Z | t u | ( r +1 )( ag ( t u ) t u ) r 1+1 dx Ω where α = kr ++11 (0 , k + 1), α < 2.Now,usingH¨olderinequality Z Ω | t u | ( 2 rr + k 1+1 )( ag ( t u ) t u ) r 1+1 dx 1 1 Z Ω | t u | ( 2 rr + k 1+1 )( δ δ 1 ) dx 1 δ Z Ω ( ag ( t u ) t u ) r +11 δ dx δ (2.12) r 1 Z Ω | t u | ( 2 r rk +1 ) dx r +1 Z Ω ag ( t u ) t udx r +1 , with δ = r + 1. Recall that by Sobolev’s imbedding theorem, 2 2 q qn< 2 n 2 ,, iiff nn > 22 k f k L q (Ω) C kr f k L 2 (Ω) f H 01 (Ω) . = Consequently, since 2 r rk +1 2, if ( n 2) (1 k ) 4 r (that is 2 r rk +1 n 2 n 2 when n > 2), we have by Sobolev’s imbedding theorem | 2 dx 12 ( 2 r rk +1 ).(2.13) Z | t u | ( 2 r rk +1 ) dx C Z Ω |r t u Ω Finally,combining ( 2.13 ) and ( 2.12 ) with ( 2.11 ), we conclude that if ( n 2) (1 k ) 4 r , Z Ω a | t u | 2 dx C k a k r L + r 1 (Ω) Z Ω |r t u | 2 dx 2 1 ( 2 rr + k 1+1 ) Z Ω ag ( t u ) t udx r +11 . 2 (2.14)
Lemma 2.3. There is C > 0 such that for a.e. t 0 , 2 Z Ω 1 a ( x ) | g ( t u ( x, t )) | 2 dx C Z Ω a ( x ) g ( t u ( x, t )) t u ( x, t ) dx r +1 . (2.15)