Global existence and decay estimates for quasilinear wave equations with nonuniform dissipative term
aa r X i v : . [ m a t h . A P ] N ov Global existence and decay estimates forquasilinear wave equations with nonuniformdissipative term
Tomonari Watanabe (Hiroshima University, Japan)
Abstract
We study global existence and decay estimates for quasilinear waveequations with dissipative terms in the Sobolev space H L × H L − ,where L ≥ [ d/
2] + 3. The linear dissipative terms depend on spacevariable coefficient, and these terms may vanish in some compact re-gion. For the proof of global existence, we need estimates of higherorder energies. To control derivatives of the dissipative coefficient, weintroduce an argument using the rescaling. Furthermore we get thedecay estimates with additional assumptions on the initial data. Toobtain the decay estimates, the rescaling argument is also needed.
Key Words and Phrases. dissipative quasilinear wave equation,space variable coefficient, time decay estimates2010
Mathematics Subject Classification Numbers.
In this paper, we consider the Cauchy problem for quasilinear wave equationswith nonuniform dissipative term in R d ( d ≥
1) :(DW) (cid:26) ( ∂ t − △ + B ( x ) ∂ t ) u ( t, x ) = N [ u, u ]( t, x ) , ( t, x ) ∈ [0 , ∞ ) × R d ,u (0 , x ) = u ( x ) , ∂ t u (0 , x ) = u ( x ) , x ∈ R d , where u = ( u , u , · · · , u d ) is a vector valued function, △ u ( t, x ) = ( △ u , △ u , · · · , △ u d ), ∂ t u = ( ∂ t u , ∂ t u , · · · , ∂ t u d ) and ∂ t u = ( ∂ t u , ∂ t u , · · · , ∂ t u d ),and the initial data ( u , u ) belongs to H L × H L − , where H L is the Sobolevspace in R d . 1n (DW) coefficient function B ( x ) is d × d symmetric matrix-valued func-tion and quasilinear term N [ u, v ]( t, x ) is defined by N [ u, v ] = ( N [ u, v ] i ) i =1 , , ··· ,d = d X j,k,l,m,n =1 N ijklmn ∂ l ( ∂ m u j ∂ n v k ) ! i =1 , , ··· ,d . Furthermore we make the following assumptions for the B and N : (B0) B ( x ) = ( B pq ( x )) p,q =1 , , ··· ,d is d × d symmetric matrix-valued functionwhose components belong to B ∞ , where B ∞ is the function space ofsmooth functions with bounded derivatives. (B1) B ( x ) is nonnegative, i.e. d X p,q =1 B pq ( x ) η p η q ≥ η, x ∈ R d ) . (B2) There exist b > R > d X p,q =1 B pq ( x ) η p η q ≥ b | η | ( | x | ≥ R, η ∈ R d ) . (N0) N ijklmn ∈ R ( i, j, k, l, m, n = 1 , , · · · , d ) . (N1) N ijklmn = N jikmln = N ikjlnm ( i, j, k, l, m, n = 1 , , · · · , d ).The main objective of this paper is to prove the global existence anddecay estimate to (DW). Throughout this paper, k · k p and k · k H l stand forthe usual L p ( R d )-norm and H l ( R d )-norm. Furthermore, we adopt h g, f i = d X i =1 Z R d f i ( x ) g i ( x ) dx as the usual L ( R d )-inner product.In the case where the coefficient function B vanishes, (DW) becomes thequasilinear wave equation. Then it is well known that no matter how smallthe initial data, there do not exist globally defined smooth solutions in general(e.g.[6]). Klainerman introduce the ”Null condition” for the nonlinear term.If the nonlinear term N has ”Null condition” then (DW) has a global smoothsolution for sufficiently smooth and small the initial data (e.g. [1], [8], [14]).2n the case where the coefficient function B ≡ Const >
0, there aremany results ([3], [7], [9], [10] etc.). When linear or semilinear version, it iswell known that the solution to (DW) has the decay estimates like L − L p estimates for heat equation (e.g. [7], [9] etc.). For general quasilinear versionincluding N , Racke [13] shows that there exists the unique global solutionand decay estimates when the initial data are sufficiently smooth and small.In the case where the coefficient function B ( x ) is nonuniform, there arealso many results too. Todorova and Yordanov consider like B ( x ) = 1 / (1 + | x | ) γ to linear version in [16]. When semilinear version like u | u | p − , we referto [17].Now we consider the nonuniform dissipative term satisfying (B1) and (B2) . In the linear case, Nakao [11] get the energy decay estimates like E ( u ( t )) = O ((1 + t ) − ), where E ( u ( t )) is the energy of u . FurthermoreIkehata [5] get the decay estimates as k u ( s ) k = O ((1 + t ) − ) and E ( u ( t )) = O ((1 + t ) − ) with additional condition for initial data. Those results extendto semilinear elastic wave version with the non-linear term N = | u | p − u in[2]. As we mentioned above, if B vanishes we need ”Null condition”. But wecan prove the global existence by assuming the dissipation effective near theinfinity, even if N has no ”null condition” and (DW) behave the quasilinearwave equations on the bounded domain. We prove the global existence asfollows: Theorem 1.1.
Let L ≥ L = [ d/
2] + 3 . Then there exists a small constant ˆ δ > such that if the initial data ( u , u ) ∈ H L × H L − satisfies k u k H L + k u k H L − ≤ ˆ δ, (1) then there exists a unique global solution to (DW) in ∩ L − L +1 j =0 C j ([0 , ∞ ); H L − j ) . In the proof to theorem 1.1, we use higher order energies (see e.g.[14])and the rescaling (see section 2). Note that if B = Const >
0, we can provetheorem 1.1 under the assumption k∇ u k H L − + k u k H L − ≤ ˆ δ instead of (1).Thus the smallness of k u k is needed to the case of nonuniform dissipativeterms B ( x ).We will prove the decay estimates with additional assumptions as follow: Theorem 1.2.
In addition to the assumptions in theorem 1.1, we assumethat one of the following (H1) - (H3) holds: (H1) d ≥ and there exists ≤ p ≤ dd +2 such that Bu + u ∈ L p , (H2) d ≥ and | · |{ Bu + u } ∈ L , H3) d = 1 or , | · |{ Bu + u } ∈ L and Z R d B ( x ) u ( x ) + u ( x ) dx = 0 .Then for any µ (0 ≤ µ ≤ L − L ) , there exists a constant E > depending on ( u , u ) such that the global solution u to (DW) satisfies following estimates: k ∂ µt u ( t ) k H L − µ + k ∂ µ +1 t u ( t ) k H L − µ − ≤ E (1 + t ) − µ − , (2) k∇ ∂ µt u ( t ) k + k ∂ µ +1 t u ( t ) k ≤ E (1 + t ) − µ − , (3) k ∂ µt u ( t ) k ∞ ≤ E (1 + t ) − µ − . (4) Furthermore if
L > L , it holds that k△ u ( t ) k ≤ E (1 + t ) − . (5)Nakao [11] obtained the decay estimates k u ( t ) k = O (1) and E ( u ( t )) = O ((1 + t ) − ) without (H1) - (H3) . Ikehata [5] obtained the decay estimates k u ( t ) k = ((1+ t ) − ) and E ( u ( t )) = O ((1+ t ) − ) with (H2) . In this paper, weassume one of the (H1) - (H3) and regularity of initial data, then we will getthe decay estimates including Ikehata [5] in quasilinear version. In addition,we can get the decay estimates correspond to the Nakao [11] when we putonly the assumption of theorem 1.1.The paper is organized as follows. In section 2 we prepare the notation,some known lemmas and the rescaling function. In section 3 we prove theglobal existence to (DW) (theorem 1.1). In section 4 we prove the decayestimate for solution to (DW) (theorem 1.2). In section 5 we prepare theenergy estimate that is used in section 3. We consider the rescaling to (DW). Let u be the solution to (DW). We define v ( t, x ) = 1 λ u ( λt, λx ) ( λ > v satisfies ∂ t v ( t, x ) − △ v ( t, x ) = λ (cid:8) ∂ t u ( λt, λx ) − △ u ( λt, λx ) (cid:9) = − λB ( λx ) ∂ t u ( λt, λx ) + λN [ u, u ] ( λt, λx )= − λB ( λx ) ∂ t v ( t, x ) + N [ v, v ]( t, x ) . So v is the solution to following the Cauchy problem (DW) λ :(DW) λ (cid:26) ( ∂ t − △ + B λ ( x ) ∂ t ) v ( t, x ) = N [ v, v ]( t, x ) , ( t, x ) ∈ [0 , ∞ ) × R d ,v (0 , x ) = v ( x ) , ∂ t v (0 , x ) = v ( x ) , x ∈ R d , where B λ ( x ) = λB ( λx ) , v ( x ) = u ( λx ) /λ, v ( x ) = u ( λx ). Now B λ satisfies4 B1 ) λ B λ ( x ) is nonnegative. (B2 ) λ There exist b > R > d X p,q =1 ( B λ ) pq ( x ) η p η q = d X p,q =1 λB pq ( λx ) η p η q ≥ λb | η | ( | x | ≥ Rλ , η ∈ R d )instead of (B1) and (B2) . Furthermore B λ satisfies (B3) λ k∇ b B λ k ∞ ≤ λ | b | +1 k∇ b B k ∞ .We consider (DW) λ for λ ≤ Lemma 2.1. (Sobolev’s lemma) There exists a constant
C > such that k f k ∞ ≤ C k f k H [ d ] +1 ( f ∈ H [ d ] +1 ) . Lemma 2.2. (Gagliardo-Nirenberg-Moser type estimate (For proof, see e.g.p.11 of [15].)) Let k ∈ Z and b, c ∈ Z d + satisfy | b | + | c | = k . There exists C > such that k∇ b f ∇ c g k ≤ C k f k ∞ k∇ b + c g k + C k∇ b + c f k k g k ∞ ≤ C k f k ∞ k g k H k + C k f k H k k g k ∞ ( f, g ∈ C ∞ ) . Next we prepare the Poincare type inequality of B λ for the proof of globalexistence. Lemma 2.3. (Poincare type inequality) There exists a constant C ≥ / such that k f k ≤ C λ h f, B λ f i + C λ k∇ f k ( f ∈ H , λ >
0) (6)
Proof.
We define U r = { x ∈ R d || x | ≤ r } . Using Poincare inequality [4], weobtain the following estimate: Z U r | f ( x ) | dx ≤ r Z U r |∇ f ( x ) | dx ( f ∈ H ( U r )) . ρ ∈ C ∞ ( R d ) be a function satisfying 0 ≤ ρ ≤ , ρ ( x ) = 1( | x | ≤ , ρ ( x ) = 0( | x | ≥ ) . For any f ∈ H ( R d ) and λ > ρ λ ( x ) = ρ ( λxR ), then because of ρ λ f ∈ H ( U Rλ ) we have k f k = Z R d | ρ λ ( x ) f ( x ) | dx + Z R d (1 − | ρ λ ( x ) | ) f ( x ) | dx = Z | x |≤ Rλ | ρ λ ( x ) f ( x ) | dx + Z | x |≥ Rλ (1 − | ρ λ ( x ) | ) f ( x ) | dx ≤ R λ Z | x |≤ Rλ |∇ ( ρ λ ( x ) f ( x )) | dx + Z | x |≥ Rλ | f ( x ) | dx ≤ R λ Z Rλ ≤| x |≤ Rλ |∇ ρ λ ( x ) | | f ( x ) | dx + 4 R λ Z | x |≤ Rλ | ρ λ ( x ) | |∇ f ( x ) | dx + Z | x |≥ Rλ | f ( x ) | dx ≤ (cid:0) R k∇ ρ k ∞ + 1 (cid:1) Z | x |≥ Rλ | f ( x ) | dx + 4 R λ Z R d |∇ f ( x ) | dx ≤ R k∇ ρ k ∞ + 1 b λ h f, B λ f i + 4 R λ k∇ f k . Hence we get (6).Finally, we introduce Hardy inequality and Gagliardo-Nirenberg inequal-ity. We need them in the proof of theorem 1.2 in section 4.
Lemma 2.4. (Hardy inequality) Let d ≥ . There exists a constant C > such that any f ∈ H satisfies (cid:13)(cid:13)(cid:13)(cid:13) f | · | (cid:13)(cid:13)(cid:13)(cid:13) ≤ C k∇ f k . (7) Lemma 2.5. (Gagliardo-Nirenberg inequality) Assume ≤ q < d and p = q − d . Then there exists a constant C > depend on p, q, d such that k g k p ≤ C k∇ g k q ( ∀ g ∈ C ∞ ( R d )) . (8) In this section we prove theorem 1.1. First we define some notations. Forany h, g : R d → R d , we define [ h ; ∇ g ] : R d → R d as follows:([ h ; ∇ g ]) i ( x ) = h ( x ) · ∇ g i ( x ) ( i = 1 , , · · · , d ) . E ( u ( t )) and higher order energies E ¯ L ( u ( t )) of u are defined by E ( u ( t )) = 12 {k ∂ t u ( t ) k + k∇ u ( t ) k } (9)and E ¯ L ( u ( t )) = X | a |≤ ¯ L − E ( ∇ a u ( t )) . (10)Moreover we define˜ N [ u, v, w ]( t ) = d X i,j,k,l,m,n =1 N ijklmn Z R d ∂ l u i ( t, x ) ∂ m v j ( t, x ) ∂ n w k ( t, x ) dx (11)and ˜ E ¯ L,µ ( u ( t )) = E ¯ L − µ ( ∂ µt u ( t )) + X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a u, ∂ µt ∇ a u, u ]( t ) . (12)The function spaces X δ,T , X δ are defined by X δ,T = (cid:8) u ∈ ∩ L − L +1 j =0 C j ([0 , T ]; H L − j ) | E L ( u ( t )) ≤ δ (0 ≤ t ≤ T ) (cid:9) (13)and X δ = (cid:8) u ∈ ∩ L − L +1 j =0 C j ([0 , ∞ ); H L − j ) | E L ( u ( t )) ≤ δ (0 ≤ t < ∞ ) (cid:9) . (14)Let L ≤ ¯ L ≤ L , µ ≤ L − L and λ >
0, we define G ¯ L,µ ( v ( t )) below. G ¯ L,µ ( v ( t )) (15)= C λ ˜ E ¯ L,µ ( v ( t )) + b (2 d − X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , ∂ µ +1 t ∇ a v ( t ) i + b (2 d − X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i + X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ]( t ) i , where C = max (cid:26)(cid:18) b Rd + C b (2 d − (cid:19) × , d, k B k ∞ b R × b (cid:27) , (16)and φ ( r ) = (cid:26) b , ( r ≤ Rλ ) b Rλr , ( r ≥ Rλ ) , h ( x ) = xφ ( | x | ) . (17)We need the energy estimate to prove the global existence.7 emma 3.1. Let L ≤ ¯ L ≤ L and µ ≤ ¯ L − L . There exists a constant C > such that for any δ, λ, T > and a local solution v ∈ X δ,T to (DW ) λ satisfy ddt G ¯ L,µ ( v ( t )) + b E ¯ L − µ ( ∂ µt v ( t )) (18) ≤ λCE ¯ L − µ ( ∂ µt v ( t )) + Cλ D ¯ L,µ ( v ( t )) + 2 k B k ∞ b R C E ¯ L − µ ( ∂ µt v ( t )) , where C > is the constant give in (16) , and D ¯ L,µ ( v ( t )) = E ¯ L − µ ( ∂ µt v ( t )) · µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) ! . (19)The proof is given in Section 5. In what follows, assuming lemma 3.1 wederive energy estimates for (DW) λ . We prove the energy estimate of (DW) λ for the global existence. Lemma 3.2.
Let L ≤ ¯ L ≤ L and µ ≤ ¯ L − L . There exists C > suchthat if δ > and λ are sufficiently small then a local solution v ∈ X δ,T to(DW ) λ satisfies C { λ k ∂ µt v ( t ) k + 1 λ E ¯ L − µ ( ∂ µt v ( t )) } ≤ G ¯ L,µ ( ∂ µt v ( t )) (20) ≤ C { λ k ∂ µt v ( t ) k + 1 λ E ¯ L − µ ( ∂ µt v ( t )) } . Proof.
Let v ∈ X δ,T . First it holds that12 E ¯ L − µ ( ∂ µt v ( t )) ≤ ˜ E ¯ L,µ ( v ( t )) ≤ E ¯ L − µ ( ∂ µt v ( t )) . (21)Because (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, v ]( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X | a |≤ ¯ L − µ − k∇ v ( t ) k ∞ k∇ ∂ µt ∇ a v ( t ) k ≤ C X | a |≤ ¯ L − µ − k∇ v ( t ) k H [ d ] +1 k∇ ∂ µt ∇ a v ( t ) k δCE ¯ L − µ ( ∂ µt v ( t )) , we choose δ sufficiently small depend on d, N then we get (21). It followsfrom lemma 2.3 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , ∂ µ +1 t ∇ a v ( t ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (22) ≤ X | a |≤ ¯ L − µ − (cid:26) λ C k ∂ µt ∇ a v ( t ) k + C λ k ∂ µ +1 t ∇ a v ( t ) k (cid:27) ≤ X | a |≤ ¯ L − µ − (cid:26) h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i + 14 λ k∇ ∂ µt ∇ a v ( t ) k + C λ k ∂ µ +1 t ∇ a v ( t ) k (cid:27) ≤ X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i + 2 C λ E ¯ L − µ ( ∂ µt v ( t )) . Using k h k ∞ ≤ b Rλ and k∇ h k ∞ ≤ b , (23)we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ]( t ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (24) ≤ X | a |≤ ¯ L − µ − d X k,j =1 (cid:12)(cid:12)(cid:12)(cid:12)Z R d ∂ µ +1 t ∇ a v k ( t ) ∂ j ∂ µt ∇ a v k ( t ) h j dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ X | a |≤ ¯ L − µ − d X k,j =1 k ∂ µ +1 t ∇ a v k ( t ) k k ∂ j ∂ µt ∇ a v k ( t ) k k h j k ∞ ≤ d X | a |≤ ¯ L − µ − (cid:26) k ∂ µ +1 t ∇ a v ( t ) k + 12 k∇ ∂ µt ∇ a v ( t ) k (cid:27) k h k ∞ ≤ b Rd λ E ¯ L − µ ( ∂ µt v ( t )) . Using (21),(22),(24), and (16) we have G ¯ L,µ ( v ( t )) ≥ λ (cid:18) C − b Rd − C b (2 d − (cid:19) E ¯ L − µ ( ∂ µt v ( t ))9 b (2 d − X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i≥ λ (cid:18) b Rd + C b (2 d − (cid:19) E ¯ L − µ ( ∂ µt v ( t ))+ b (2 d − X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i . So there exists a constant C such that v satisfies1 λ E ¯ L − µ ( ∂ µt v ( t )) + X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i ≤ CG ¯ L,µ ( v ( t )) . (25)Furthermore using (25) and lemma 2.3, we have k ∂ µt v ( t ) k ≤ C λ h B λ ∂ µt v ( t ) , ∂ µt v ( t ) i + C λ k∇ ∂ µt v k (26) ≤ C λ X | a |≤ ¯ L − µ − h B λ ∂ µt ∇ a v ( t ) , ∂ µt ∇ a v ( t ) i + 2 C λ E ¯ L − µ ( ∂ µt v ( t )) ≤ C λ X | a |≤ ¯ L − µ − h B λ ∂ µt ∇ a v ( t ) , ∂ µt ∇ a v ( t ) i + 1 λ E ¯ L − µ ( ∂ µt v ( t )) ≤ C Cλ G ¯ L,µ ( v ( t )) . From (25) and (26), it follows that there exists a constant
C > v satisfies 1 C (cid:26) λ k ∂ µt v ( t ) k + 1 λ E ¯ L − µ ( ∂ µt v ( t )) (cid:27) ≤ G ¯ L,µ ( v ( t )) . (27)On the other hand using (21), (22) and (24) , we get G ¯ L,µ ( v ( t )) (28) ≤ λ (cid:18) C λ + C b (2 d − b Rd (cid:19) E ¯ L − µ ( ∂ µt v ( t ))+ 3 b (2 d − X | a |≤ ¯ L − µ − h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i≤ Cλ E ¯ L − µ ( ∂ µt v ( t )) + C k B λ k ∞ X | a |≤ ¯ L − µ − k ∂ µt ∇ a v ( t ) k ≤ Cλ E ¯ L − µ ( ∂ µt v ( t )) + λC k ∂ µt v ( t ) k .
10o combining (27) and (28), we get (20). This completes the proof of lemma3.2.
Lemma 3.3.
Let L ≤ ¯ L ≤ L . There exist sufficiently small constants λ and δ such that if v ∈ X δ,T is a local solution to (DW ) λ then v satisfies ddt G ¯ L, ( v ( t )) + b E ¯ L ( v ( t )) ≤ . (29) Proof.
Let v ∈ X δ,T be a local solution to (DW) λ . From (16) C satisfies C ≥ d , so we can use lemma 3.1. Using lemma 3.1 for µ = 0 and D ¯ L, ( v ( t )) = E L ( ∂ µt v ( t )) E ¯ L ( ∂ µt v ( t )) ≤ δE ¯ L ( ∂ µt v ( t )) , we have ddt G ¯ L, ( v ( t )) + b E ¯ L ( v ( t )) ≤ C ( λ + δλ ) E ¯ L ( v ( t )) + 2 k B k ∞ b R C E ¯ L ( v ( t )) . From (16) we obtain 2 k B k ∞ b R C ≤ b . On the other hand we can choose a sufficiently small constants λ and δ suchthat C ( λ + δλ ) ≤ b . So we can choose sufficiently small constants λ, δ such that ddt G ¯ L, ( v ( t )) + b E ¯ L ( v ( t )) ≤ , which completes the proof of lemma 3.3. Corollary 3.4.
Let L ≤ ¯ L ≤ L , and λ and δ be sufficiently small constantsin lemma 3.3. Then there exists a constant C ∗ depending on λ such that forany T > and a local solution v ∈ X δ,T to (DW ) λ satisfies k v ( t ) k + E ¯ L ( v ( t )) + Z t E ¯ L ( v ( s )) ds ≤ C ∗ {k v (0) k + E ¯ L ( v (0)) } (30) and k v ( t ) k H L + k ∂ t v ( t ) k H L − ≤ C ∗ {k v (0) k H L + k ∂ t v (0) k H L − } , ( t ∈ [0 , T ]) . (31)11 roof. Integrating (29) over [0 , t ] we, get G ¯ L, ( v ( t )) + b Z t E ¯ L ( v ( s )) ds ≤ G ¯ L,µ ( v (0)) . (32)Then using lemma 3.2, there exists a constant C > C { λ k v ( t ) k + 1 λ E l ( v ( t )) } + b Z t E l ( v ( s )) ds ≤ C { λ k v (0) k + 1 λ E ¯ L ( v (0)) } . We rearrange coefficient and define C ∗ depend on λ , it holds that (30).(31) is clear because of (30). We can prove the local existence theorem to (DW) λ as the argument similarto [10] and [15]. Lemma 3.5. (Local existence theorem) Let L ≥ L = [ d/
2] + 3 , λ > and ( v , v ) ∈ H L × H L − . For any sufficiently small constant ε > there existconstants < t and < η ≤ such that if k∇ v k H L − + k v k H L − ≤ ηε (33) then (DW ) λ has the unique local solution v ∈ ∩ L − L +1 j =0 C j ([0 , t ]; H L − j ) andthe v satisfies E L ( v ( t )) ≤ ε (0 ≤ t ≤ t ) . (34)Using lemma 3.5 and corollary 3.4, we can prove the global existencetheorem. Theorem 3.6. (Global existence theorem to (DW ) λ jLet L ≥ L = [ d/
2] + 3 , λ and δ are sufficiently small constants. There exists a small constant δ ∗ > such that if the initial data ( v , v ) ∈ H L × H L − satisfies k v k H L + k v k H L − ≤ δ ∗ (35) then (DW ) λ has the unique global solution v ∈ X δ .Proof. Let λ and δ > < ε ≤ δ , 0 < t and 0 < η ≤ δ ∗ = min (cid:26) ηε , ηε C ∗ (cid:27) , C ∗ is given in corollary 3.4. We assume that ( v , v ) ∈ H L × H L − satisfies k v k H L + k v k H L − ≤ δ ∗ . Because of k∇ v k H L − + k v k H L − ≤ δ ∗ ≤ ηε , lemma 3.5 yields that there exists v ∈ ∩ L − L +1 j =0 C j ([0 , t ]; H L − j ) such that v is a unique local solution to (DW) λ and satisfies E L ( v ( t )) ≤ ε ≤ δ (0 ≤ t ≤ t ) . Because of v ∈ X δ,t , we can use corollary 3.4. So it holds that k v ( t ) k H L + k ∂ t v ( t ) k H L − ≤ C ∗ {k v k H L + k v k H L − } ≤ ηε ( t ∈ [0 , t ]) . Thus we can use lemma 3.5 in t = t . The solution v is uniquely extendedto ∩ L − L +1 j =0 C j ([0 , t ]; H L − j ) and satisfies E L ( v ( t )) ≤ ε ≤ δ (0 ≤ t ≤ t ) . Because of v ∈ X δ, t we can use corollary 3.4 again. So v satisfies k v ( t ) k H L + k ∂ t v ( t ) k H L − ≤ C ∗ {k v k H L + k v k H L − } ≤ ηε ( t ∈ [0 , t ]) . Thus we can use lemma 3.5 in t = 2 t .Repeating this argument, we can uniquely extend v to a global solutionto (DW) λ , furthermore because of corollary 3.4 it holds that E L ( v ( t )) ≤ δ ( t ∈ [0 , ∞ )) . This completes the proof of theorem 3.6.
Proof of theorem1.1
Let λ and δ ∗ are the small constants in theorem 3.6. We define ˆ δ = λ d +1 δ ∗ and assume the initial data ( u , u ) ∈ H L × H L − satisfies k u k H L + k u k H L − ≤ ˆ δ. Now we define v ( x ) = 1 λ u ( λx ) , v ( x ) = u ( λx ) , v , v ) satisfy k v k H L + k v k H L − = X | a |≤ L λ | a |− Z R d |∇ a u ( λx ) | dx + X | a |≤ L − λ | a | Z R d |∇ a u ( λx ) | dx = X | a |≤ L λ | a |− − d Z R d |∇ a u ( x ) | dx + X | a |≤ L − λ | a |− d Z R d |∇ a u ( x ) | dx ≤ λ − d − { X | a |≤ L Z R d |∇ a u ( x ) | dx + X | a |≤ L − Z R d |∇ a u ( x ) | dx } = λ − d − {k u k H L + k u k H L − } ≤ δ ∗ . From theorem 3.6, there exists a unique global solution v to (DW) λ in ∩ L − L +1 j =0 C j ([0 , ∞ ); H L − j ). We define u ( t, x ) = λv ( tλ , xλ ) , then u ∈ ∩ L − L +1 j =0 C j ([0 , ∞ ); H L − j ) and the u satisfies (DW).As regard to uniqueness, if u and u ′ are solutions to (DW) then rescalingfunctions u λ and u ′ λ are solutions to (DW) λ . From theorem 3.6 we got theuniqueness of (DW) λ , so we obtain u λ = u ′ λ , thus u = u ′ . The goal of this section is to show theorem 4.1. We say that f satisfies theproperty (H1) ′ , (H2) ′ or (H3) ′ if and only if (H1) ′ d ≥ ≤ p ≤ dd +2 such that f ∈ L p , (H2) ′ d ≥ | · | f ∈ L , (H3) ′ d = 1 or 2 , | · | f ∈ L and Z R d f ( x ) dx = 0.We prove the decay estimates for (DW) λ as follow: Theorem 4.1.
In addition to the assumptions in theorem 3.6, we assumethat B λ v + v satisfies one of the (H1) ′ - (H3) ′ . Then for any i (0 ≤ i ≤ − L ) , there exists a constant E depending on λ, v and v such that theglobal solution v ∈ X δ to (DW ) λ satisfies (1+ t ) i +1 {k ∂ it v ( t ) k + E L − i ( ∂ it v ( t )) } + Z t (1+ s ) i +1 E L − i ( ∂ it v ( s )) ds ≤ E (36) and (1 + t ) i +2 E ( ∂ it v ( t )) + Z t (1 + s ) i +2 h ∂ i +1 t v ( s ) , B λ ∂ i +1 t v ( s ) i ≤ E . (37)Using theorem 4.1, we can prove theorem 1.2. Proof of theorem 1.2
We assume that theorem 4.1 is true. From theorem 1.1 there exists a constantˆ δ such that if the initial data ( u , u ) satisfies k u k H L + k u k H L − ≤ ˆ δ, then (DW) has a unique global solution u ∈ ∩ L − L +1 j =0 C j ([0 , ∞ ); H L − j ). Nowwe define v = 1 λ u ( λt, λx ) then v satisfies(DW) λ (cid:26) ( ∂ t − △ + B λ ( x ) ∂ t ) v ( t, x ) = N [ v, v ]( t, x ) , ( t, x ) ∈ [0 , ∞ ) × R d ,v ( , x ) = v ( x ) , ∂ t v λ (0 , x ) = v , x ∈ R d , where v ( x ) = λ u ( λx ) and v ( x ) = u ( λx ). If Bu + u satisfies (H1) , (H2) or (H3) , then B λ v + v satisfies (H1) ′ , (H2) ′ or (H3) ′ . Thus we can usetheorem 4.1. For any µ (0 ≤ µ ≤ L − L ) there exists a constant E dependingon λ, v and v such that v satisfies k ∂ µt v ( τ ) k + E L − µ ( ∂ µt v ( τ )) ≤ E (1 + τ ) − µ − and E ( ∂ µt v ( τ )) ≤ E (1 + τ ) − µ − . Replacing v ( t, x ) = 1 λ u ( λt, λx ) and t = λτ , we get (2) and (3).Next using (2) and lemma 2.1, estimate (4) is clear.Finally we prove (5). Let L < L . The global solution u to (DW) satisfies k△ u ( t ) k = k ∂ t u ( t ) + ∂ t u ( t ) − N [ u, u ]( t ) k (38) ≤ k ∂ t u ( t ) k + k ∂ t u ( t ) k + k N [ u, u ]( t ) k . L < L , we can use (2) and (3) to µ = 1. So we obtain k ∂ t u k ≤ E (1 + t ) − , k ∂ t u k ≤ E (1 + t ) − . Furthermore using (2) and (3), we obtain k N [ u, u ]( t ) k ≤ d X i,j,k,l,m,n =1 N ijklmn k ∂ l ( ∂ m u j ∂ n u k ) k ≤ C k∇ u ( t ) k ∞ k∇ u ( t ) k ≤ CE L ( u ( t )) E ( u ( t )) ≤ CE (1 + t ) − . So (5) holds from (38).
Let v be the global solution to (DW) λ and define w ( t, x ) = Z t v ( s, x ) ds. (39)Then w satisfies ∂ t w ( t, x ) − △ w ( t, x ) + B λ ( x ) ∂ t w ( t, x )= Z t N [ v, v ]( τ, x ) dτ + B λ ( x ) v ( x ) + v ( x ) , ( t, x ) ∈ [0 , ∞ ) × R d ,w (0 , x ) = 0 , ∂ t w (0 , x ) = v ( x ) , x ∈ R d . (40)We remark ∂ t w = v and E L − ( w ( t )) are well-defined in [0 , ∞ ) because ofcorollary 3.4.We will prove the energy estimate of w under the assumption which B λ v + v satisfies one of the (H1) ′ - (H3) ′ . So we prepare the next lemma. Lemma 4.2.
Let f ∈ H L − satisfies one of the (H1) ′ - (H3) ′ . Then for any g ∈ H L − , there exists a constant E depending on f such that h g, f i ≤ E k∇ g k H L − . (41)16 roof. First we assume that f satisfies (H1) ′ . Then f ∈ L dd +2 because f ∈ H L − ⊂ L ∞ and f ∈ L p (1 ≤ p ≤ dd +2 ). From lemma 2.5,we obtain g ∈ L dd − and there exists a constant C > g satisfies k g k L dd − ≤ C k∇ g k . So using H¨older inequality, we get h g, f i ≤ k f k L dd +2 k g k L dd − ≤ C k f k L dd +2 k∇ g k ≤ C k f k L dd +2 k∇ g k H L − . Thus we get (41).Next we assume that f satisfies (H2) ′ . Then using lemma 2.4, we get h g, f i ≤ k| · | f k (cid:13)(cid:13)(cid:13)(cid:13) g | · | (cid:13)(cid:13)(cid:13)(cid:13) ≤ C k| · | f k k∇ g k ≤ C k| · | f k k∇ g k H L − . Thus we get (41).Finally we assume that f satisfies (H3) ′ . Because of g ∈ H L − ⊂ C ( R d ), we have | g ( x ) − g (0) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ddθ g ( θx ) dθ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z x · ∇ g ( θx ) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x |k∇ g k ∞ a.e.x. So using Z R d f ( x ) dx = 0, we obtain |h g, f i| = (cid:12)(cid:12)(cid:12)(cid:12)Z R d ( g ( x ) − g (0)) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d | ( g ( x ) − g (0) || f ( x ) | dx ≤ k∇ g k ∞ Z R d | x || f ( x ) | dx ≤ C k∇ g k H L − k| · | f k . Thus we get (41).We need the estimate R ∞ k v ( t ) k dt < ∞ for proof of prop 4.3. In orderto prove this property, we use the idea in Ikehata [5]. Proposition 4.3.
In addition to the assumptions theorem 3.6, we assume B λ v + v satisfies one of the (H1) ′ - (H3) ′ . Then there exists a constant E depending on λ, v and v such that the global solution v ∈ X δ to (DW ) λ satisfies Z t k v ( s ) k ds ≤ E , ( t ∈ [0 , ∞ )) . (42)17 roof. Let v be the global solution to (DW) λ and we define w by (39). Using(40), we have ddt E L − ( w ( t ))= X | a |≤ L − h ∂ t ∇ a w ( t ) , ∂ t ∇ a w ( t ) i − X | a |≤ L − h ∂ t ∇ a w ( t ) , △∇ a w ( t ) i = − X | a |≤ L − h ∂ t ∇ a w ( t ) , ∇ a ( B λ ∂ t w ( t )) i + X | a |≤ L − h ∂ t ∇ a w ( t ) , Z t ∇ a N [ v, v ] dτ i + X | a |≤ L − h ∂ t ∇ a w ( t ) , ∇ a ( B λ v + v ) i = − X | a |≤ L − h ∂ t ∇ a w ( t ) , B λ ∂ t ∇ a w ( t )) i− X ≤| a |≤ L − X b ≤ ab =0 (cid:18) ab (cid:19) h ∂ t ∇ a w ( t ) , ∇ b B λ ∂ t ∇ a − b w ( t )) i + X | a |≤ L − ddt h∇ a w ( t ) , Z t ∇ a N [ v, v ] dτ i − X | a |≤ L − h∇ a w ( t ) , ∇ a N [ v, v ]( t ) i + X | a |≤ L − ddt h∇ a w ( t ) , ∇ a ( B λ v + v ) i . So we integrating it over [0 , t ], we get E L − ( w ( t )) + X | a |≤ L − Z t h∇ a v ( s ) , B λ ∇ a v ( s ) i ds (43) ≤ k v k H L − − X ≤| a |≤ L − X b ≤ ab =0 (cid:18) ab (cid:19) Z t h∇ a v ( s ) , ∇ b B λ ∇ a − b v ( s )) i ds + X | a |≤ L − h∇ a w ( t ) , Z t ∇ a N [ v, v ]( s ) ds i− X | a |≤ L − Z t h∇ a w ( s ) , ∇ a N [ v, v ]( s ) i ds + X | a |≤ L − h∇ a w ( t ) , ∇ a ( B λ v + v ) i = 12 k v k H L − + A + A + A + A . We estimate from A to A . Using lemma 2.3 , (B3 ) λ and smallness of λ ,18e get | A | ≤ X ≤| a |≤ L − X b ≤ ab =0 (cid:18) ab (cid:19) Z t k∇ a v ( s ) k k∇ b B λ k ∞ k∇ a − b v ( s ) k ds ≤ λ C X ≤| a |≤ L − X b ≤ ab =0 Z t k∇ a v ( s ) k k∇ a − b v ( s ) k ds ≤ C X ≤| a |≤ L − Z t k∇ a v ( s ) k ds + λ C X ≤| a |≤ L − X b ≤ ab =0 Z t k∇ a − b v ( s ) k ds ≤ C Z t E L − ( v ( s )) ds + λ C X ≤| a |≤ L − X b ≤ ab =0 (cid:26)Z t h∇ a − b v ( s ) , B λ ∇ a − b v ( s ) i ds + Z t k∇∇ a − b v ( s ) k ds (cid:27) ≤ C Z t E L ( v ( s )) ds + 14 X | a |≤ L − Z t h∇ a v ( s ) , B λ ∇ a v ( s ) i ds. Next we define M ( t ) = sup ≤ s ≤ t E L − ( w ( s )) . (44)Using M ( t ) and lemma 2.2, we have | A | = X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12) h∇ a w ( t ) , Z t ∇ a N [ v, v ]( s ) ds i (cid:12)(cid:12)(cid:12)(cid:12) = X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 N ijklmn Z t Z R d ∇ a w i ( t, x ) ∂ l ∇ a ( ∂ m v j ( s, x ) ∂ n v k ( s, x )) dxds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 N ijklmn Z t Z R d ∂ l ∇ a w i ( t, x ) ∇ a ( ∂ m v j ( s, x ) ∂ n v k ( s, x )) dxds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X | a |≤ L − Z t k∇∇ a w ( t ) k d X j,k,m,n =1 k∇ a ( ∂ m v j ( s ) ∂ n v k ( s )) k ds ≤ C ( M ( t )) Z t E L ( v ( s )) ds | A | = X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12)Z t h∇ a w ( s ) , ∇ a N [ v, v ]( s ) ds i (cid:12)(cid:12)(cid:12)(cid:12) ≤ X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 N ijklmn Z t Z R d ∇ a w i ( s, x ) ∂ l ∇ a ( ∂ m v j ( s, x ) ∂ n v k ( s, x )) dxds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X | a |≤ L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 N ijklmn Z t Z R d ∂ l ∇ a w i ( s, x ) ∇ a ( ∂ m v j ( s, x ) ∂ n v k ( s, x )) dxds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X | a |≤ L − Z t k∇∇ a w ( s ) k d X j,k,m,n =1 k∇ a ( ∂ m v j ( s ) ∂ n v k ( s )) k ds ≤ C ( M ( t )) Z t E L ( v ( s )) ds. Using lemma 4.2, we get A = h w ( t ) , B λ v + v i + X ≤| a |≤ L − h∇ a w ( t ) , ∇ a ( B λ v + v ) i≤ E k∇ w ( t ) k H L − + X ≤| a |≤ L − k∇ a w ( t ) k k∇ a ( B λ v + v ) k ≤ E E L − ( w ( t )) ≤ E + 14 E L − ( w ( t )) , where E depend on λ, v and v . Corollary 3.4 implies that there exists aconstant C ∗ > Z t E L ( v ( s )) ds ≤ C ∗ {k v (0) k + E L ( v (0)) } . From (43) and estimates of the terms A − A , we get E L − ( w ( t )) + X | a |≤ L − Z t h∇ a v ( s ) , B λ ∇ a v ( s ) i ds ≤ E + E ( M ( t )) . (45)From (45) we have M ( t ) ≤ E + E ( M ( t )) ( t ∈ [0 , ∞ )) .
20t means that M ( t ) is bounded in [0 , ∞ ). So we get the following estimatefrom (45). Z t h v ( s ) , B λ v ( s ) i ds ≤ E . (46)Finally using (46), lemma 2.3 and Corollary 3.4, we have Z t k v ( t ) k ds ≤ Cλ Z t h v ( s ) , B λ v ( s ) i ds + Cλ Z t k∇ v ( s ) k ds ≤ E λ + E λ . Thus we get (42).We prove theorem 4.1 by induction. First we prove theorem 4.1 for i = 0. Theorem 4.4.
In addition to the assumption theorem 3.6, we assume that B λ v + v satisfies one of the (H1) ′ - (H3) ′ . Then there exists constant E depending on λ, v and v such that the global solution v ∈ X δ to (DW ) λ satisfies (1 + t ) (cid:8) k v ( t ) k + E L ( v ( t )) (cid:9) + Z t (1 + s ) E L ( v ( s )) ds ≤ E (47) and (1 + t ) E ( v ( t )) + Z t (1 + s ) h ∂ t v ( s ) , B λ ∂ t v ( s ) i ≤ E . (48) Proof.
First we prove (47). Using lemma 3.3 for ¯ L = L , we obtain ddt { (1 + t ) G L, ( v ( t )) } = G L, ( v ( t )) + (1 + t ) ddt G L, ( v ( t )) ≤ G L, ( v ( t )) − b (1 + t )8 E L ( v ( t )) . Integrating it over [0 , t ] and using lemma 3.2, we get(1 + t ) C (cid:26) λ k v ( t ) k + 1 λ E L ( v ( t )) (cid:27) + b Z t (1 + s ) E L ( v ( s )) ds ≤ C (cid:26) λ k v (0) k + 1 λ E L ( v (0)) + λ Z t k v ( s ) k ds + 1 λ Z t E L ( v ( s )) ds (cid:27) . So there exists a constant E which depend on λ, v and v such that(1 + t ) (cid:8) k v ( t ) k + E L ( v ( t )) (cid:9) + Z t (1 + s ) E L ( v ( s )) ds (49)21 E + E Z t k v ( s ) k ds + E Z t E L ( v ( s )) ds. Using proposition 4.3 and corollary 3.4, we get the following estimate from(49): (1 + t ) (cid:8) λ k v ( t ) k + E L ( v ( t )) (cid:9) + Z t (1 + s ) E L ( v ( s )) ds ≤ E + E + E C ∗ {k v (0) k + E L ( v (0)) } . So rearranging E if we need, we get (47).Next we prove (48). For the global solution v to (DW) λ holds that ddt (cid:8) (1 + t ) E ( v ( t )) (cid:9) (50)= 2(1 + t ) E ( v ( t )) + (1 + t ) ddt E ( v ( t ))= 2(1 + t ) E ( v ( t )) + (1 + t ) {h ∂ t v ( t ) , ∂ t v ( t ) i − h ∂ t v ( t ) , △ v ( t ) i} = 2(1 + t ) E ( v ( t )) − (1 + t ) h ∂ t v ( t ) , B λ ∂ t v ( t ) i + (1 + t ) h ∂ t v ( t ) , N [ v, v ]( t ) i . Using (50) and h ∂ t v ( t ) , N [ v, v ]( t ) i ≤ CE ( v ( t )) E L ( v ( t )) , we get ddt (cid:8) (1 + t ) E ( v ( t )) (cid:9) + (1 + t ) h ∂ t v ( t ) , B λ ∂ t v ( t ) i (51) ≤ t ) E ( v ( t )) + C (1 + t ) E ( v ( t )) E L ( v ( t )) . Now we define M ( t ) = sup ≤ s ≤ t (1 + s ) E ( v ( s )) . (52)Integrating (51) over [0 , ∞ ) and using (47), we obtain(1 + t ) E ( v ( t )) + Z t (1 + s ) h ∂ t v ( s ) , B λ ∂ t v ( s ) i ds (53) ≤ E ( v (0)) + 2 Z t (1 + s ) E ( v ( s )) ds + C Z t (1 + s ) E ( v ( s )) E L ( v ( s )) ds ≤ E ( v (0)) + 2 Z t (1 + s ) E ( v ( s )) ds + C ( M ( t )) Z t (1 + s ) E L ( v ( s )) ds ≤ E + E ( M ( t )) . M ( t ) ≤ E + E ( M ( t )) , which means that M ( t ) is bounded in [0 , ∞ ). So it holds from (53) that(1 + t ) E ( v ( t )) + Z t (1 + s ) h ∂ t v ( s ) , B λ ∂ t v ( s ) i ≤ E . Thus we get (48).Next assuming the decay estimate of ∂ it v for 0 ≤ i ≤ µ −
1, we show thedecay estimate of ∂ µt v . For the purpose, we need the following lemma: Lemma 4.5.
In addition to the assumption theorem 3.6, we assume that B λ v + v satisfies one of the (H1) ′ - (H3) ′ . Let ≤ µ ≤ L − L and assumethat for any ≤ i ≤ µ − estimates (36) and (37) in theorem 4.1 hold.Then there exists a constant E depending on λ, v and v such that theglobal solution v ∈ X δ to (DW ) λ satisfies ddt G L,µ ( v ( t )) + b E L − µ ( ∂ µt v ( t )) ≤ E µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) − µ − ν ) − , (54) if λ and δ in theorem 3.6 are chosen small enough.Proof. We use (18) in lemma 3.1. From the assumption of induction, (36)for i with 0 ≤ i ≤ µ − D L,µ ( v ( t )) = E L − µ ( ∂ µt v ( t )) · µ X ν =0 E L ( ∂ µ − νt v ( t )) E L − ν ( ∂ νt v ( t )) ! ≤ δE L − µ ( ∂ µt v ( t )) + E L − µ ( ∂ µt v ( t )) · µ − X ν =0 E L ( ∂ µ − νt v ( t )) E L − ν ( ∂ νt v ( t )) ! ≤ δE L − µ ( ∂ µt v ( t )) + 14 δ µ − X ν =0 E L ( ∂ µ − νt v ( t )) E L − ν ( ∂ νt v ( t )) ! ≤ δE L − µ ( ∂ µt v ( t )) + µ δ µ − X ν =0 E L ( ∂ µ − νt v ( t )) E L − ν ( ∂ νt v ( t )) ≤ δE L − µ ( ∂ µt v ( t )) + µE δ µ − X ν =0 E L − ( µ − ν ) ( ∂ µ − νt v ( t ))(1 + t ) − ν − = 2 δE L − µ ( ∂ µt v ( t )) + µE δ µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) − µ − ν ) − .
23o we get D L,µ ( v ( t )) ≤ δE L − µ ( ∂ µt v ( t )) + E δ µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) − µ − ν ) − . (55)The above estimate and lemma 3.1 imply that ddt G L,µ ( v ( t )) + b E L − µ ( ∂ µt v ( t )) (56) ≤ λCE L − µ ( ∂ µt v ( t )) + Cλ D
L,µ ( v ( t )) + 2 k B k ∞ b R C E L − µ ( ∂ µt v ( t )) ≤ (cid:18) λC + 2 δCλ + 2 k B k ∞ b R C (cid:19) E L − µ ( ∂ µt v ( t ))+ CE λδ µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) − µ − ν ) − . From (16), it follows that 2 k B k ∞ b R C ≤ b . Choosing λ and δ > λC + 2 δCλ ≤ b E suchthat ddt G L,µ ( v ( t )) + b E L − µ ( ∂ µt v ( t )) ≤ E µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) − µ − ν ) − . This is (54). Hence we obtain lemma 4.5.We complete theorem 4.1. When i = 0 we already proved (theorem 4.4),so we assume 1 ≤ µ ≤ L − L and for any 0 ≤ i ≤ µ − i = µ .First we prove (36) for i = µ . Lemma 4.5 yields that ddt { (1 + t ) µ +1 G L,µ ( v ( t )) } (57)= (2 µ + 1)(1 + t ) µ G L,µ ( v ( t )) + (1 + t ) µ +1 ddt G L,µ ( v ( t ))24 (2 µ + 1)(1 + t ) µ G L,µ ( v ( t )) − (1 + t ) µ +1 b E L − µ ( ∂ µt v ( t ))+ E µ X ν =1 E L − ν ( ∂ νt v ( t ))(1 + t ) ν . Integrating (57) over [0 , t ], we get(1 + t ) µ +1 G L,µ ( v ( t )) + b Z t (1 + s ) µ +1 E L − µ ( ∂ µt v ( s )) ds (58) ≤ G L,µ ( v (0)) + (2 µ + 1) Z t (1 + s ) µ G L,µ ( v ( s )) ds + E µ X ν =1 Z t (1 + s ) ν E L − ν ( ∂ νt v ( s )) ds. From (20) and (57), it follows that there exists a constant E depending on λ, v and v such that(1 + t ) µ +1 {k ∂ µt v ( t ) k + E L − µ ( ∂ µt v ( t )) } + Z t (1 + s ) µ +1 E L − µ ( ∂ µt v ( s )) ds ≤ E + E Z t (1 + s ) µ k ∂ µt v ( s ) k ds + E Z t (1 + t ) µ E L − µ ( ∂ µt v ( s )) ds + E µ − X ν =1 Z t (1 + s ) ν E L − ν ( ∂ νt v ( s )) ds. (59)We estimate the right-side of (59). Using lemma 2.3 and (37) for i = µ − Z t (1 + s ) µ k ∂ µt v ( s ) k ds (60) ≤ C λ Z t (1 + s ) µ h ∂ µt v ( s ) , B λ ∂ µt v ( s ) i ds + C λ Z t (1 + s ) µ k∇ ∂ µt v ( s ) k ds ≤ C E λ + 2 C λ Z t (1 + s ) µ E L − µ ( ∂ µt v ( s )) ds. From the assumption of induction, it follows that (36) for i = ν with ν ≤ µ − µ − X ν =1 Z t (1 + s ) ν E L − ν ( ∂ νt v ( s )) ds ≤ ( µ − E . (61)25sing (59), (60) and (61), we obtain(1 + t ) µ +1 {k ∂ µt v ( t ) k + E L − µ ( ∂ µt v ( t )) } + Z t (1 + s ) µ +1 E L − µ ( ∂ µt v ( s )) ds ≤ E + E Z t (1 + t ) µ E L − µ ( ∂ µt v ( s )) ds. (62)We choose a constant t ∗ such that 2 E (1 + t ∗ ) − ≤ E Z t (1 + s ) µ E L − µ ( ∂ µt v ( s )) ds (63)= E Z t ∗ (1 + s ) µ E L − µ ( ∂ µt v ( s )) ds + E Z tt ∗ (1 + s ) µ +1 (1 + s ) − E L − µ ( ∂ µt v ( s )) ds ≤ E (1 + t ∗ ) µ +1 µ + 1 sup ≤ s ≤ t ∗ E L − µ ( ∂ µt v ( s )) + 12 Z t (1 + t ) µ +1 E L − µ ( ∂ µt v ( s )) ds. Now E (1 + t ∗ ) µ +1 µ + 1 sup ≤ s ≤ t ∗ E L − µ ( ∂ µt v ( s )) can include E because it is a con-stant depend on λ, u and u . So using (62) and (63), we get (36) to i = µ .Next we prove (37) to i = µ . For the solution v to (DW) λ holds that ddt (cid:8) (1 + t ) µ +2 E ( ∂ µt v ( t )) (cid:9) = (2 µ + 2)(1 + t ) µ +1 E ( ∂ µt v ( t )) + (1 + t ) µ +2 ddt E ( ∂ µt v ( t )) ≤ (2 µ + 2)(1 + t ) µ +1 E L − µ ( ∂ µt v ( t )) − (1 + t ) µ +2 h ∂ µ +1 t v ( t ) , B λ ∂ µ +1 t v ( t ) i +(1 + t ) µ +2 h ∂ µ +1 t v ( t ) , ∂ µt N [ v, v ]( t ) i . Using above estimate and h ∂ µ +1 t v ( t ) , ∂ µt N [ v, v ]( t ) i≤ C k ∂ µ +1 t v ( t ) k µ X ν =0 (cid:18) µν (cid:19) k ∂ νt ∇ v ( t ) k ∞ k ∂ µ − νt ∇ v ( t ) k ≤ CE ( ∂ µt v ( t )) µ X ν =0 E L − ν ( ∂ νt v ( t )) E L − ( µ − ν ) ( ∂ µ − νt v ( t )) , we obtain ddt (cid:8) (1 + t ) µ +2 E ( ∂ µt v ( t )) (cid:9) + (1 + t ) µ +2 h ∂ µ +1 t v ( t ) , B λ ∂ µ +1 t v ( t ) i C (1 + t ) µ +1 E L − µ ( ∂ µt v ( t )) (64)+ C (1 + t ) µ +2 E ( ∂ µt v ( t )) µ X ν =0 E L − ν ( ∂ νt v ( t )) E L − ( µ − ν ) ( ∂ µ − νt v ( t )) . Now we define M µ ( t ) = sup ≤ s ≤ t (1 + s ) µ +2 E ( ∂ µt v ( s )) . Integrating (64) over [0 , t ] and using (36) for i = ν , we obtain(1 + t ) µ +2 E ( ∂ µt v ( t )) + Z t (1 + s ) µ +2 h ∂ µ +1 t v ( s ) , B λ ∂ µ +1 t v ( s ) i ds (65) ≤ E ( ∂ µt v (0)) + C Z t (1 + s ) µ +1 E L − µ ( ∂ µt v ( s )) ds + C Z t (1 + s ) µ +2 E ( ∂ µt v ( s )) µ X ν =0 E L − ν ( ∂ νt v ( s )) E L − ( µ − ν ) ( ∂ µ − νt v ( s )) ds ≤ E ( ∂ µt v (0)) + C Z t (1 + s ) µ +1 E L − µ ( ∂ µt v ( s )) ds + C M µ ( t )) × µ X ν =0 (cid:26)Z t (1 + s ) ν +1 E L − ν ( ∂ νt v ( s )) ds + Z t (1 + s ) µ − ν )+1 E L − ( µ − ν ) ( ∂ µ − νt v ( s )) ds (cid:27) ≤ E ( ∂ µt v (0)) + CE + 2( µ + 1) CE ( M µ ( t )) . From (65), we get the following estimate:(1+ t ) µ +2 E ( ∂ µt v ( t ))+ Z t (1+ s ) µ +2 h ∂ µ +1 t v ( s ) , B λ ∂ µ +1 t v ( s ) i ds ≤ E + E ( M µ ( t )) . (66)From (66), we have M µ ( t ) ≤ E + E ( M µ ( t )) . It means that M µ ( t ) is bounded in [0 , ∞ ). So it holds from (66) that(1 + t ) µ +2 E ( ∂ µt v ( t )) + Z t (1 + s ) µ +2 h ∂ µ +1 t v ( s ) , B λ ∂ µ +1 t v ( s ) i ds ≤ E . This is (37) for i = µ . Thus from induction, we obtain theorem 4.1.27 Proof of lemma 3.1
We prove lemma 3.1. First we prepare the estimates of nonlinear terms.
Lemma 5.1. (Estimates of nonlinear term) Let L ≤ ¯ L ≤ L , ≤ µ ≤ ¯ L − L and D ¯ L,µ is defined by (19) . Then there exists a constant
C > such thatfor any T, δ > , < λ ≤ and the local solution v ∈ X δ,T to (DW ) λ satisfy X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, ∂ t v ]( t ) ≤ CD ¯ L,µ ( v ( t )) , (67) X | a |≤ ¯ L − µ − µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v ( t ) , N [ ∂ νt ∇ a v, ∂ µ − νt v ]( t ) i ≤ CD ¯ L,µ ( v ( t )) , (68) X | a |≤ ¯ L − µ − X b + c = ab,c =0 h ∂ µ +1 t ∇ a v ( t ) , ∂ µt N [ ∇ b v, ∇ c v ]( t ) i ≤ CD ¯ L,µ ( v ( t )) , (69) X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µt ∇ a v ( t ) , ∂ µt N [ ∇ b v, ∇ c v ]( t ) i ≤ CD ¯ L,µ ( v ( t )) (70) and X | a |≤ ¯ L − µ − h ∂ µt ∇ a N [ v, v ]( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i ≤ Cλ D ¯ L,µ ( v ( t )) . (71) Proof.
First, we prove (67). Using lemma 2.1, we have X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, ∂ t v ]( t ) ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v ( t ) k k∇ ∂ µt ∇ a v ( t ) k k∇ ∂ t v ( t ) k ∞ ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v ( t ) k k∇ ∂ µt ∇ a v ( t ) k k ∂ t v ( t ) k H [ d ] +2 ≤ CE ¯ L − µ ( ∂ µt v ( t )) E ¯ L − µ ( ∂ µt v ( t )) E L ( v ( t )) ≤ CE ¯ L − µ ( ∂ µt v ( t )) µ X ν =0 E ¯ L − ν ( ∂ νt v ( t )) E L ( ∂ µ − νt v ( t )) = CD ¯ L,µ ( v ( t )) . Next, we prove (68). We remark that if | a | ≤ ¯ L − µ − ≤ ν ≤ µ − | a | + 1 ≤ ¯ L − ν −
1. Hence it follows that X | a |≤ ¯ L − µ − µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v ( t ) , N [ ∂ νt ∇ a v, ∂ µ − νt v ]( t ) i C X | a |≤ ¯ L − µ − k ∂ µ +1 t ∇ a v ( t ) k µ − X ν =0 d X j,k,l,m,n =1 k ∂ l ( ∂ m ∂ νt ∇ a v j ( t ) ∂ n ∂ µ − νt v k ( t )) k ≤ C X | a |≤ ¯ L − µ − k ∂ µ +1 t ∇ a v ( t ) k × µ − X ν =0 d X j,k,l,m,n =1 (cid:8) k ∂ l ∂ m ∂ νt ∇ a v j ( t ) k k ∂ n ∂ µ − νt v k ( t ) k ∞ + k ∂ m ∂ νt ∇ a v j ( t ) k k ∂ l ∂ n ∂ µ − νt v k ( t ) k ∞ (cid:9) ≤ CE ¯ L − µ ( ∂ µt v ( t )) µ − X ν =0 n E ¯ L − ν ( ∂ νt v ( t )) E L ( ∂ µ − νt v ( t ))+ E ¯ L − ν ( ∂ νt v ( t )) E L ( ∂ µ − νt v ( t )) o ≤ CE ¯ L − µ ( ∂ µt v ( t )) µ X ν =0 E ¯ L − ν ( ∂ νt v ( t )) E L ( ∂ µ − νt v ( t )) = CD ¯ L,µ ( v ( t )) . Next, we prove (69). For any | a | ≤ ¯ L − µ − b + c = a and b, c = 0 it holdthat k ∂ µt N [ ∇ b v, ∇ c v ]( t ) k ≤ C µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) . (72)Because if | a | ≤ ¯ L − µ − b + c = a and b, c = 0 then we can decompose c ofthe form: c = c ′ + c ′′ , | c ′ | = | c | − | c ′′ | = 1. So using lemma 2.1 and lemma2.2 we obtain k ∂ µt ( ∂ l ∂ m ∇ b v i ( t ) ∂ n ∇ c v k ( t )) k ≤ C µ X ν =0 k ∂ νt ∂ l ∂ m ∇ b v j ( t ) ∂ µ − νt ∂ n ∇ c ′′ ∇ c ′ v k ( t ) k ≤ C µ X ν =0 n k∇ ∂ νt v ( t ) k ∞ k∇ ∇ b + c ′ ∂ µ − νt v ( t ) k + k∇ ∇ b + c ′ ∂ νt v ( t ) k k∇ ∂ µ − νt v ( t ) k ∞ o ≤ C µ X ν =0 n k∇ ∂ νt v ( t ) k H [ d ] +1 k∇ ∇ b + c ′ ∂ µ − νt v ( t ) k + k∇ ∇ b + c ′ ∂ νt v ( t ) k k∇ ∂ µ − νt v ( t ) k H [ d ] +1 o ≤ C µ X ν =0 k∇ ∂ µ − νt v ( t ) k H [ d ] +1 k∇ ∇ b + c ′ ∂ νt v ( t ) k C µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) . Similarly we obtain k ∂ µt ( ∂ m ∇ b v i ∂ l ∂ n ∇ c v k ) k ≤ C µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) . So we get (72). It follows from (72) that X | a |≤ ¯ L − µ − X b + c = ab,c =0 h ∂ µ +1 t ∇ a v ( t ) , ∂ µt N [ ∇ b v, ∇ c v ]( t ) i≤ X | a |≤ ¯ L − µ − X b + c = ab,c =0 k ∂ t ∂ µt ∇ a v ( t ) k k ∂ µt N [ ∇ b v, ∇ c v ]( t ) k ≤ CE ¯ L − µ ( ∂ µt v ( t )) µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) = CD ¯ L,µ ( v ( t )) . Next, we prove (70). Using lemma 2.1 and lemma 2.2 we have X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µt ∇ a v ( t ) , ∂ µt N [ ∇ b v, ∇ c v ]( t ) i≤ C X | a |≤ ¯ L − µ − X b + c = a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 Z R d ∂ µt ∇ a v i ( t, x ) ∂ µt ∂ l ( ∂ m ∇ b v j ( t, x ) ∂ n ∇ c v k ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C X | a |≤ ¯ L − µ − X b + c = a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i,j,k,l,m,n =1 Z R d ∂ l ∂ µt ∇ a v i ( t, x ) ∂ µt ( ∂ m ∇ b v j ( t, x ) ∂ n ∇ c v k ( t, x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v ( t ) k × µ X ν =0 (cid:8) k∇ ∂ µ − νt v ( t ) k ∞ k∇∇ a ∂ νt v ( t ) k + k∇∇ a ∂ µ − νt v ( t ) k k∇ ∂ νt v ( t ) k ∞ (cid:9) ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v ( t ) k µ X ν =0 k∇ ∂ µ − νt v ( t ) k H [ d ] +1 k∇∇ a ∂ νt v ( t ) k ≤ CE ¯ L − µ ( ∂ µt v ( t )) µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − µ ( ∂ νt v ) ≤ CD ¯ L,µ ( v ( t )) . Finally, we prove (71). X | a |≤ ¯ L − µ − h ∂ µt ∇ a N [ v, v ] , [ h ; ∇ ∂ µt ∇ a v ] i X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i = X | a |≤ ¯ L − µ − X b + c = ab,c =0 (cid:18) ab (cid:19) h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i +2 X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ µt ∂ l ( ∂ m ∇ a v j ∂ n v k ) h p ∂ p ∂ µt ∇ a v i dx = X | a |≤ ¯ L − µ − X b + c = ab,c =0 (cid:18) ab (cid:19) h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i +2 X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 µ − X ν =0 (cid:18) µν (cid:19) Z R d ∂ l ( ∂ m ∂ νt ∇ a v j ∂ n ∂ µ − νt v k ) h p ∂ p ∂ µt ∇ a v i dx +2 X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ l ( ∂ m ∂ µt ∇ a v j ∂ n v k ) h p ∂ p ∂ µt ∇ a v i dx = X | a |≤ ¯ L − µ − X b + c = ab,c =0 (cid:18) ab (cid:19) h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i +2 X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 µ − X ν =0 (cid:18) µν (cid:19) Z R d ∂ l ( ∂ m ∂ νt ∇ a v j ∂ n ∂ µ − νt v k ) h p ∂ p ∂ µt ∇ a v i dx − X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ m ∂ µt ∇ a v j ∂ n v k ∂ l h p ∂ p ∂ µt ∇ a v i dx − X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ m ∂ µt ∇ a v j ∂ n v k h p ∂ l ∂ p ∂ µt ∇ a v i dx = X | a |≤ ¯ L − µ − X b + c = ab,c =0 (cid:18) ab (cid:19) h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i +2 X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 µ − X ν =0 (cid:18) µν (cid:19) Z R d ∂ l ( ∂ m ∂ νt ∇ a v j ∂ n ∂ µ − νt v k ) h p ∂ p ∂ µt ∇ a v i dx − X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ m ∂ µt ∇ a v j ∂ n v k ∂ l h p ∂ p ∂ µt ∇ a v i dx X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ p ( ∂ m ∂ µt ∇ a v j ∂ l ∂ µt ∇ a v i ) ∂ n v k h p dx = J + J + J + J , where we use (N1) . For J and J , it follows from (23) that | J | ≤ C X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 d X p =1 µ − X ν =0 Z R d | ∂ l ( ∂ m ∂ νt ∇ a v j ∂ n ∂ µ − νt v k ) h p ∂ p ∂ µt ∇ a v i | dx ≤ C X | a |≤ ¯ L − µ − d X p =1 µ − X ν =0 k∇ ∂ νt ∇ a v k k∇ ∂ µ − νt v k ∞ k h p k ∞ k ∂ p ∂ µt ∇ a v k + C X | a |≤ ¯ L − µ − d X p =1 µ − X ν =0 k∇ ∂ νt ∇ a v k k∇ ∂ µ − νt v k ∞ k h p k ∞ k ∂ p ∂ µt ∇ a v k ≤ C k h k ∞ µ − X ν =0 E ¯ L − ν ( ∂ νt v ( t )) E L ( ∂ µ − νt v ( t )) E ¯ L − µ ( ∂ µt v ( t )) ≤ Cλ D ¯ L,µ ( v ( t ))and | J | ≤ C X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 d X p =1 Z R d | ∂ m ∂ µt ∇ a v j ∂ n v k ∂ l h p ∂ p ∂ µt ∇ a v i | dx ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v k k∇ v k ∞ k∇ h k ∞ k∇ ∂ µt ∇ a v k ≤ CE ¯ L − µ ( ∂ µt v ( t )) E ¯ L − µ ( ∂ µt v ( t )) E L ( v ( t )) ≤ CD µ ( v ( t )) ≤ Cλ D ¯ L,µ ( v ( t )) . For J , from (23) and (72), we get | J | ≤ C X | a |≤ ¯ L − µ − X b + c = ab,c =0 |h ∂ µt N [ ∇ b v, ∇ c v ] , [ h ; ∇ ∂ µt ∇ a v ] i|≤ C X | a |≤ ¯ L − µ − X b + c = ab,c =0 kh ∂ µt N [ ∇ b v, ∇ c v ] k k h k ∞ k∇ ∂ µt ∇ a v k ≤ C k h k ∞ E ¯ L − µ ( ∂ µt v ( t )) µ X ν =0 E L ( ∂ µ − νt v ( t )) E ¯ L − ν ( ∂ νt v ( t )) ≤ Cλ D ¯ L,µ ( v ( t )) . For J , (72) implies | J | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ p ( ∂ m ∂ µt ∇ a v j ∂ l ∂ µt ∇ a v i ) ∂ n v k h p dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 N ijklmn d X p =1 Z R d ∂ m ∂ µt ∇ a v j ∂ l ∂ µt ∇ a v i ∂ p ( ∂ n v k h p ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 d X p =1 Z R d | ∂ m ∂ µt ∇ a v j ∂ l ∂ µt ∇ a v i ∂ p ∂ n v k h p | dx + C X | a |≤ ¯ L − µ − d X i,j,k,l,m,n =1 d X p =1 Z R d | ∂ m ∂ µt ∇ a v j ∂ l ∂ µt ∇ a v i ∂ n v k ∂ p h p | dx ≤ C X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v k k∇ ∂ µt ∇ a v k (cid:0) k∇ v k ∞ k h k ∞ + k∇ v k ∞ k∇ h k ∞ (cid:1) ≤ C (cid:18) λ (cid:19) E ¯ L − µ ( ∂ µt v ( t )) E ¯ L − µ ( ∂ µt v ( t )) E L ( v ( t )) ≤ Cλ D ¯ L,µ ( v ( t )) . Hence we obtain (71), which complete the proof of lemma 5.1.Because of dissipation effect (B2 ) λ , we are expected to decrease the en-ergy in | x | >>
1. Next lemma corresponds to this phenomenon.
Lemma 5.2.
Let L ≤ ¯ L ≤ L , ≤ µ ≤ ¯ L − L . Then there exists a constant C > such that for any T, δ > , < λ ≤ and the local solution v ∈ X δ,T to (DW ) λ satisfy ddt ˜ E ¯ L,µ ( v ( t )) + X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i (73) ≤ Cλ E ¯ L − µ ( ∂ µt v ( t )) + CD ¯ L,µ ( v ( t )) and ddt X | a |≤ ¯ L − µ − (cid:26) h ∂ µt ∇ a v ( t ) , ∂ µ +1 t ∇ a v ( t ) i + 12 h ∂ µt ∇ a v ( t ) , B λ ∂ µt ∇ a v ( t ) i (cid:27) − X | a |≤ ¯ L − µ − k ∂ µ +1 t ∇ a v ( t ) k + X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v ( t ) k (74) ≤ Cλ E ¯ L − µ ( ∂ µt v ( t )) + CD ¯ L,µ ( v ( t )) . Proof.
Let L ≤ ¯ L ≤ L and 0 ≤ µ ≤ ¯ L − L . First we show (73). Wecalculate ddt ˜ E ¯ L,µ ( v ( t )) = X | a |≤ ¯ L − µ − ddt E ( ∂ µt ∇ a v ( t ))+ X | a |≤ ¯ L − µ − ddt ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, v ]( t ) . (75)33or the second terms of (75), for any | a | ≤ ¯ L − µ −
1, we need the followingequality: h ∂ µ +1 t ∇ a v, ∂ µt N [ ∇ a v, v ] i (76)= h ∂ µ +1 t ∇ a v, N [ ∂ µt ∇ a v, v ] i + µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i = d X i,j,k,l,m,n =1 N ijklmn Z R d ∂ µ +1 t ∇ a v i ∂ l ( ∂ m ∂ µt ∇ a v j ∂ n v k ) dx + µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i = − d X i,j,k,l,m,n =1 N ijklmn Z R d ∂ l ∂ µ +1 t ∇ a v i ∂ m ∂ µt ∇ a v j ∂ n v k dx + µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i = − ddt d X i,j,k,l,m,n =1 N ijklmn Z R d ∂ l ∂ µt ∇ a v i ∂ m ∂ µt ∇ a v j ∂ n v k dx + 12 d X i,j,k,l,m,n =1 N ijklmn Z R d ∂ l ∂ µt ∇ a v i ∂ m ∂ µt ∇ a v j ∂ n ∂ t v k dx + µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i = − ddt ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, v ] + 12 ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, ∂ t v ]+ µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i , where we use (N1) . To handle the first terms of (75), we apply ∂ µt ∇ a to(DW) λ . We get ∂ µ +2 t ∇ a v − △ ∂ µt ∇ a v + ∇ a ( B λ ∂ µ +1 t v ) = X b + c = a (cid:18) ab (cid:19) ∂ µt N [ ∇ b v, ∇ c v ] , (77)which yields that X | a |≤ ¯ L − µ − ddt E ( ∂ µt ∇ a v ) (78)34 − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∇ a ( B λ ∂ µ +1 t v ) i + X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µ +1 t ∇ a v, ∂ µt N [ ∇ b v, ∇ c v ] i . Combining (75), (76), (78) and lemma 5.1, we get ddt ˜ E ¯ L,µ ( ∂ µt v ) (79)= X | a |≤ ¯ L − µ − ddt E ( ∂ µt ∇ a v ) + X | a |≤ ¯ L − µ − ddt ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, v ]= − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∇ a ( B λ ∂ µ +1 t v ) i + X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µ +1 t ∇ a v, ∂ µt N [ ∇ b v, ∇ c v ] i− X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∂ µt N [ ∇ a v, v ] i + X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, ∂ t v ]+2 X | a |≤ ¯ L − µ − µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i = − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∇ a ( B λ ∂ µ +1 t v ) i + X | a |≤ ¯ L − µ − X b + c = ab,c =0 (cid:18) ab (cid:19) h ∂ µ +1 t ∇ a v, ∂ µt N [ ∇ b v, ∇ c v ] i + X | a |≤ ¯ L − µ − ˜ N [ ∂ µt ∇ a v, ∂ µt ∇ a v, ∂ t v ]+2 X | a |≤ ¯ L − µ − µ − X ν =0 (cid:18) µν (cid:19) h ∂ µ +1 t ∇ a v, N [ ∂ νt ∇ a v, ∂ µ − νt v ] i≤ − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∇ a ( B λ ∂ µ +1 t v ) i + CD ¯ L,µ ( v ) . Since for λ ≤
1, we have − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, ∇ a ( B λ ∂ µ +1 t v ) i = − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i X | a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) h ∂ µ +1 t ∇ a v, ∇ b B λ ( x ) ∂ µ +1 t ∇ a − b v i≤ − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + C X | a |≤ ¯ L − µ − X b ≤ ab =0 k∇ b B λ k ∞ k ∂ µ +1 t ∇ a v k k ∂ µ +1 t ∇ a − b v k ≤ − X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + Cλ E ¯ L − µ ( ∂ µt v ) , from (79), we obtain (73).Next we prove (74). Using (77) and (70) in lemma 5.1, we have ddt X | a |≤ ¯ L − µ − h ∂ µt ∇ a v, ∂ µ +1 t ∇ a v i + 12 X | a |≤ ¯ L − µ − h ∂ µt ∇ a v, B λ ∂ µt ∇ a v i (80)= X | a |≤ ¯ L − µ − (cid:8) k ∂ µ +1 t ∇ a v k + h ∂ µt ∇ a v, ∂ µ +2 t ∇ a v i + h ∂ µt ∇ a v, B λ ∂ µ +1 t ∇ a v i (cid:9) = X | a |≤ ¯ L − µ − k ∂ µ +1 t ∇ a v k − k∇ ∂ µt ∇ a v k − X b ≤ ab =0 (cid:18) ab (cid:19) h ∂ µt ∇ a v, ∇ b B λ ∂ µ +1 t ∇ a − b v i + X | a |≤ ¯ L − µ − X b + c = a (cid:18) ab (cid:19) h ∂ µt ∇ a v, ∂ µt N [ ∇ b v, ∇ c v ] i≤ X | a |≤ ¯ L − µ − k ∂ µ +1 t ∇ a v k − X | a |≤ ¯ L − µ − k∇ ∂ µt ∇ a v k − X b ≤ ab =0 (cid:18) ab (cid:19) h ∂ µt ∇ a v, ∇ b B λ ∂ µ +1 t ∇ a − b v i + CD ¯ L,µ ( v ) . For λ ≤
1, we have X | a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) |h ∂ µt ∇ a v, ∇ b B λ ∂ µ +1 t ∇ a − b v i| = X ≤| a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) |h ∂ µt ∇ a v, ∇ b B λ ∂ µ +1 t ∇ a − b v i|≤ X ≤| a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) k∇ b B λ k ∞ k ∂ µt ∇ a v k k ∂ µ +1 t ∇ a − b v k ≤ λ CE ¯ L − µ ( ∂ µt v ) , | x | ≤ Rλ since B λ ( x ) may notstrictly positive. So we use the local energy decay property in (DW) λ . Welead the estimate which corresponding this property by using the argumentof Nakao[11] and Ikehata[5]. Lemma 5.3.
Let L ≤ ¯ L ≤ L , ≤ µ ≤ ¯ L − L . Then there exists aconstant C > such that for any K, T, δ > , < λ ≤ and the localsolution v ∈ X δ,T to (DW ) λ satisfy ddt X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i (81)+ X | a |≤ ¯ L − µ − Z R d (cid:26) dφ ( | x | ) + | x | φ ′ ( | x | )2 (cid:27) | ∂ µ +1 t ∇ a v ( t, x ) | dx + X | a |≤ ¯ L − µ − Z R d (cid:26) φ ( | x | ) + | x | φ ′ ( | x | ) − dφ ( | x | ) + | x | φ ′ ( | x | )2 (cid:27) |∇ ∂ µt ∇ a v ( t, x ) | dx ≤ λCE ¯ L − µ ( ∂ µt v ( t )) + Cλ D ¯ L,µ ( v ( t ))+ K X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i + 2 k B k ∞ b R λK E ¯ L − µ ( ∂ µt v ( t )) , where φ, h are defined by (17) .Proof. Let | a | ≤ ¯ L − µ −
1. We apply ∂ µt ∇ a to (DW) λ and take inner productthe equation by [ h ; ∇ ∂ µt ∇ a v ] we obtain h ∂ µ +2 t ∇ a v, [ h ; ∇ ∂ µt ∇ a v ] i − h△ ∂ µt ∇ a v, [ h ; ∇ ∂ µt ∇ a v ] i (82)+ h∇ a ( B λ ∂ µ +1 t v ) , [ h ; ∇ ∂ µt ∇ a v ] i = h ∂ µt ∇ a N [ v, v ] , [ h ; ∇ ∂ µt ∇ a v ] i . Noting (82), h ∂ µ +2 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i = ddt h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i − d X k =1 d X i =1 Z R d ∂ µ +1 t ∇ a v k ( t ) h i ∂ i ∂ µ +1 t ∇ a v k ( t ) dx = ddt h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i − d X k =1 d X i =1 Z R d ∂ i | ∂ µ +1 t ∇ a v k ( t ) | h i dx = ddt h ∂ µ +1 t ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i + 12 Z R d | ∂ µ +1 t ∇ a v ( t ) | div hdx − h△ ∂ µt ∇ a v ( t ) , [ h ; ∇ ∂ µt ∇ a v ( t )] i = d X k =1 d X i =1 Z R d ∇ ∂ µt ∇ a v k · ∇ ( h i ∂ i ∂ µt ∇ a v k ) dx = d X k =1 d X i =1 Z R d ∇ ∂ µt ∇ a v k · ∇ h i ∂ i ∂ µt ∇ a v k dx + d X k =1 d X i =1 Z R d ∇ ∂ µt ∇ a v k · ∇ ∂ i ∂ µt ∇ a v k h i dx = d X k =1 d X i =1 d X j =1 Z R d ∂ j ∂ µt ∇ a v k ∂ j h i ∂ i ∂ µt ∇ a v k dx + 12 d X k =1 d X i =1 Z R d ∂ i |∇ ∂ µt ∇ a v k | h i dx = d X i,j,k =1 Z R d ∂ j ∂ µt ∇ a v k ∂ j h i ∂ i ∂ µt ∇ a v k dx − Z R d |∇ ∂ µt ∇ a v | div hdx, we obtain ddt h ∂ µ +1 t ∇ a v, [ h ; ∇ ∂ µt ∇ a v ] i + 12 Z R d | ∂ µ +1 t ∇ a v | div hdx (83) − Z R d |∇ ∂ µt ∇ a v | div hdx + d X i,j,k =1 Z R d ∂ j ∂ µt ∇ a v k ∂ j h i ∂ i ∂ µt ∇ a v k dx = −h∇ a ( B λ ∂ µ +1 t v ) , [ h ; ∇ ∂ µt ∇ a v ] i + h ∂ µt ∇ a N [ v, v ] , [ h ; ∇ ∂ µt ∇ a v ] i . Now we remark that it holds that ∂h i ∂x j = δ ij φ ( | x | ) + φ ′ ( | x | ) x i x j | x | (84)and div h ( x ) = dφ ( | x | ) + φ ′ ( | x | ) | x | ( x ∈ R d ) . (85)Furthermore using φ ′ ( r ) ≤ d X i,j,k =1 Z R d ∂ j ∂ µt ∇ a v k ∂ j h i ∂ i ∂ µt ∇ a v k dx = d X i,k =1 Z R d ∂ i ∂ µt ∇ a v k φ ( | x | ) ∂ µt ∂ i ∇ a v k dx + d X i,j,k =1 Z R d ∂ j ∂ µt ∇ a v k φ ′ ( | x | ) x i x j | x | ∂ i ∂ µt ∇ a v k dx = Z R d |∇ ∂ µt ∇ a v | φ ( | x | ) dx + d X k =1 Z R d | x · ∇ ∂ µt ∇ a v k | φ ′ ( | x | ) 1 | x | dx Z R d { φ ( | x | ) + | x | φ ′ ( | x | ) }|∇ ∂ µt ∇ a v | dx. This estimate and (83) imply that ddt h ∂ µ +1 t ∇ a v, [ h ; ∇ ∂ µt ∇ a v ] i + Z R d (cid:26) ( dφ ( | x | ) + φ ′ ( | x | ) | x | )2 (cid:27) | ∂ µ +1 t ∇ a v | dx Z R d (cid:26) φ ( | x | ) + | x | φ ′ ( | x | ) − dφ ( | x | ) + φ ′ ( | x | ) | x | (cid:27) |∇ ∂ µt ∇ a v | dx (86) ≤ −h B λ ∂ µ +1 t ∇ a v, [ h ; ∇ ∂ µt ∇ a v ( t )] i − X b ≤ ab =0 (cid:18) ab (cid:19) h∇ b B λ ∂ µ +1 t ∇ a − b v, [ h ; ∇ ∂ µt ∇ a v ] i + h ∂ µt ∇ a N [ v, v ] , [ h ; ∇ ∂ µt ∇ a v ] i . Let estimate for the right side of (86). First, since B λ is a nonnegativesymmetric matrix, there exists a nonnegative symmetric matrix S λ such that S λ = B λ . Using (B3 ) λ and (23), for any | a | ≤ ¯ L − µ − K > |h B λ ∂ µ +1 t ∇ a v, [ h ; ∇ ∂ µt ∇ a v ] i| = |h S λ ∂ µ +1 t ∇ a v, S λ [ h ; ∇ ∂ µt ∇ a v ] i|≤ K k S λ ∂ µ +1 t ∇ a v k + 1 K k S λ [ h ; ∇ ∂ µt ∇ a v ] k = K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + 1 K h [ h ; ∇ ∂ µt ∇ a v ] , B λ [ h ; ∇ ∂ µt ∇ a v ] i≤ K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + 1 K k B λ k ∞ k h k ∞ k∇ ∂ µt ∇ a v k ≤ K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + k B k ∞ b R λK k∇ ∂ µt ∇ a v k . So it holds that − X | a |≤ ¯ L − µ − h B λ ∂ µ +1 t ∇ a v, [ h ; ∇∇ a ∂ µt v ] i (87) ≤ X | a |≤ ¯ L − µ − K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + 2 k B k ∞ b R λK E ¯ L − µ ( ∂ µt v ) . Second, using (B3 ) λ and (23), for λ ≤ X | a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) h∇ b B λ ∂ µ +1 t ∇ a − b v, [ h ; ∇ ∂ µt ∇ a v ] i (88)39 X | a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) d X i,j,k =1 (cid:12)(cid:12)(cid:12)(cid:12)Z R d ∇ b ( B λ ) ij ∂ µ +1 t ∇ a − b v j h k ∂ k ∂ µt ∇ a v i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ X | a |≤ ¯ L − µ − X b ≤ ab =0 (cid:18) ab (cid:19) d X i,j,k =1 k∇ b B λ k ∞ k ∂ µ +1 t ∇ a − b v j k k h k k ∞ k ∂ k ∇ a ∂ µt v i k ≤ λCE ¯ L − µ ( ∂ µt v ) . We already got the estimate of h ∂ µt ∇ a N [ v, v ] , [ h ; ∇ ∂ µt ∇ a v ] i in lemma 5.1.Combining estimates (86), (87), (88) and (71), we get (81). This completesthe proof of lemma 5.3. Proof of lemma 3.1
Let K = C λ . Calculating K × (73) + b (2 d − × (74) + (81) we get ddt G ¯ L,µ ( v ( t )) + X | a |≤ ¯ L − µ − (cid:8) K h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i (89) − b (2 d − k ∂ µ +1 t ∇ a v ( t ) k + Z R d dφ + | x | φ ′ | ∂ µ +1 t ∇ a v ( t ) | dx (cid:27) + X | a |≤ ¯ L − µ − Z R d (cid:18) b (2 d − φ + | x | φ ′ − dφ + | x | φ ′ (cid:19) |∇ ∂ µt ∇ a v ( t ) | dx ≤ C ( λ K + λ b (2 d − λ ) E ¯ L − µ ( ∂ µt v ( t )) + C ( K + b (2 d − λ ) D ¯ L,µ ( v ( t ))+ K X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i + 2 k B k ∞ b R λK E ¯ L − µ ( ∂ µt v ( t )) . From (17), it holds that rφ ′ ( r ) = (cid:26) , ( r ≤ Rλ ) − φ ( r ) , ( r ≥ Rλ ) . (90)Using (B2 ) λ , (90), K ≥ dλ and φ ≥
0, we obtain K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i − b (2 d − k ∂ µ +1 t ∇ a v k (91)+ Z R d (cid:26) dφ + | x | φ ′ (cid:27) | ∂ µ +1 t ∇ a v | dx K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + Z | x |≤ Rλ (cid:18) − b (2 d − db (cid:19) | ∂ µ +1 t ∇ a v | dx + Z | x |≥ Rλ (cid:18) λb K − b (2 d − d − φ (cid:19) | ∂ µ +1 t ∇ a v | dx ≥ K h ∂ µ +1 t ∇ a v, B λ ∂ µ +1 t ∇ a v i + b k ∂ µ +1 t ∇ a v ( t ) k for any | a | ≤ ¯ L − µ −
1. Since we have Z R d (cid:18) b (2 d − φ + | x | φ ′ − dφ + | x | φ ′ (cid:19) |∇ ∂ µt ∇ a v | dx (92)= Z | x |≤ Rλ (cid:18) b (2 d − b − db (cid:19) |∇ ∂ µt ∇ a v | dx + Z | x |≥ Rλ (cid:18) b (2 d − − ( d − b Rλ | x | (cid:19) |∇ ∂ µt ∇ a v | dx ≥ b Z | x |≤ Rλ |∇ ∂ µt ∇ a v | dx + Z | x |≥ Rλ (cid:18) b (2 d − − b ( d − (cid:19) |∇ ∂ µt ∇ a v | dx ≥ b k∇ ∂ µt ∇ a v k , estimates (89), (91) and (92) imply ddt G ¯ L,µ ( v ( t )) + X | a |≤ ¯ L − µ − K h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i + b E ¯ L − µ ( ∂ µt v ( t )) ≤ C ( λ K + λ b (2 d − λ ) E L − µ ( ∂ µt v ( t )) + C ( K + b (2 d − λ ) D ¯ L,µ ( v ( t ))+ K X | a |≤ ¯ L − µ − h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i + 2 k B k ∞ b R λK E ¯ L − µ ( ∂ µt v ( t )) . (93)Finally using λ ≤ K = C λ and h ∂ µ +1 t ∇ a v ( t ) , B λ ∂ µ +1 t ∇ a v ( t ) i ≥
0, we get(18), which completes the proof of lemma 3.1.
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