Lifespan of smooth solutions for timelike extremal surface equation in de Sitter spacetime
aa r X i v : . [ m a t h . A P ] N ov Lifespan of smooth solutions for timelike extremal surfaceequation in de Sitter spacetime
De-Xing Kong and Chang-Hua Wei ∗ Department of Mathematics, Zhejiang UniversityHangzhou 310027, China
Abstract
In this paper, we study the generalized timelike extremal surface equation in the de Sitter space-time, which plays an important role in both mathematics and physics. Under the assumption of smallinitial data with compact support, we investigate the lower bound of lifespan of smooth solutions byweighted energy estimates.
Key words and phrases : timelike extremal surface equation, de Sitter spacetime, smooth solution,lifespan, weighted energy estimate. : 35L40, 35L65.
In this paper, we investigate the lifespan of smooth solutions for timelike extremal surface equation in thede Sitter spacetime under the assumption of small initial data. This kind of equation plays an importantrole in general relativity, the theory of black hole, particle physics, fluid mechanics and so on.
The simplest family of black hole spacetimes with positive cosmological constant is the so-called Schwarzschild-de Sitter. If the cosmological constant Λ > M , g ) to the Einstein vacuum field equation R µν − g µν = − Λ g µν (1.1) ∗ Corresponding author: [email protected]. M , called “mass of black hole”, where R µν and R are the Ricci curvature and scalarcurvature of the manifold M , respectively.The line element has the form in local coordinates ds = − (1 − Mr −
13 Λ r ) dt + (1 − Mr −
13 Λ r ) − dr + r dσ S , (1.2)where dσ S denotes the standard metric on the unit 2-sphere and r is the standard distance in theEuclidean space.In the present paper, we set M = 0 to ignore the influence of the black hole. Then by the Lamaˆ i tre-Robertson transformation [23], (1.2) reads ds = − dt + e tR ( dx + dy + dz ) . (1.3)The new coordinates t, x, y, z can take all values from −∞ to ∞ and R is the “radius” of the universe.This is a special case of the line element of the Robertson-Walker space. For more details with respectto this spacetime, one can refer to [15].The de Sitter line element in the higher dimensional analogue to the de Sitter space is ds = − dt + e t ( n X i =1 dx i ) , (1.4)where we have set R = 2 for simplicity.The timelike extremal surface equation corresponding to the de Sitter spacetime under the abovecoordinates (1.4) reads (cid:3) g ϕ − ϕ t = Q ( ϕ, e t Q ( ϕ, ϕ ))2(1 + e t Q ( ϕ, ϕ )) , (1.5)where ϕ = ϕ ( t, x , · · · , x n ) is the unknown function which corresponds to hypersurface, Q ( ϕ, ψ ) is thenull form of the de Sitter universe, which is given by Q ( ϕ, ψ ) = − ϕ t ψ t + e − t ( n X i =1 ϕ i ψ i ) . (1.6)Here and hereafter, without confusion, we denote ∂ t ϕ and ∂ x i ϕ by ϕ t and ϕ i , respectively, (cid:3) g denotesthe covariant wave operator, which is given by (cid:3) g = 1 p | det g | ∂ µ ( p | det g | g µν ∂ ν ) , (1.7)where the Einstein’s summation convention has been used, namely, the same upper and lower indexmeans summation. In the following, we will use this convention without a hint.Due to the important significances in both mathematics and physics, up to now, a lot of resultson minimal surfaces in the Euclidean R n and the Riemannian manifolds have been obtained. One canrefer to two excellent books: Coding and Minicozzi [10] and Osserman [24]. Since the metric of Lorentz2anifolds is not positive, a surface in these physical spacetimes may include the following four types:timelike, spacelike, lightlike and mixed types. For the global existence of extremal surface equation inMinkowski spacetime, one can refer to Lindblad [19] and Brandle [3]. For the case in curved spaces, Gu[12] proves that in the isothermal coordinates, the equations of the motion of relativistic strings can belocally written as a form of harmonic map from Minkowski to the Lorentz manifold. He and Kong [14]study the spherical solution of the relativistic membrane in Schwarzschild universe.An important structure for the global existence of extremal surface equation in the Minkowski space-time is the null condition, which is first studied by Klainerman [16] and Christodoulou [5], respectively.For static curved spacetimes, Luk [21] investigates the global existence for nonlinear wave equations onslowly rotating Kerr spacetime satisfying the null condition, which is also suitable for the Schwarzschildspacetime. Other important models satisfying the null condition are the wave maps in curved spacetimes,which are interesting in geometry and physics, see [4, 11]. For more information on null conditions, onecan refer to Alinhac [1]. Obviously, the nonlinear terms of the extremal surface equation (1.5) satisfiesthe null condition, since Q ( ϕ, ψ ) satisfies null condition by definition.Recently, the wave equations in the background of de Sitter spacetime become the focus of interest foran increasing number of mathematicians. For linear wave equations, fundamental solutions to the Cauchyproblem with or without source terms are obtained, see [25, 27]. For semilinear case, the global existenceand blowup results for the Klein-Gordon equation have been arrived, see [26, 28]. To our knowledge, thepresent work is the first work on nonlinear wave equations in the background of de Sitter spacetime andwe believe that it will play an important role in the study of the nonlinear stability of de Sitter universe. Instead of considering (1.5), we shall consider Cauchy problems for the following wave equations (cid:3) g ϕ − ϕ t = Q ( ϕ, ϕ ) (1.8)and (cid:3) g ϕ − ϕ t = Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) , (1.9)with the following initial data t = 0 : ϕ (0 , x , · · · , x n ) = ǫf ( x , · · · , x n ) , ϕ t (0 , x , · · · , x n ) = ǫg ( x , · · · , x n ) , (1.10)where f, g ∈ C ∞ ( R n ) and α ≤ Remark 1.1
When α = 1 , (1.9) is nothing but (1.5). Before we state our main results, we firstly give the definition of the lifespan of solution.3 efinition 1.1
The lifespan T ( ǫ ) is the supremum of T > such that the Cauchy problem (1.8), (1.10)(or (1.9), (1.10)) has a smooth solution on [0 , T ] × R n . Theorem 1.1
There exists a positive constant ǫ such that, for any ǫ ∈ [0 , ǫ ] , the Cauchy problem (1.8),(1.10) has a unique global smooth solution on [0 , ∞ ) × R n . Theorem 1.2
There exist positive constants C and ǫ , such that for ǫ ∈ [0 , ǫ ] , the lifespan of theCauchy problem (1.9), (1.10) satisfies T ( ǫ ) = ∞ , α < , C ǫ , α = 1 , (1.11) where C depends on ǫ and α . Remark 1.2
The results of Theorem 1.1 and 1.2 still hold if the dissipative term ϕ t does not appear inequations (1.8) and (1.9). This paper is organized as follows. In Section 2, we investigate the basic equation for the motion ofrelativistic membrane in the de Sitter spacetime and derive an interesting nonlinear wave equation. InSection 3, we get a pointwise decay estimate for the linear wave equation with a dissipative term by themethod of weighted energy estimates. In Section 4, we prove the global existence of the model equation(1.8). Section 5 is devoted to the lifespan of the Cauchy problem (1.9)-(1.10). Section 6 gives somediscussions.
The de Sitter metric is given by (1.4), i.e., ds = − dt + e t ( n X i =1 dx i ) . Consider the motion of a relativistic membrane in the de Sitter spacetime( t, x , · · · , x n ) → ( t, x , · · · , x n , ϕ ( t, x , · · · , x n )) . In the coordinates ( t, x , · · · , x n ), the induced metric of the submanifold M reads as ds = ( dt, dx , · · · , dx n ) G ( dt, dx , · · · , dx n ) T , (2.1)4here G = g g · · · g n g g · · · g n ·· ·· · · · ·· g n g n · · · g nn , in which g = − e t ϕ t , g i = g i = e t ϕ t ϕ i and g ii = e t + e t ϕ i , g ij = e t ϕ i ϕ j for i, j = 1 , · · · , n .We assume that the submanifold M is timelike, i.e.,∆ := det G = e nt [ − − n X i =1 ϕ i + e t ϕ t ] < . (2.2)This is equivalent to 1 + n X i =1 ϕ i − e t ϕ t > . Thus the area element of M is dA = √− ∆ dtdx · · · dx n . (2.3)The submanifold M is called to be extremal if ϕ = ϕ ( t, x , · · · , x n ) is a critical point of the area functional I ( ϕ ) = Z · · · Z √− ∆ dtdx · · · dx n . (2.4)By direct calculations, the corresponding Euler-Lagrange equation reads as ∂∂t − e nt e t ϕ t p P ni =1 ( ϕ i ) − e t ϕ t ! + n X j =1 ∂∂x j e nt ϕ j p P ni =1 ( ϕ i ) − e t ϕ t ! , (2.5)which is equivalent to − ϕ tt − n + 22 ϕ t + e − t n X i =1 ϕ ii + ϕ t ∂ t ( P ni =1 ϕ i − e t ϕ t )2(1 + P ni =1 ϕ i − e t ϕ t ) − n X j =1 e − t ϕ j ∂ x j ( P ni =1 ϕ i − e t ϕ t )2(1 + P ni =1 ϕ i − e t ϕ t ) = 0 . (2.6)By direct calculatuons, the linear wave equation in de Sitter spacetime is (cid:3) g ϕ = − ϕ tt − n ϕ t + e − t n X i =1 ϕ ii . (2.7)Denote Q ( ϕ, ψ ) = − ϕ t ψ t + e − t n X i =1 ϕ i ψ i , (2.8)5hen by (2.7) and (2.8), (2.6) can be rewritten as (cid:3) g ϕ − ϕ t = Q ( ϕ, e t Q ( ϕ, ϕ ))2(1 + e t Q ( ϕ, ϕ )) . This is nothing but (1.5).Before we state the structure enjoyed by (1.5), we generalize the definition of the null condition inthe Minkowski spacetime to the de Sitter spacetime, which can be found in Alinhac [1].Define g µν by g µν g µλ = δ νλ , where δ νλ is the Kronecker symbol. Definition 2.1
We say that a quadratic form A µν ϕ µ ψ ν satisfies the null condition in general Lorentz manifold ( M , g ) , if the coefficients A µν satisfy A µν ξ µ ξ ν = 0 , whenever ξ is a null vector, namely, g ( ξ, ξ ) = g µν ξ µ ξ ν = 0 . Lemma 2.1
The nonlinear term Q ( ϕ, ψ ) satisfies the null condition in the sense of Definition (2.1).Proof. It is easy to see that Q ( ϕ, ψ ) = g µν ϕ µ ψ v satisfies the null condition. Thus, the lemma holdsobviously by Definition 2.1.Another important property for (2.5) is the linear degeneracy of its characteristics. In order toillustrate this phenomenon, we first recall the definition of linear degeneracy and genuine nonlinearity(see [17, 18]).Consider the following quasilinear hyperbolic systems u t + n X k =1 A k ( u ) u x k = B ( u ) , (2.9)where u = ( u , · · · , u n ) T is the unknown vector function, A k ( u ) = ( a kij ( u )) is an n × n matrix withsuitably smooth elements a kij ( u ) ( i, j = 1 , · · · , n ), B ( u ) = ( B ( u ) , · · · , B n ( u )) T is a given smooth vectorfunction, which denotes the source term. Define A ( u ; ξ ) = n X k =1 A k ( u ) ξ k , (2.10)where ξ = ( ξ , · · · , ξ n ) is any unit vector in the Euclidean space.By hyperbolicity, for any given u on the domain under consideration, A ( u ; ξ ) has n real eigenval-ues λ ( u ; ξ ) , · · · , λ n ( u ; ξ ) and a complete system of left (resp. right) eigenvectors. For i = 1 , · · · , n ,6et l i ( u ; ξ ) = ( l i ( u ; ξ ) , · · · , l in ( u ; ξ )) (resp. r i ( u ; ξ ) = ( r i ( u ; ξ ) , · · · , r in ( u ; ξ ) ) T ) be a left (resp. right)eigenvector corresponding to λ i ( u ; ξ ): l i ( u ; ξ ) A ( u ; ξ ) = λ i ( u ; ξ ) l i ( u ; ξ ) (resp. A ( u ; ξ ) r i ( u ; ξ ) = λ i ( u ; ξ ) r i ( u ; ξ )) . (2.11)We have det | l ij ( u ; ξ ) | 6 = 0 (equivalently , det | r ij ( u ; ξ ) | ) = 0 (2.12)Then Definition 2.2 λ i ( u ; ξ ) ( i ∈ { , · · · , n } ) is said to be genuinely nonlinear, if for every state u and anyunit vector ξ , it holds that ∇ λ i ( u ; ξ ) r i ( u ; ξ ) = 0 , (2.13) λ i ( u ; ξ ) is called to be linearly degenerate, if for every state u and any unit vector ξ , it holds that ∇ λ i ( u ; ξ ) r i ( u ; ξ ) ≡ . (2.14) The system (2.9) is genuinely nonlinear (resp. linearly degenerate), if all λ i ( i = 1 , · · · , n ) are genuinelynonlinear (resp. linearly degenerate). Based on the above definition, we have
Lemma 2.2
System (2.5) is linearly degenerate in the sense of P. D. Lax.Proof.
Set τ = − e − t , τ ∈ [ − , . (2.15)We have ∂ τ = dtdτ ∂ t = e t ∂ t , then ϕ τ = e t ϕ t . Thus, in the ( τ, x , · · · , x n ) coordinates, (2.5) can be rewritten as − ∂ τ ϕ τ q P ni =1 ϕ x i − ϕ τ + n X j =1 ∂ x j ϕ x j q P ni =1 ϕ x i − ϕ τ = − n + 1 τ ϕ τ q P ni =1 ϕ x i − ϕ τ . (2.16)The principle term of (2.16) is nothing but the timelike extremal surface equation in the Minkowskispacetime R n , which is linearly degenerate obviously. One can refer to [13]. Thus, the lemma isproved.If the solution of (2.5) takes the following form ϕ ( t, x ) = ϕ ( t, x , · · · , x n ) , (2.17)7here x = P ni =1 ξ i x i and ξ = ( ξ , · · · , ξ n ) is the unit vector, then (2.5) can be reduced to − ∂ t e n +12 t e t ϕ t p ϕ x − e t ϕ t ! + ∂ x e nt ϕ x p ϕ x − e t ϕ t ! = 0 . (2.18)Under the ( τ, x ) coordinate, (2.19) can be rewritten as − ∂ τ ϕ τ p ϕ x − ϕ τ ! + ∂ x ϕ x p ϕ x − ϕ τ ! = − n + 1 τ ϕ τ p ϕ x − ϕ τ . (2.19) Remark 2.1
In the ( τ, x ) coordinates, the principle term of (2.20) is nothing but the classical Born-Infeldequation [2]. Remark 2.2
Different from the timelike extremal surface equation in the Minkowski spacetime R n ,(2.17) has an extra dissipative term, since in the ( τ, x , · · · , x n ) coordinates, the coefficient of the sourceterm − n +1 τ > , but the singularity appears as τ tends to zero. Remark 2.3
Equation (2.16) can be derived as the timelike extremal surface equation in the coordinates ( τ, x , · · · , x n ) , where τ is defined by (2.15). In fact, in this coordinate frame, the metric (1.4) of deSitter spacetime becomes ds = 4 τ ( − dτ + n X i =1 dx i ) . (2.20) In this section, we investigate the pointwise decay estimates of the following linear wave equation in deSitter spacetime (cid:3) g ϕ − ϕ t = 0 . (3.1)It will play a key role in the study of nonlinear cases. Before we state our main results of this section,we introduce the following notations k u ( x ) k L = (cid:18)Z R n | u ( x ) | dx (cid:19) , k u ( x ) k L ∞ := ess sup | u ( x ) | and k u ( x ) k H s = s X i =0 ( k D i u ( x ) k L ) ! , where s is an integer.Define the energy momentum tensor corresponding to the equation (3.1) by T µν ( ϕ ) = ∂ µ ϕ∂ ν ϕ − g µν |∇ ϕ | , (3.2)8here |∇ ϕ | = g κλ ∂ κ ϕ∂ λ ϕ = − ϕ t + n X i =1 e − t ϕ i . For a vector field V = V µ ∂ µ , define the compatible currents J Vµ ( ϕ ) = T µν ( ϕ ) V ν (3.3)and K V ( ϕ ) = Π Vµν T µν ( ϕ ) , (3.4)where Π Vµν is the deformation tensor defined byΠ
Vµν = 12 ( ∇ µ V ν + ∇ ν V µ ) , (3.5)in which ∇ denotes the covariant derivative and ∇ µ V ν = g ( ∇ µ V, ∂ ν ) . For a constant t -slice, the induced volume form is defined by dV ol t = e nt dx · · · dx n . (3.6) Remark 3.1
In above notations, raising and lowering of indices in this paper is always done with respectto the metric g of the de Sitter spacetime. With above notations, by direct calculations, we have for i, j = 1 , · · · , n and i = jT tt ( ϕ ) = ϕ t + 12 |∇ ϕ | = 12 ( ϕ t + n X i =1 e − t ϕ i ) , (3.7) T ti = ϕ t ϕ i , T ij = ϕ i ϕ j (3.8)and T ii = ϕ i − e t |∇ ϕ | = 12 ( e t ϕ t + ϕ i − X j = i ϕ j ) . (3.9)The following lemma is easy and can be found in [6, 7, 8, 9, 20, 21]. Lemma 3.1
For the equation (cid:3) g ϕ = f , it holds that ∇ µ T µν = (cid:3) g ϕϕ ν , ∇ µ J Vµ ( ϕ ) = K V ( ϕ ) + (cid:3) g ϕ · V ( ϕ ) . (3.10) Proof.
By direct calculations, we have ∇ µ T µν ( ϕ ) = ∇ µ ( ∂ µ ϕ∂ ν ϕ − g µν ∂ λ ϕ∂ λ ϕ )= (cid:3) g ϕ∂ ν ϕ + ∂ µ ϕ ∇ µ ∂ ν ϕ − g µν ∂ λ ϕ ∇ µ ∂ λ ϕ = (cid:3) g ϕϕ ν ∇ µ J Vµ ( ϕ ) = ∇ µ ( V ν T µν ( ϕ ))= ∇ µ V ν T µν ( ϕ ) + V ν ∇ µ T µν ( ϕ )= K V ( ϕ ) + (cid:3) g ϕV ( ϕ ) . Thus, the lemma is proved.The energy density e ( V, υ ) of the mapping ϕ at time t with respect to the past oriented timelike vectorfield V is the nonnegative number e ( V, υ ) = J Vα υ α = T αβ ( ϕ ) V β υ α (3.11)with υ α the components of the past oriented unit normal υ = − ∂ t .Taking the past oriented vector field V , by Lemma 3.1 and divergence theorem, we easily get thefollowing lemma Lemma 3.2
The following energy identity holds in the domain D = { ≤ τ ≤ t } Z Σ t J Vα υ α dV ol t − Z Σ J Vα υ α dV ol = Z t Z Σ τ ( K V ( ϕ ) + (cid:3) g ϕV ( ϕ )) dV ol τ dτ. (3.12)From now on, we take the vector field V = − ∂ t , thenΠ Vµν = 12 ( ∇ µ V ν + ∇ ν V µ )= 12 ( g ( ∇ µ ( − ∂ t ) , ∂ ν ) + g ( ∇ ν ( − ∂ t ) , ∂ µ ))= −
12 ( g νκ Γ κµt + g µκ Γ κνt ) . (3.13)So, for i, j = 1 , · · · , n , Π Vii = − g iκ Γ κit = − g ii Γ iit = − e t . (3.14)And for i = j , Π Vij = 0 , Π V i = 0 and Π V = 0 . (3.15)Here Γ kij denotes the connection coefficients, which are given byΓ kij = 12 g km ( ∂g im ∂x j + ∂g jm ∂x i − ∂g ij ∂x m ) , where we have assumed that x = t .By (3.4), (3.9), (3.14) and (3.15), we obtain K − ∂ t ( ϕ ) = Π − ∂ t µν T µν ( ϕ )= g µµ g νν Π − ∂ t µν T µν ( ϕ ) = g ii g ii Π − ∂ t ii T ii ( ϕ )= e − t n X i =1 ( − e t )[ 12 ( e t ϕ t + ϕ i − X j = i ϕ j )]= 14 [( n − n X i =1 e − t ϕ i − nϕ t ] . (3.16)10y (3.11), e ( V, υ ) = T αβ ( ϕ ) V α υ β = T tt ( ϕ ) . (3.17)Denote D = { ∂ , · · · , ∂ n } and D I = ∂ I · · · ∂ I n n , where I = ( I , · · · , I n ) with | I | = P nj =1 | I j | . For the constant t -slice, define E | I | ,I ( t ) = 12 Z Σ t ( ∂ I t D I ϕ ) t dV ol t , E | I | ,I ( t ) = 12 Z Σ t ( n X i =1 e − t ( ∂ I t D I ϕ ) i ) dV ol t (3.18)and E | I | ,I ( t ) = E | I | ,I ( t ) + E | I | ,I ( t ) . (3.19)Then, by above calculations and Lemma 3.2, the following zero-th order energy identity holds. Lemma 3.3
The energy identity (3.12) can be rewritten as E , ( t ) − E , (0) = Z t [ −
12 ( n + 4) E , ( τ ) + 12 ( n − E , ( τ )] dτ (3.20) Proof.
By direct calculations, from (3.12), (3.16)-(3.18) and Z t Z Σ τ ( − ϕ τ ) dV ol τ dτ = − Z t E , ( τ ) dτ, we get (3.20) immediately. Corollary 3.1
It holds that ddt E , ( t ) = −
12 ( n + 4) E , ( t ) + 12 ( n − E , ( t ) . (3.21)Based on the geometry of de Sitter spacetime with the metric given by (1.4), it is easy to see that theoperator D is a killing vector field, which means thatΠ Dµν = 0 . Thus, the structure of the equation (3.1) will not change if we take D J as a commutator, namely (cid:3) g ( D J ϕ ) − ( D J ϕ ) t = 0 . (3.22)By (3.22) and Corollary 3.1, for I = 0, we have Corollary 3.2
It holds that ddt E | J | , ( t ) = −
12 ( n + 4) E | J | , ( t ) + 12 ( n − E | J | , ( t ) , (3.23) for arbitrary J . f | I | ,I = E | I | ,I e − n t , f | I | ,I = E | I | ,I e − n t and f | I | ,I = E | I | ,I e − n t . (3.24)Then we obtain Lemma 3.4 f | I | , is uniformly bounded, provided that E | I | , (0) is bounded for arbitrary I .Proof. By (3.24), ddt f | I | , ( t ) = ddt E | I | , ( t ) e − n t + 2 − n E | I | , ( t ) e − n t = e − n t [ −
12 ( n + 4) E | I | , ( t ) + 12 ( n − E | I | , ( t ) + 2 − n E | I | , ( t ) + E | I | , ( t ))]= − ( n − e − n t E | I | , ( t ) ≤ . (3.25)This proves Lemma 3.4.For I >
0, by (3.22), it holds that
Lemma 3.5 (cid:3) g ( ∂ I t D J ϕ ) − ( ∂ I t D J ϕ ) t = e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i D J ϕ ) , (3.26) where C M ( M = 0 , · · · , I − are constants depending on M .Proof. Denote D J ϕ = v, it suffices to prove (cid:3) g ( ∂ I t v ) − ( ∂ I t v ) t = e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i v ) . (3.27)By (3.22), it holds that (cid:3) g v − v t = 0 . (3.28)Since ∂ t is not a killing vector field, it does not commutate with the operator (cid:3) g , by a direct calculation,it holds that [ (cid:3) g , ∂ t ] = [ − ∂ t − n ∂ t + e − t ( n X i =1 ∂ i ) , ∂ t ] = e − t ( n X i =1 ∂ i ) . (3.29)We prove this lemma by the method of induction.When I = 1, it holds that (cid:3) g ( ∂ t v ) = [ (cid:3) g , ∂ t ] v + ∂ t ( (cid:3) g v )= e − t ( n X i =1 ∂ i v ) + ( ∂ t v ) t , (3.30)i.e., (cid:3) g ( ∂ t v ) − ( ∂ t v ) t = e − t ( n X i =1 ∂ i v ) . (3.31)12hus, the lemma holds for I = 1.Suppose the lemma holds for I −
1, namely, (cid:3) g ( ∂ I − t v ) − ( ∂ I − t v ) t = e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i v ) , (3.32)then (cid:3) g ( ∂ I t v ) = [ (cid:3) g , ∂ t ]( ∂ I − t v ) + ∂ t ( (cid:3) g ( ∂ I − t v ))= e − t ( n X i =1 ∂ i ∂ I − t v ) + ∂ t ( ∂ I − t v ) t + e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i v ) ! = e − t ( n X i =1 ∂ I − t ∂ i v ) + ( ∂ I t v ) t − e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i v ) + e − t ( n X i =1 I − X M =0 C M ∂ M +1 t ∂ i v )= ( ∂ I t v ) t + e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i v ) , (3.33)thus, the lemma holds for arbitrary I .By Lemmas 3.2, 3.4, 3.5 and Corollary 3.2, for I >
0, it holds that
Lemma 3.6 f | I | ,I ( t ) is uniformly bounded, and it holds that for arbitrary I , f | I | ,I ( t ) ≤ C I,I ( I X k =0 X | l | + k ≤| I | + I f | l | ,k (0)) , (3.34) where C I,I is a constant depending only on I, I .Proof. By Lemmas 3.2, 3.5 and Corollary 3.2, it is obvious that ddt E | I | ,I ( t ) = −
12 ( n + 4) E | I | ,I ( t ) + 12 ( n − E | I | ,I ( t ) − Z Σ t e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i D | I | ϕ ) ∂ I +1 t D | I | ϕdV ol t . (3.35)As Lemma 3.4, it holds that ddt f | I | ,I ( t ) = ddt E | I | ,I ( t ) e − n t + 2 − n E | I | ,I ( t ) e − n t ≤ | e − n t Z Σ t e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i D | I | ϕ ) ∂ I +1 t D | I | ϕdV ol t | . (3.36)Now, we prove the lemma by induction.For I = 1, by Lemma 3.4 and H¨older inequality, it holds that ddt f | I | , ( t ) ≤ | e − n t Z Σ t e − t ( n X i =1 ∂ i D | I | ϕ ) ∂ t D | I | ϕdV ol t |≤ e − t ( f | I | +1 , ( t )) ( f | I | , ( t )) . (3.37)13hus, by Lemma 3.4, it holds that (cid:16) f | I | , ( t ) (cid:17) ≤ (cid:16) f | I | , (0) (cid:17) + Z ∞ e − t ( f | I | +1 , (0)) dt, (3.38)it implies that the lemma holds for I = 1.Suppose that the lemma holds for N ≤ I −
1, i.e., f | I | ,N ( t ) ≤ C I,N ( N X k =0 X | l | + k ≤| I | + N f | l | ,k (0)) for N ≤ I − , (3.39)by (3.36) and H¨older inequality, we have ddt f | I | ,I ( t ) ≤ I − X M =0 C M e − t ( f | I | +1 ,M ( t )) ! ( f | I | ,I ( t )) , (3.40)thus, by (3.39) and (3.40), the lemma holds for arbitrary I . Remark 3.2
The quantities f | I | ,I (0) and E | I | ,I can be derived directly from the initial data and theequation. Define e | I | ,I ( t ) = 12 k ( ∂ I t D I ϕ ) t k L = 12 Z R n ( ∂ I t D I ϕ ) t dx · · · dx n , (3.41) e | I | ,I ( t ) = 12 k n X i =1 ( ∂ I t D I ϕ ) i k L = 12 Z R n n X i =1 ( ∂ I t D I ϕ ) i dx · · · dx n (3.42)and e | I | ,I ( t ) = e | I | ,I ( t ) + e | I | ,I ( t ) . (3.43)By (3.18), (3.41) and (3.42), we have E | I | ,I ( t ) = 12 Z R n ( ∂ I t D I ϕ ) t e nt dx · · · dx n = e nt e | I | ,I ( t ) (3.44)and E | I | ,I ( t ) = 12 Z R n n X i =1 ( ∂ I t D I ϕ ) i e − t e nt dx · · · dx n = e ( n − t e | I | ,I ( t ) . (3.45)Thus, we obtain easily Lemma 3.7
The following decay estimates hold e | I | ,I ( t ) ≤ e − t f | I | ,I ( t ) , e | I | ,I ( t ) ≤ f | I | ,I ( t ) . (3.46)In what follows, we will use the following Sobolev embedding theorem on R n Lemma 3.8 If u = u ( x , · · · , x n ) ∈ H s for any s > n , then there exists a constant C s such that u ∈ L ∞ ( R n ) , and it holds that k u k L ∞ ≤ C s k u k H s . (3.47)14y Lemmas (3.7) and (3.8), we easily obtain Lemma 3.9
For any | I | ≥ | J | + ⌈ n + 1 ⌉ and i = 1 , · · · , n , it holds that k ( ∂ I t D J ϕ ) t ( t ) k L ∞ ≤ C I e − t ( | I | X | M | =0 f | M | ,I ( t )) for I ≥ and k ( D J ϕ ) i ( t ) k L ∞ ≤ C I ( | I | X | M | =0 f | M | , ( t )) . (3.49) provided that f | M | ,I ( t ) ( | M | = 0 , · · · , | I | ) is bounded. Here ⌈ a ⌉ stands for the smallest integer larger than a . Remark 3.3
By the discussions above, we observe that the dissipative term ϕ t does not affect the decayrate in this procedure. In this section, we shall consider the global existence of the following equation (cid:3) g ϕ = ϕ t + Q ( ϕ, ϕ ) , where Q ( ϕ, ϕ ) is defined by (l.6) and satisfies Definition 2.1.Since the equation can be reduced into a symmetric hyperbolic system, the local existence and unique-ness theorem holds, provided that the initial data belongs to the Sobolev space H s for s > n + 1, whichcan be found in Majda [22] and Alinhac [1].The pointwise decay estimates derived in the last section will play a key role in the proof of the globalexistence and the lower bound of the lifespan for nonlinear wave equations. We will prove the maintheorem by continity method and take the nonlinear terms as the disturbances. Before proving Theorem1.1, we need the following lemmas, which state the structure enjoyed by null condition. Lemma 4.1
The null structure is conserved under D -derivatives, namely, the following holds (cid:3) g ( D I ϕ ) = ( D I ϕ ) t + X | I | + | I | = | I | C I Q ( D I ϕ, D I ϕ ) , (4.1) where C I is a constant depending on I .Proof. Since the vector field D is killing vector field corresponding to the operator (cid:3) g and is commutablewith ∂ t , it suffices to prove D I Q ( ϕ, ϕ ) = X | I | + | I | = | I | C I Q ( D I ϕ, D I ϕ ) .
15n what follows, we prove it by induction.When | I | = 1, we have DQ ( ϕ, ϕ ) = D ( − ϕ t + n X i =1 e − t ϕ i ) = − ϕ t ( Dϕ ) t + n X i =1 e − t ϕ i ( Dϕ ) i = 2 Q ( Dϕ, ϕ ) . Thus, the lemma holds for | I | = 1.Suppose that the lemma holds for | I | −
1, i.e., it holds that D I − Q ( ϕ, ϕ ) = X | ˜ I | + | ˜ I | = | I |− C I − Q ( D ˜ I ϕ, D ˜ I ϕ ) , then D I Q ( ϕ, ϕ ) = D ( D I − Q ( ϕ, ϕ ))= D ( X | ˜ I | + | ˜ I | = | I |− C I − Q ( D ˜ I ϕ, D ˜ I ϕ ))= D X | ˜ I | + | ˜ I | = | I |− C I − ( − ( D ˜ I ϕ ) t ( D ˜ I ϕ ) t + n X i =1 e − t ( D ˜ I ϕ ) i ( D ˜ I ϕ ) i ) = X | ˜ I | + | ˜ I | = | I |− C I − − ( DD ˜ I ϕ ) t ( D ˜ I ϕ ) t + n X i =1 e − t ( DD ˜ I ϕ ) i ( D ˜ I ϕ ) i ! + X | ˜ I | + | ˜ I | = | I |− C I − − ( D ˜ I ϕ ) t ( DD ˜ I ϕ ) t + n X i =1 e − t ( D ˜ I ϕ ) i ( DD ˜ I ϕ ) i ! = X | I | + | I | = | I | C I Q ( D I ϕ, D I ϕ ) . (4.2)Thus, the lemma holds.For derivatives with respect to t , we have the following Lemma 4.2
It holds that (cid:3) g ( ∂ I t D I ϕ ) − ( ∂ I t D I ϕ ) t = e − t ( n X i =1 I − X M =0 C M ∂ Mt ∂ i D I ϕ )+ X I
01 + I
02 = I | I | + | I | = | I | C I ,I ,I ,I Q ( ∂ I t D I ϕ, ∂ I t D I ϕ )+ e − t ( n X i =1 X ˜ I
01 + ˜ I ≤ I − | I | + | I | = | I | C ˜ I , ˜ I ,I ,I ∂ ˜ I t ∂ i D I ϕ∂ ˜ I t ∂ i D I ϕ ) , (4.3) where C M , C I ,I ,I ,I and C ˜ I , ˜ I ,I ,I are constants.Proof. Denote D I ϕ = v, D I ϕ = w.
16y Lemmas 3.5 and 4.1, it suffices to prove ∂ I t Q ( v, w ) = X I + I = I C I ,I Q ( ∂ I t v, ∂ I t w ) + e − t ( n X i =1 X ˜ I +˜ I ≤ I − C ˜ I , ˜ I ∂ ˜ I t ∂ i v∂ ˜ I t ∂ i w ) . (4.4)As Lemmas 3.5 and 4.1, we prove (4.4) by induction.When I = 1, it holds that ∂ t Q ( v, w ) = ∂ t − ∂ t v∂ t w + e − t ( n X i =1 ∂ i v∂ i w ) ! = − ∂ t v∂ t w − ∂ t v∂ t w + e − t n X i =1 ( ∂ i ∂ t v∂ i w + ∂ i v∂ i ∂ t w ) ! − e − t ( n X i =1 ∂ i v∂ i w )= Q ( ∂ t v, w ) + Q ( v, ∂ t w ) − e − t ( n X i =1 ∂ i v∂ i w ) . (4.5)Thus, the lemma holds for I = 1.Suppose the lemma holds for I −
1, then ∂ I t Q ( v, w ) = ∂ t ( ∂ I − t Q ( v, w ))= ∂ t X I + I = I − C I ,I Q ( ∂ I t v, ∂ I t w ) + e − t ( n X i =1 X ˜ I +˜ I ≤ I − ∂ ˜ I t ∂ i v∂ ˜ I t ∂ i w ) = X I + I = I − C I ,I (cid:16) Q ( ∂ I +1 t v, ∂ I t w ) + Q ( ∂ I t v, ∂ I +1 t w ) (cid:17) − e − t ( n X i =1 ∂ I t ∂ i v∂ I t ∂ i w ) − e − t ( n X i =1 X ˜ I +˜ I ≤ I − ∂ ˜ I t ∂ i v∂ ˜ I t ∂ i w )+ e − t n X i =1 X ˜ I +˜ I ≤ I − ∂ ˜ I +1 t ∂ i v∂ ˜ I t ∂ i w + n X i =1 X ˜ I +˜ I ≤ I − ∂ ˜ I t ∂ i v∂ ˜ I +1 t ∂ i w = X I + I = I C I ,I Q ( ∂ I t v, ∂ I t w ) + e − t ( n X i =1 X ˜ I +˜ I ≤ I − ∂ ˜ I t ∂ i v∂ ˜ I t ∂ i w ) (4.6)Thus, the lemma holds for arbitrary I and I .Now we prove Theorem 1.1. Proof of Theorem 1.1.
We prove the theorem by the following three steps.Step 1: Define F ( t ) = X I + | I |≤ N f | I | ,I ( t ) 0 ≤ t < T, (4.7)for N ≥ n + 4, where f | I | ,I ( t ) is defined by (3.24) whenever ϕ ∈ C ∞ ([0 , T ) × R n ) solves (1.8), (1.10) on[0 , T ) × R n for some T >
0. By (1.10), there exists a positive constant C depends only on the initialdata f , g and their derivatives such that F (0) ≤ C ǫ. (4.8)17hen by the pointwise decay estimates of last section, when N ≥ max {| J | + | J | − , | J | − } + ⌈ n + 1 ⌉ ,it holds that k ∂ J t D J ϕ k L ∞ ≤ Ce − t F ( t ) , when | J | ≥ k D J ϕ k L ∞ ≤ CF ( t ) , (4.10)where C is a constant coming from the Sobolev embedding theorem.Step 2: Energy estimatesSuppose | J | + M ≤ N , by Corollary 3.2, Lemmas 3.9, 4.2 and (3.36), we have ddt f | J | ,M ( t ) ≤ | e − n t Z Σ t e − t ( n X i =1 M − X K =0 C K ∂ Kt ∂ i D | J | ϕ ) ∂ M +1 t D | J | ϕdV ol t | + | e − n t Z Σ t X I
01 + I
02 = M | J | + | J | = | J | C M ,M ,J ,J Q ( ∂ M t D J ϕ, ∂ M t D J ϕ ) ∂ M +1 t D | J | ϕdV ol t | + | e − n t Z Σ t e − t ( n X i =1 X ˜ M
01 + ˜ M ≤ M − | J | + | J | = | J | C ˜ M , ˜ M ,J ,J ∂ ˜ M t ∂ i D J ϕ∂ ˜ M t ∂ i D J ϕ ) ∂ M +1 t D | J | ϕdV ol t |≤ e − t M − X K =0 | C K | ( f | J | +1 ,K ( t )) ( f | J | ,M ( t )) ! + X M
01 + M
02 = M | J | + | J | = | J | C M ,M ,J ,J ( k ∂ M +1 t D J ϕ k L ∞ + e − t k ∂ M t ∂ i D J ϕ k L ∞ ) × (cid:16) f | J | ,M ( t ) (cid:17) (cid:16) f | J | ,M ( t ) (cid:17) + n X i =1 X ˜ M
01 + ˜ M ≤ M − | J | + | J | = | J | C ˜ M , ˜ M ,J ,J e − t k ∂ ˜ M t ∂ i D J ϕ k L ∞ (cid:16) f | J | , ˜ M ( t ) (cid:17) (cid:16) f | J | ,M ( t ) (cid:17) (4.11)where we have assumed without loss of generality M + | J | ≤ M + | J | and ˜ M + | J | ≤ ˜ M + | J | Summing | J | and M , which satisfy | J | + M ≤ N , we easily get that ddt F ( t ) ≤ C N ( e − t + k ∂ M +1 t D J ϕ k L ∞ + e − t k ∂ M t ∂ i D J ϕ k L ∞ + e − t k ∂ ˜ M t ∂ i D J ϕ k L ∞ ) F ( t ) , (4.12)where M + | J | ≤ N M + | J | ≤ N − . By Lemma 3.9, if M + | J | + 1 + n + 1 ≤ N + n + 2 ≤ N , i.e., N ≥ n + 4, it holds that ddt F ( t ) ≤ C N ( e − t + e − t F ( t )) F ( t ) . (4.13)18tep 3: Boot-strapSet E = { t ∈ [0 , T ) : F ( s ) ≤ Aǫ for all 0 ≤ s ≤ t } . By (4.8), E is not empty. Since F ( t ) is continuous in t , E is relatively closed in [0 , T ). Thus, it sufficesto prove that E is relatively open such that the following holds.For any T , set E = [0 , T ) . In order to prove E is open, we fix t ∈ E with t < T . Since F ( t ) is continuous, there exists t > t such that F ( t ) ≤ Aǫ for 0 ≤ t ≤ t . (4.14)We shall prove F ( t ) ≤ Aǫ for 0 ≤ t ≤ t , (4.15)provided ǫ is sufficiently small. By (4.13), in the domain [0 , t ], we have ddt F ( t ) ≤ C N ( e − t + e − t (2 Aǫ ) ) F ( t ) . (4.16)Thus, we obtain F ( t ) ≤ F (0) e R t C N ( e − t + e − t (2 Aǫ ) ) dt ≤ C ǫe R ∞ C N ( e − t + e − t (2 Aǫ ) ) dt . (4.17)If ǫ is sufficiently small, then the following holds obviously e R ∞ C N ( e − t + e − t (2 Aǫ ) ) dt ≤ AC , (4.18)provided A is sufficiently large. Thus, we can get that the solution of the Cauchy problem (1.8), (1.10)exists globally by the standard continuity method. This completes the proof. Remark 4.1
From the above procedure, we can see that the null condition Q ( ϕ, ϕ ) plays a key role,especially the coefficients e − t before ϕ i . In fact, e − t − δ is enough for the global existence result of (1.8),provided that δ > . In next section, we will clarify the influence of this term. In this section, we consider the more complicated and representative case (1.9), which generalizes thetimelike extremal surface equation and is interesting in the fields of both mathematics and physics. Sincethe nonlinearity is higher than the components of (1.8), we must generalize the corresponding energyestimates. As before, we study the structure enjoyed by the nonlinear term after differentiated severaltimes by D . 19efine S ( I ) := ( I , I , I , I m , I m ) : | I | + | I | + | I | + | I m | + | I m | ≤ | I | max {| I | , | I |} ≤ | I | − , for 0 ≤ j ≤ | I | , ≤ m ≤ j . (5.1) Lemma 5.1
Differentiate the nonlinear term Q ( ϕ,e αt Q ( ϕ,ϕ ))2(1+ e αt Q ( ϕ,ϕ )) for | I | times by D , we have D I Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) = Q ( ϕ, e αt Q ( ϕ, D I ϕ ))(1 + e αt Q ( ϕ, ϕ ))+ | I | X j =0 X S ( I ) j Y m =0 G ( e αt Q ( ϕ, ϕ )) Q ( D I ϕ, e αt Q ( D I ϕ, D I ϕ )) e jαt Q ( D I m ϕ, D I m ϕ ) , (5.2) where G ( e αt Q ( ϕ, ϕ )) is a smooth function depending on e αt Q ( ϕ, ϕ ) and when m = 0 , the terms containingthe index m in the product terms do not appear.Proof. The proof is by induction on I and using Lemma 4.1 repeatedly.When | I | = 1, we have D Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ ))= Q ( ϕ, e αt Q ( ϕ, Dϕ ))(1 + e αt Q ( ϕ, ϕ )) + Q ( Dϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) − Q ( ϕ, e αt Q ( ϕ, ϕ )) e αt Q ( ϕ, Dϕ )2(1 + e αt Q ( ϕ, ϕ )) = Q ( ϕ, e αt Q ( ϕ, Dϕ ))(1 + e αt Q ( ϕ, ϕ )) + X j =0 X S (1) G ( e αt Q ( ϕ, ϕ )) Q ( D I ϕ, e αt Q ( D I ϕ, D I ϕ )) e jαt Q ( D I j ϕ, D I j ϕ ) . (5.3)Thus, the lemma holds for | I | = 1.Suppose that the lemma holds for | J | = | I | −
1, then for | I | D I Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) = D (cid:18) D I − Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) (cid:19) = D Q ( ϕ, e αt Q ( ϕ, D J ϕ ))(1 + e αt Q ( ϕ, ϕ ))+ D | J | X j =0 X S ( J ) j Y m =0 G ( e αt Q ( ϕ, ϕ )) Q ( D J ϕ, e αt Q ( D J ϕ, D J ϕ )) e jαt Q ( D J m ϕ, D J m ϕ ) = Q ( ϕ, e αt Q ( ϕ, D I ϕ ))1 + e αt Q ( ϕ, ϕ ) + Q ( Dϕ, e αt Q ( ϕ, D J ϕ ))1 + e αt Q ( ϕ, ϕ )+ Q ( ϕ, e αt Q ( Dϕ, D J ϕ ))1 + e αt Q ( ϕ, ϕ ) − Q ( ϕ, e αt Q ( ϕ, D J ϕ )) e αt Q ( ϕ, Dϕ )(1 + e αt Q ( ϕ, ϕ )) + | J | X j =0 X S ( J ) D j Y m =0 G ( e αt Q ( ϕ, ϕ )) Q ( D J ϕ, e αt Q ( D J ϕ, D J ϕ )) e jαt Q ( D J m ϕ, D J m ϕ ) ! = Q ( ϕ, e αt Q ( ϕ, D I ϕ ))1 + e αt Q ( ϕ, ϕ )+ | I | X j =0 X S ( I ) j Y m =0 G ( e αt Q ( ϕ, ϕ )) Q ( D I ϕ, e αt Q ( D I ϕ, D I ϕ )) e jαt Q ( D I m ϕ, D I m ϕ ) . (5.4)The last equality comes from the product role. Thus, the lemma is proved.For derivatives with respect to t , the following lemmas play key roles.20 emma 5.2 It holds that ∂ Jt Q ( u, e αt Q ( v, w )) = X J + J + J ≤ J C ( J , J , J ) Q ( ∂ J t u, e αt Q ( ∂ J t v, ∂ J t w ))+ n X i =1 X J + J + J ≤ J − C ( J , J , J ) Q ( ∂ J t u, e ( α − t ∂ J t v i ∂ J t w i )+ n X i =1 X J + J + J ≤ J − C ( J , J , J ) e ( α − t ∂ J t u i ∂ i (cid:16) Q ( ∂ J t v, ∂ J t w ) (cid:17) + n X i,j =1 X J + J + J ≤ J − C ( J , J , J ) e ( α − t ∂ J t u i ∂ J t v j ∂ J t w j , (5.5) where J, J i ( i = 1 , · · · , are non-negative integers and C ( J ) denotes a constant depending on J .Proof. For simplicity, we neglect the constants in the proof. By (4.4), it holds that ∂ Jt Q ( u, e αt Q ( v, w ))= X a + a = J Q (cid:0) ∂ a t u, ∂ a t ( e αt Q ( v, w )) (cid:1) + n X i =1 X a + a ≤ J − e − t ∂ a t u i ∂ a t ∂ i ( e αt Q ( v, w ))= X a + a = J Q ∂ a t u, X a + a = a ∂ a t ( e αt ) ∂ a t Q ( v, w ) ! + n X i =1 X a + a ≤ J − e − t ∂ a t u i X a + a = a ∂ a t ( e αt ) ∂ a t ∂ i Q ( v, w )= X a + a = J Q ∂ a t u, X a + a = a X a + a = a ∂ a t ( e αt ) Q ( ∂ a t v, ∂ a t w ) ! + X a + a = J Q ∂ a t u, n X i =1 X a + a ≤ a − X a + a = a ∂ a t ( e αt ) e − t ∂ a t v i ∂ a t w i + n X i =1 X a + a ≤ J − e − t ∂ a t u i X a + a = a ∂ a t ( e αt ) ∂ i X a + a = a Q ( ∂ a t v, ∂ t a w ) ! + n X i =1 X a + a ≤ J − e − t ∂ a t u i X a + a = a ∂ a t ( e αt ) ∂ i n X j =1 X a + a ≤ a − e − t ∂ a t v j ∂ a t w j . (5.6)Rearrange the indices a i ( i = 1 , · · · ,
16) of (5.6), the lemma holds.The following lemma can be derived by a simple induction.
Lemma 5.3
For J = 0 , ∂ Jt [ G ( v )] is a linear combination of terms [ D m G ]( v ) ∂ β t v∂ β t v · · · ∂ β m t v where ≤ m ≤ J, m X i =1 β m = J. (5.7)Define 21 ( I, J ) := ( β , · · · , β m , α , · · · , α m , ˜ α , · · · , ˜ α m , I , I , I , J , J , J ) :max {| I | + J , | I | + J } ≤ | I | + J − P mi =1 β i + | α i | + | ˜ α i | + | I | + | I | + | I | + J + J + J ≤ | I | + J , (5.8) S ( I, J ) := ( β , · · · , β m , α , · · · , α m , ˜ α , · · · , ˜ α m , I , I , I , J , J , J ) : P mi =1 β i + | α i | + | ˜ α i | + | I | + | I | + | I | + J + J + J ≤ | I | + J − (5.9)and S ( I, J ) := ( β , · · · , β m , α , · · · , α m , ˜ α , · · · , ˜ α m , I , I , I , J , J , J ) : P mi =1 β i + | α i | + | ˜ α i | + | I | + | I | + | I | + J + J + J ≤ | I | + J − . (5.10)Combing Lemmas 5.1-5.3, the following lemma holds obviously Lemma 5.4
For J ≥ and | I | ≥ , it holds that ∂ Jt D I Q ( ϕ, e αt Q ( ϕ, ϕ ))2(1 + e αt Q ( ϕ, ϕ )) = Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))(1 + e αt Q ( ϕ, ϕ ))+ X S ( I,J ) m Y i =1 G ( e αt Q ( ϕ, ϕ )) ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )) Q ( ∂ J t D I ϕ, e αt Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ))+ n X k =1 X S ( I,J ) m Y i =1 G ( e αt Q ( ϕ, ϕ )) ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )) Q ( ∂ J t D I ϕ, e ( α − t ∂ J t D I ϕ k ∂ J t D I ϕ k )+ n X k =1 X S ( I,J ) m Y i =1 G ( e αt Q ( ϕ, ϕ )) ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )) e ( α − t ( ∂ J t D I ϕ k ) ∂ k (cid:16) Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ) (cid:17) + n X k,j =1 X S ( I,J ) m Y i =1 G ( e αt Q ( ϕ, ϕ )) ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )) e ( α − t ( ∂ J t D I ϕ k )( ∂ J t D I ϕ j )( ∂ J t D I ϕ j ):= Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))(1 + e αt Q ( ϕ, ϕ )) + R, (5.11) where G ( e αt Q ( ϕ, ϕ )) denotes the set of smooth functions depending on e αt Q ( ϕ, ϕ ) and R stands for theremaining terms. Remark 5.1
The term Q ( ϕ,e αt Q ( ϕ,ϕ ))1+ e αt Q ( ϕ,ϕ ) contains the highest order derivatives. Remark 5.2
From now on, without loss of generality, we assume that α + | α | ≤ β + | β | whenever theyappear in Q ( ∂ α t D α ϕ, ∂ β t D β ϕ ) simultaneously. Now, we need to derive the energy inequality of the following equation (cid:3) g ( ∂ Jt D I ϕ ) − ( ∂ Jt D I ϕ ) t = e − t ( n X i =1 J − X M =0 C M ∂ Mt ∂ i D I ϕ )+ Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))1 + e αt Q ( ϕ, ϕ ) + R, (5.12)22here R is defined by (5.11).The following lemma will play a key role in the proof of Theorem 1.2. Lemma 5.5
The following generalized energy inequality holds in the existence domain of the solution ofCauchy problem (1.9)-(1.10) F ( t ) ≤ (cid:18) F (0) + Z t ( C ( e − τ + e ( α − τ F ( τ )) F ( τ ) dτ (cid:19) , (5.13) provided the initial data is sufficiently small and F ( t ) is defined by (4.7), C is a constant depend only on α and N .Proof. We will prove the lemma by three steps:Step 1: Energy estimates for (5.12).Taking the vector field V = − ∂ t and by Lemma 3.2, we have the following E | I | ,J ( t ) − E | I | ,J (0) = Z t Z Σ τ ( K V ( ∂ Jτ D I ϕ ) + (cid:3) g ( ∂ Jτ D I ϕ ) V ( ∂ Jτ D I ϕ )) dV ol τ dτ. (5.14)Which is equivalent to ddt E | I | ,J ( t ) = Z Σ t ( K V ( ∂ Jt D I ϕ ( t )) + (cid:3) g ( ∂ Jt D I ϕ ) V ( ∂ Jt D I ϕ )) dV ol t . (5.15)Then, as (4.11), we get ddt f | I | ,J ( t )= e − n t ddt E | I | ,J ( t ) + 2 − n e − n t E | I | ,J ( t )= e − n t Z Σ t (cid:0) K V ( ∂ Jt D I ϕ ( t )) + (cid:3) g ( ∂ Jt D I ϕ ) V ( ∂ Jt D I ϕ ) (cid:1) dV ol t + 2 − n e − n t E | I | ,J ( t )= e − n t Z Σ t e − t ( n X i =1 J − X M =0 C M ∂ Mt ∂ i D I ϕ ) + Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))1 + e αt Q ( ϕ, ϕ ) + R ! V ( ∂ Jt D I ϕ ) dV ol t + e − n t Z Σ t (cid:0) K V ( ∂ Jt D I ϕ ( t )) + ( ∂ Jt D I ϕ ) t V ( ∂ Jt D I ϕ ) (cid:1) dV ol t + 2 − n e − n t E | I | ,J ( t ) ≤ e − n t Z Σ t e − t ( n X i =1 J − X M =0 C M ∂ Mt ∂ i D I ϕ ) + Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))1 + e αt Q ( ϕ, ϕ ) + R ! V ( ∂ Jt D I ϕ ) dV ol t (5.16)The last inequality holds according to Lemma 3.4. By (5.16), we have to estimate the following integralterm containing the second order derivatives of ∂ Jt D I ϕ . Z R n Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))1 + e αt Q ( ϕ, ϕ ) V ( ∂ Jt D I ϕ ) e n t e − n t dx · · · dx n . (5.17)Step 2: Estimates for (5.17). 23y (1.6) Q ( ϕ, e αt Q ( ϕ, ∂ Jt D I ϕ ))= − ϕ t ∂ t " e αt − ϕ t ( ∂ Jt D I ϕ ) t + e − t n X i =1 ϕ i ( ∂ Jt D I ϕ ) i ! + e − t n X j =1 ϕ j ∂ j " e αt − ϕ t ( ∂ Jt D I ϕ ) t + e − t n X i =1 ϕ i ( ∂ Jt D I ϕ ) i ! = e αt ( ϕ t ) ( ∂ Jt D I ϕ ) tt + e ( α − t n X i,j =1 ϕ i ϕ j ( ∂ Jt D I ϕ ) ij − e ( α − t n X i =1 ϕ t ϕ i ( ∂ Jt D I ϕ ) ti + αe αt ( ϕ t ) ( ∂ Jt D I ϕ ) t + e αt ϕ t ϕ tt ( ∂ Jt D I ϕ ) t − n X i =1 ( α − e ( α − t ϕ t ϕ i ( ∂ Jt D I ϕ ) i − n X i =1 e ( α − t ϕ t ϕ ti ( ∂ Jt D I ϕ ) i − n X j =1 e ( α − t ϕ j ϕ tj ( ∂ Jt D I ϕ ) t + e ( α − t n X i,j =1 ϕ j ϕ ij ( ∂ Jt D I ϕ ) i := A + B + D + P, (5.18)where A = e αt ( ϕ t ) ( ∂ Jt D I ϕ ) tt , (5.19) B = e ( α − t n X i,j =1 ϕ i ϕ j ( ∂ Jt D I ϕ ) ij . (5.20) D = − e ( α − t n X i =1 ϕ t ϕ i ( ∂ Jt D I ϕ ) it (5.21)and P = αe αt ( ϕ t ) ( ∂ Jt D I ϕ ) t + e αt ϕ t ϕ tt ( ∂ Jt D I ϕ ) t − n X i =1 ( α − e ( α − t ϕ t ϕ i ( ∂ Jt D I ϕ ) i − n X i =1 e ( α − t ϕ t ϕ ti ( ∂ Jt D I ϕ ) i − n X j =1 e ( α − t ϕ j ϕ tj ( ∂ Jt D I ϕ ) t + e ( α − t n X i,j =1 ϕ j ϕ ij ( ∂ Jt D I ϕ ) i (5.22)Denote ∂ Jt D I ϕ by v , then by (5.17), (5.19) and integrating by parts Z R n A e αt Q ( ϕ, ϕ ) ( − v t ) e nt e − n t dx · · · dx n = Z R n e αt ( ϕ t ) ( v ) tt e αt Q ( ϕ, ϕ ) ( − v t ) e nt e − n t dx · · · dx n = − ddt Z R n e αt ϕ t e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n + A + A + A , (5.23)where A = 12 Z R n ( α + 1) e αt ϕ t e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n , (5.24) A = Z R n e αt ϕ t ϕ tt e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n (5.25)24nd A = − Z R n e αt ϕ t [ e αt Q ( ϕ, ϕ )] t (1 + e αt Q ( ϕ, ϕ )) v t e nt e − n t dx · · · dx n . (5.26)By (5.17), (5.20) and integrating by parts, it holds that Z R n B e αt Q ( ϕ, ϕ ) ( − v t ) e nt e − n t dx · · · dx n = Z R n e ( α − t ϕ i ϕ j v ij e αt Q ( ϕ, ϕ ) ( − v t ) e nt e − n t dx · · · dx n = 12 ddt n X i,j =1 Z R n e ( α − t ϕ i ϕ j e αt Q ( ϕ, ϕ ) v i v j e nt e − n t dx · · · dx n + X κ =1 B κ , (5.27)where B = − n X i,j =1 Z R n ( α − e ( α − t ϕ i ϕ j e αt Q ( ϕ, ϕ ) v i v j e nt e − n t dx · · · dx n , (5.28) B = − n X i,j =1 Z R n e ( α − t ϕ it ϕ j e αt Q ( ϕ, ϕ ) v i v j e nt e − n t dx · · · dx n , (5.29) B = 12 n X i,j =1 Z R n e ( α − t ϕ i ϕ j [ e αt Q ( ϕ, ϕ )] t (1 + e αt Q ( ϕ, ϕ )) v i v j e nt e − n t dx · · · dx n , (5.30) B = n X i,j =1 Z R n e ( α − t ϕ ij ϕ j e αt Q ( ϕ, ϕ ) v i v t e nt e − n t dx · · · dx n , (5.31) B = n X i,j =1 Z R n e ( α − t ϕ i ϕ jj e αt Q ( ϕ, ϕ ) v i v t e nt e − n t dx · · · dx n , (5.32)and B = − n X i,j =1 Z R n e ( α − t ϕ i ϕ j [ e αt Q ( ϕ, ϕ )] j (1 + e αt Q ( ϕ, ϕ )) v i v t e nt e − n t dx · · · dx n . (5.33)By (5.17), (5.21) and integrating by parts, we have Z R n D e αt Q ( ϕ, ϕ ) ( − v t ) e nt e − n t dx · · · dx n = n X i =1 Z R n e ( α − t ϕ t ϕ i e αt Q ( ϕ, ϕ ) ( v it v t ) e nt e − n t dx · · · dx n := D + D + D , (5.34)where D = − n X i =1 Z R n e ( α − t ϕ ti ϕ i e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n . (5.35) D = − n X i =1 Z R n e ( α − t ϕ t ϕ ii e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n (5.36)and D = n X i =1 Z R n e ( α − t ϕ t ϕ i [ e αt Q ( ϕ, ϕ )] i (1 + e αt Q ( ϕ, ϕ )) v t e nt e − n t dx · · · dx n (5.37)25ow, suppose that N ≥ n + 6 and F ( t ) defined by (4.7) is small enough. According to Lemma 3.7 andRemark 3.2, we easily obtain for any J + | I | ≤ N | e αt Q ( ϕ, ϕ ) | = | e αt ( − ϕ t + n X i =1 e − t ϕ i ) | ≤ Ce ( α − t F ( t ) (5.38)and | ∂ t ( e αt Q ( ϕ, ϕ )) | ≤ | αe αt Q ( ϕ, ϕ ) | + | e αt Q ( ∂ t ϕ, ϕ ) | + n X i =1 | e ( α − t ϕ i | ≤ Ce ( α − t F ( t ) (5.39)then 12 ≤ e αt Q ( ϕ, ϕ ) ≤ , (5.40) | A | ≤ k ( α + 1) e αt ϕ t e αt Q ( ϕ, ϕ ) k L ∞ F ( t ) ≤ Ce ( α − t F ( t ) F ( t ) . (5.41) | A | ≤ k e αt ϕ t ϕ tt e αt Q ( ϕ, ϕ ) k L ∞ F ( t ) ≤ Ce ( α − t F ( t ) F ( t ) . (5.42) | A | ≤ k e αt ϕ t [ e αt Q ( ϕ, ϕ )] t (1 + e αt Q ( ϕ, ϕ )) k L ∞ F ( t ) ≤ Ce α − t F ( t ) F ( t ) . (5.43)Similarly, we have | B | ≤ k ( α − e ( α − t ϕ i ϕ j e αt Q ( ϕ, ϕ ) k L ∞ F ( t ) ≤ Ce ( α − t F ( t ) F ( t ) . (5.44) | B | ≤ Ce ( α − ) t F ( t ) F ( t ) , | B | ≤ Ce α − t F ( t ) F ( t ) . (5.45) | B | ≤ Ce ( α − ) t F ( t ) F ( t ) F ( t ) , | B | ≤ Ce ( α − ) t F ( t ) F ( t ) F ( t ) . (5.46)and | B | ≤ Ce (2 α − ) t F ( t ) F ( t ) F ( t ) . (5.47)For D i ( i = 1 , , | D i | ≤ Ce ( α − ) t F ( t ) F ( t ) i = 1 , | D | ≤ Ce (2 α − ) t F ( t ) F ( t ) . (5.49)At last, we estimate the term Z R n P ( − v t ) e nt e − n t dx · · · dx n := X κ =1 P κ , (5.50)where P κ is defined orderly by the six parts of P . As before, we have | P | ≤ Ce ( α − t F ( t ) F ( t ) , | P | ≤ Ce ( α − t F ( t ) F ( t ) . (5.51) | P | ≤ Ce ( α − t F ( t ) F ( t ) F ( t ) , | P | ≤ Ce ( α − ) t F ( t ) F ( t ) F ( t ) (5.52)26nd | P | ≤ Ce ( α − ) t F ( t ) F ( t ) , | P | ≤ Ce ( α − t F ( t ) F ( t ) . (5.53)In the last step, we estimate the remaining term Z R n R ( − v t ) e − n t e nt dx · · · dx n , (5.54)where R is defined by (5.11).Step 3: Estimates for (5.54).Before estimating (5.54), we expand every terms of R .For Q (cid:16) ∂ J t D I ϕ, e αt Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ) (cid:17) , it holds that Q (cid:16) ∂ J t D I ϕ, e αt Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ) (cid:17) = αe αt ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) t + e αt ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) tt ( D I ϕ ) t + e αt ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) tt − n X i =1 ( α − e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) i ( ∂ J t D I ϕ ) i − n X i =1 e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) it ( ∂ J t D I ϕ ) i − n X i =1 e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) i ( ∂ J t D I ϕ ) it − n X j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ ) tj ( ∂ J t D I ϕ ) t − n X j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ ) tj + n X i,j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ ) ij ( ∂ J t D I ϕ ) i + n X i,j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ ) i ( ∂ J t D I ϕ ) ij := X λ =1 H λ , (5.55)where H λ ( λ = 1 , · · · ,
10) are defined orderly.For Q ( ∂ J t D I ϕ, e ( α − t ∂ J t D I ϕ i ∂ J t D I ϕ i ), it holds that Q ( ∂ J t D I ϕ, e ( α − t ∂ J t D I ϕ i ∂ J t D I ϕ i )= − ( α − e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ i ) − e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ i ) t ( ∂ J t D I ϕ i ) − e ( α − t ( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ i ) t + n X j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ i ) j ( ∂ J t D I ϕ i )+ n X j =1 e ( α − t ( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ i )( ∂ J t D I ϕ i ) j := X λ =1 O λ . (5.56)27or e ( α − t ( ∂ J t D I ϕ i ) ∂ i (cid:16) Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ) (cid:17) , it holds that e ( α − t ( ∂ J t D I ϕ i ) ∂ i (cid:16) Q ( ∂ J t D I ϕ, ∂ J t D I ϕ ) (cid:17) = − e ( α − t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ i ) t ( ∂ J t D I ϕ ) t − e ( α − t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ ) t ( ∂ J t D I ϕ i ) t + n X j =1 e ( α − t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ i ) j ( ∂ J t D I ϕ ) j + n X j =1 e ( α − t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ ) j ( ∂ J t D I ϕ i ) j := X λ =1 Q λ . (5.57)Denote X = e ( α − t ( ∂ J t D I ϕ i )( ∂ J t D I ϕ j )( ∂ J t D I ϕ j ) . (5.58)At last, for ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )), it holds that ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ ))= e αt X β i + β i ≤ β i ∂ β i t ( D α i ϕ ) t ∂ β i t ( D ˜ α i ϕ ) t + e ( α − t n X j =1 X β i + β i ≤ β i ∂ β i t ( D α i ϕ ) j ∂ β i t ( D ˜ α i ϕ ) j := X λ =1 Y λ , (5.59)where we have omit the constant coefficients, which do not affect the main result. We estimate (5.54) bythe following four cases according to the index.Case I: when P mi =1 β i + α i + ˜ α i = 0, we have Z R n R ( − v t ) e − n t e nt dx · · · dx n ≤ Z R n | G (cid:0) e αt Q ( ϕ, ϕ ) (cid:1) | ( X λ =1 X S ( I,J ) | H λ | + X λ =1 X S ( I,J ) | O λ | ) e − n t e nt dx · · · dx n + Z R n | G (cid:0) e αt Q ( ϕ, ϕ ) (cid:1) ( X λ =1 X S ( I,J ) | Q λ | + X S ( I,J ) | X | ) e − n t e nt dx · · · dx n (5.60)We will use the following principle to estimate these product terms here and hereafter. Principle : Since max { J + | I | , J + | I |} ≤ | I | + J − P i =1 ( | I i | + J i ) ≤ | I | + J , there mustbe at most one term that exceeds | I | , we use L norm to control this term and use L ∞ norm to controlother terms.According the above principle and step 2, when | I | + J ≥ | I | + J , by Lemma 3.9, since N + 2 + n + 1 ≤ N , it holds that Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − t F ( t ) F ( t ) i = 1 , · · · , R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) F ( t ) i = 5 , Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) F ( t ) F ( t ) i = 7 , · · · ,
10 (5.63) Z R n | G ( e αt Q ( ϕ, ϕ )) O i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − t F ( t ) i = 1 (5.64) Z R n | G ( e αt Q ( ϕ, ϕ )) O i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) i = 2 , , , Z R n | G ( e αt Q ( ϕ, ϕ )) Q i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) i = 1 , , , Z R n | G ( e αt Q ( ϕ, ϕ )) Xv t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) . (5.67)When | I | + J ≥ | I | + J , we have Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − t F ( t ) F ( t ) i = 1 , · · · , Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − t F ( t ) F ( t ) F ( t ) i = 4 (5.69) Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) F ( t ) F ( t ) i = 5 , ,
10 (5.70) Z R n | G ( e αt Q ( ϕ, ϕ )) H i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) F ( t ) i = 6 , , Z R n | G ( e αt Q ( ϕ, ϕ )) O i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − t F ( t ) i = 1 (5.72) Z R n | G ( e αt Q ( ϕ, ϕ )) O i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) i = 2 , , , Z R n | G ( e αt Q ( ϕ, ϕ )) Q i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) i = 1 , , , Z R n | G ( e αt Q ( ϕ, ϕ )) Xv t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) (5.75)In the above calculations, we have used the fact that G ( e αt Q ( ϕ, ϕ )) is uniformly bounded, provided F ( t ) is sufficiently small and the uniform constant C depends only on α and N . Thus, by (5.61)-(5.75),we conclude that in this case Z R n R ( − v t ) e − n t e nt dx · · · dx n ≤ Ce ( α − t F ( t ) . (5.76)Case II: when P mi =1 β i + α i + ˜ α i = j >
0, and max {| I | + J , | I | + J , | I | + J } ≥ | I | + J .In this case, we have to do some additional estimates on the Y λ terms by the L ∞ norm and get someadditional decay. By direct calculations, we have | Y λ | ≤ Ce ( α − t F ( t ) . (5.77)29hen, by (5.76) and (5.77), we conclude that Z R n R ( − v t ) e − n t e nt dx · · · dx n ≤ Ce α − t F ( t ) . (5.78)Case III: when P mi =1 β i + α i + ˜ α i = j >
0, there exists some i such that | α i | + | ˜ α i | + β i ≥ | I | + J .In this case, we use the principle of Case I, we estimate (5.55)-(5.58) by L ∞ norm, by direct calculations,we have | H i | ≤ Ce ( α − ) t F ( t ) , i = 1 , · · · , | H i | ≤ Ce ( α − t F ( t ) , i = 5 , · · · ,
10 (5.80) | O i | ≤ Ce ( α − ) t F ( t ) , i = 1 (5.81) | O i | ≤ Ce ( α − t F ( t ) , i = 2 , , , | Q i | ≤ Ce ( α − t F ( t ) , i = 1 , , , | X | ≤ Ce ( α − t F ( t ) . (5.84)For the term ∂ β i t ( e αt Q ( D α i ϕ, D ˜ α i ϕ )), where | α i | + β i + | ˜ α i | ≥ | I | + J , we have Z R n | Y i v t | e nt e − n t dx · · · dx n ≤ Ce ( α − ) t F ( t ) F ( t ) . (5.85)Then, by (5.79)-(5.85), we conclude that in this case, we have Z R n R ( − v t ) e − n t e nt dx · · · dx n ≤ Ce α − t F ( t ) . (5.86)Case IV: max {| I | + J , | I | + J , | I | + J , β i + | α i | + | ˜ α i |} ≤ | I | + J .This case is easy to deal with, we can estimate it by Case II and Case III.Combing Cases I-IV, it holds that Z R n R ( − v t ) e − n t e nt dx · · · dx n ≤ Ce ( α − t F ( t ) (5.87)provided F ( t ) is small enough.By the above three steps and (4.13), summing all | I | , J satisfying | I | + J ≤ N , we have ddt ( F ( t ) + 12 Z R n e αt ϕ t e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n − n X i,j =1 Z R n e ( α − t ϕ i ϕ j e αt Q ( ϕ, ϕ ) v i v j e nt e − n t dx · · · dx n ) ≤ C N ( e − t + e ( α − t F ( t )) F ( t ) . (5.88)The following holds by direct calculations | Z R n e αt ϕ t e αt Q ( ϕ, ϕ ) v t e nt e − n t dx · · · dx n | ≤ Ce ( α − t F ( t ) ≤ F ( t ) (5.89)30nd | n X i,j =1 Z R n e ( α − t ϕ i ϕ j e αt Q ( ϕ, ϕ ) v i v j e nt e − n t dx · · · dx n ) | ≤ Ce ( α − t F ( t ) ≤ F ( t ) , (5.90)provided F ( t ) is sufficiently small. By (5.88)-(5.96), we obtain F ( t ) ≤ (cid:18) F (0) + Z t ( C N ( e − τ + e ( α − τ F ( τ )) F ( τ ) dτ (cid:19) . Thus, the lemma is proved.
Proof of Theorem 1.2.
As in the last section, what we have to do is to close the boot-strap assumption in the existence domain[0 , T ). Define E ( t ) as Section 4, we will prove that for any t ∈ [0 , T ), F ( t ) ≤ Aǫ implies F ( t ) ≤ Aǫ ,provided ǫ is small enough. By (4.4), Lemma 5.2 and Gronwall’s lemma, we have F ( t ) ≤ F (0) e R t C ( e − τ + e ( α − τ (2 Aǫ )) dτ ≤ C ǫ e R t C N ( e − τ + e ( α − τ (2 Aǫ )) dτ . (5.91)By (5.91), we have to choose ǫ sufficiently small such that e R t C N ( e − τ + e ( α − τ (2 Aǫ )) dτ ≤ A C . (5.92)When α <
1, we have e R t C N ( e − τ + e ( α − τ (2 Aǫ )) dτ ≤ e R ∞ C N ( e − τ + e ( α − τ (2 Aǫ )) dτ ≤ A C (5.93)Provided ǫ ≤ − α A and A ≥ C e CN is sufficiently large.When α = 1, by (5.92), we have T ( ǫ ) = ln( C )2 AC N ǫ , (5.94)where C = A C e CN is a positive constant. Combining (5.93) and (5.94), Theorem 1.2 holds. Remark 5.3
From the procedure we used to prove the main Theorem 1.2, we see that the role played bythe term e αt is important on the lifespan of the solution in curved spacetime. In this paper, we use the method of vector fields to prove the well-posedness of nonlinear wave equations ina curved spacetime. By this method, we get the exponential decay for the spacetime derivatives, which isimportant to the nonlinear problems. Since de Sitter spacetime is a special case of the Robertson-Walkerspacetime, whose metric has the following form ds = − dt + a ( t )( n X i =1 dx i ) , (6.1)31here a ( t ) is an appropriate factor, we can get the similar results for the general Robertson-Walkerspacetime depending on the choice of a ( t ).Since de Sitter spacetime is a special Lorentz spacetime, which is conformally flat. There maybe somemore decay properties should be explored to improve the results of the present paper, such as conformalinequality, Morawetz inequality and so on, which have played a vital role in the Minkowski spacetime R n . We will discuss these inequalities in our forthcoming paper.At last, it is interesting to study the wave equation satisfying null condition in the domain containingthe black hole region on Schwarzschild-de Sitter spacetime and this maybe give a clue to the study ofthe stability of this spacetime as solutions to the Einstein equations, which is still an open problem ingravitational physics. Acknowledgements.
This work was supported in part by the NNSF of China (Grant Nos.:11271323,91330105) and the Zhejiang Provincial Natural Science Foundation of China (Grant No.: LZ13A010002).
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