Boundary value problems in nonsmooth domains have been studied in differential aspects. Up to now, elliptic boundary value problems in domains with point singularities have been thoroughly investigated (see, e.g, 1 2 and the extensive bibliography in this book). We are concerned with initial boundary value problems for hyperbolic equations and systems in domains with conical points. These problems with the Dirichlet boundary conditions were investigated in 3 4 5 in which the unique existence, the regularity and the asymptotic behaviour near the conical points of the solutions are established. The Neumann boundary problem for general second-order hyperbolic systems with the coefficients independent of time in domains with conical points was studied in 6 . In the present paper we consider the Cauchy-Neumann (the second initial) boundary value problem for higher-order strongly hyperbolic systems in domains with conical points.

Our paper is organized as follows. Section 2 is devoted to some notations and the formulation of the problem. In Section 3 we present the results on the unique existence and the regularity in time of the generalized solution. The global regularity of the solution is dealt with in Section 4.

Let Ω be a bounded domain in ℝ ⁿ , n ≥ 2, with the boundary ∂Ω. We suppose that ∂Ω is an infinitely differentiable surface everywhere except the origin, in a neighborhood of which Ω coincides with the cone K = {x : x/|x| ∈ G}, where G is a smooth domain on the unit sphere S ^n-1. For each t, 0 < t ≤ ∞, denote Q_t = Ω × (0, t), Ω _t = Ω × {t}. Especially, we set Q = Q _∞, Γ = ∂Ω\{0}, S = Γ × [0, +∞).

For each multi-index p = (p ₁,..., p_n ) ∈ ℕ ⁿ , we use notations |p| = p ₁ + ... + p_n , D p = ∂ | p | ∂ x 1 p 1 … ∂ x n p n . For a complex-valued vector function u = (u ₁ ,..., u_s ) defined on Q, we denote D u p = ( D u 1 p , ... , D u s p ) , u t j = ∂ j u ∂ t j = ( ∂ j u 1 ∂ t j , … , ∂ j u s ∂ t j ) , | u | = ( ∑ j = 1 s | u j | 2 ) 1 2 . .

Let us introduce the following functional spaces used in this paper. Let l denote a nonnegative integer.

H^l (Ω) - the usual Sobolev space of vector functions u defined in Ω with the norm

u H l ( Ω ) = ∫ Ω ∑ | p | ≤ l | D p u | 2 d x 1 2 < ∞ .

H l - 1 2 ( Γ ) - the space of traces of vector functions from H^l (Ω) on Γ with the norm

u H l - 1 2 ( Γ ) = i n f v H l ( Ω ) : v ∈ H l ( Ω ) , v | Γ = u .

H ^l,0 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

u H l , 0 ( Q , γ ) = ∫ Q ∑ | p | ≤ l | D p u | 2 e - 2 γ t d x d t 1 2 < ∞ .

Especially, we set L ₂(Q, γ) = H ^0,0(Q, γ).

H ^l,1 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

u H l , 1 ( Q , γ ) = ∫ Q ∑ | p | ≤ l | D p u | 2 + | u t | 2 e - 2 γ t d x d t 1 2 < ∞ .

V 2 , α l ( Ω ) - the closure of C 0 ∞ ( Ω ¯ \ { 0 } ) with respect to the norm

u V 2 , α l ( Ω ) = ∑ | p | ≤ l ∫ Ω r 2 ( α + | p | - l ) | D p u | 2 d x 1 2 ,

where r = | x | = ∑ k = 1 n x k 2 1 2 .

H α l ( Ω ) ( α ∈ ℝ ) -the weighted Sobolev space of vector functions u defined in Ω with the norm

u H α l ( Ω ) = ∑ | p | ≤ l ∫ Ω r 2 α | D p u | 2 d x 1 2 .

If l ≥ 1, then V α l - 1 2 ( Γ ) , H α l - 1 2 ( Γ ) denote the spaces consisting of traces of functions from respective spaces V 2 , α l ( Ω ) , H α l ( Ω ) on the boundary Γ with the respective norms

u V α l - 1 2 ( Γ ) = i n f v V 2 , α l ( Ω ) : v ∈ V 2 , α l ( Ω ) , v | Γ = u , (1) u H α l - 1 2 ( Γ ) = i n f v H α l ( Ω ) : v ∈ H α l ( Ω ) , v | Γ = u . (2) (3)

H α l , 1 ( Q , γ ) ( α , γ ∈ ℝ ) - the weighted Sobolev space of vector functions u defined in Q with the norm

u H α l , 1 ( Q , γ ) = ∫ Q ∑ | p | ≤ l r 2 α | D p u | 2 + | u t | 2 e - 2 γ t d x d t 1 2 < ∞ .

From the definitions it follows the continuous imbeddings

V 2 , α l ( Ω ) ⊂ H α l ( Ω )

and

V 2 , α + k l + k ( Ω ) ⊂ V 2 , α l ( Ω )

for arbitrary nonnegative integers l, k and real number α. It is also well known (see [ 2 , Th. 7.1.1]) that if α < - n 2 or α > l - n 2 then

V 2 , α l ( Ω ) ≡ H α l ( Ω )

with the norms being equivalent.

Now we introduce the differential operator

L u = L ( x , t , D ) u = ∑ | p | , | q | ≤ m ( - 1 ) | p | D p ( a p q D q u ) ,

where a_pq = a_pq (x, t) are the s × s matrices with the bounded complex-valued components in Q ¯ . We assume that a p q = ( − 1 ) | p | + | q | a q p * for all |p|, |q| ≤ m, where a q p * is the transposed conjugate matrix to a_pq . This means the differential operator L is formally self-adjoint. We assume further that there exists a positive constant μ such that

∑ | p | = | q | = m a p q ( x , t ) η q η p ¯ ≥ μ ∑ | p | = m η p 2

for all η_p ∈ℂ ^s , |p| = m, and all ( x , t ) ∈ Q ̄ .

Let v be the unit exterior normal to S. It is well known that (see, e.g., [ 7 , Th. 9.47]) there are boundary operators N_j = N_j (x, t, D), j = 1, 2,..., m on S such that integration equality

∫ Ω L u v ̄ d x = ∑ | p | , | q | ≤ m ∫ Ω a p q D q u D p v ¯ d x + ∑ j = 1 m ∫ ∂ Ω N j u ∂ j - 1 v ¯ ∂ ν j - 1 d s

holds for all u , v ∈ C ∞ ( Ω ¯ ) and for all t ∈ [0, ∞). The order of the operator N_j is 2m - j for j = 1, 2,..., m.

In this paper, we consider the following problem:

u t t + L u = f in Q ,

N j u = 0 on S , j = 1 , … , m ,

u | t = 0 = u t | t = 0 = 0 on Ω .

A complex vector-valued function u ∈ H ^m,1(Q, γ) is called a generalized solution of problem (2.6)-(2.8) if and only if u| _{t = 0} = 0 and the equality

∫ Q u t η ̄ t d x d t + ∑ | p | , | q | ≤ m ∫ Q a p q D q u D p η ¯ d x d t = ∫ Q f η ̄ d x d t

holds for all η(x, t) ∈ H ^m,1(Q) satisfying η(x, t) = 0 for all t ≥ T for some positive real number T.

First, we introduce some notations which will be used in the proof of Theorems 3.3 and 3.4. For each vector function u,v defined in Ω and each nonnegative integer k,

| u | k , Ω = ∫ Ω ∑ | p | = k | D p u | 2 d x 1 2 , ( u , v ) Ω = ∫ Ω u v ̄ d x .

For vector functions u and v defined in Q and τ > 0, we set

| u | k , Q τ = ( ∫ 0 τ | u ( · , t ) | k , Ω 2   d t ) 1 2 ,       | u | k , Ω τ = | u ( · , τ ) | k , Ω ,       ( u , v ) Ω τ = ( u ( · , τ ) , v ( · , τ ) ) Ω ,

B t k ( t , u , v ) = ∑ | p | , | q | ≤ m ∫ Ω ∂ k a p q ∂ t k ( ⋅ , t ) D q u ( ⋅ , t ) D p v ( ⋅ , t ) ¯ d x , B t k τ ( u , v ) = ∫ 0 τ B t k ( t , u , v ) d t .

Especially, we set

B ( t , u , v ) = B t 0 ( t , u , v ) and B τ ( u , v ) = B t 0 τ ( u , v ) .

From the formally self-adjointness of the operator L, we see that

B ( τ , u , v ) ¯ = B ( τ , v , u ) .

Next, we introduce the following Gronwall-Bellman and interpolation inequalities as two fundamental tools to establish the theorems on the unique existence and the regularity in time.

Lemma 3.1 ( 8 , Lemma 3.1) Assume u, α, β are real-valued continuous on an interval [a, b], β is nonnegative and integrable on [a, b], α is nondecreasing satisfying

u ( τ ) ≤ α ( τ ) + ∫ a τ β ( t ) u ( t ) d t f o r a l l a ≤ τ ≤ b .

Then

u ( τ ) ≤ α ( τ ) exp ∫ a τ β ( t ) d t f o r   a l l a ≤ τ ≤ b .

From [ 9 , Th. 4.14], we have the following lemma.

Lemma 3.2. For each positive real number ε and each integer j, 0 < j < m, there exists a positive real number C = C (Ω, m, ε) which is dependent on only Ω, m and ε such that the inequality

| u | j , Ω 2 ≤ ε | u | m , Ω 2 + C | u | 0 , Ω 2

holds for all u ∈ H^m (Ω).

Now we state and prove the main theorems of this section.

Theorem 3.3. Let h be a nonnegative integer. Assume that all the coefficients a_pq together with their derivatives with respect to t are bounded on Q ¯ . Then there exists a positive real number γ ₀ such that for each γ > γ ₀, if f ∈ L ₂(Q, σ) for some nonnegative real number σ, the problem (2.6)-(2.8) has a unique generalized solution u in the space H ^m,1(Q, γ + σ) and

u H m , 1 ( Q , γ + σ ) 2 ≤ C f L 2 ( Q , σ ) 2 ,

where C is a constant independent of u and f.

Proof. The uniqueness is proved by similar way as in [ 4 , Th. 3.2]. We omit the detail here. Now we prove the existence by Galerkin approximating method. Suppose { φ k } k = 1 ∞ is an orthogonal basis of H^m (Ω) which is orthonormal in L ₂(Ω). Put

u N ( x , t ) = ∑ k = 1 N c k N ( t ) φ k ( x ) ,

where ( c k N ( t ) ) k = 1 N are the solution of the system of the following ordinary differential equations of second order:

( u t t N , φ l ) Ω t + B ( t , u N , φ l ) = ( f , φ l ) Ω t ,       l = 1 , … , N ,

with the initial conditions

c k N ( 0 ) = 0 , d c k N d t ( 0 ) = 0 , k = 1 , … , N .

Let us multiply (3.5) by d c k N ( t ) d t , take the sum with respect to l from 1 to N, and integrate the obtained equality with respect to t from 0 to τ (0 < τ < ∞) to receive

( u t t N , u t N ) Ω t + B ( t , u N , u t N ) = ( f , u t N ) Ω t .

Now adding this equality to its complex conjugate, then using (3.1) and the integration by parts, we obtain

| u t N | 0 , Ω τ 2 + B ( τ , u N , u N ) = B t τ ( u N , u N ) + 2 Re ( f , u t N ) Q τ .

With noting that, for some positive real number ρ,

ρ | u N | 0 , Ω τ 2 = 2 Re ρ ( u N , u t N ) Q τ ,

we can rewrite (3.8) as follows

| u t N | 0 , Ω τ 2 + B 0 ( τ , u N , u N ) + ρ | u | 0 , Ω τ 2 = B t τ ( u N , u N ) - ∑ | p | , | q | ≤ m | p | + | q | < 2 m - 1 ∫ Ω τ a p q D q u N D p u N ¯ d x + 2 Re ρ ( u N , u t N ) Q τ + 2 Re ( f , u t N ) Q τ .

By (2.4), the left-hand side of (3.9) is greater than

| u t N ( ⋅ , τ ) | 0 , Ω 2 + μ | u N ( ⋅ , τ ) | m , Ω 2 + ρ u ( ⋅ , τ ) 0 , Ω 2 .

We denote by I, II, III, IV the terms from the first, second, third, and forth, respectively, of the right-hand sides of (3.9). We will give estimations for these terms. Firstly, we separate I into two terms

∑ | p | = | q | = m ∫ Q τ ∂ a p q ∂ t D q u N D p u N ¯ d x d t + ∑ |p|,|q|≤m | p | + | q | ≤ 2 m - 1 ∫ Q τ ∂ a p q ∂ t D q u N D p u N ¯ d x d t ≡ I 1 + I 2 .

Put

μ 1 = sup { | ∂ a p q ∂ t ( x , t ) | : p = q = m , ( x , t ) ∈ Q ¯ } and m ′ = ∑ | p | = m 1 .

Then, by the Cauchy inequality, we have

I 1 ≤ μ 1 ∑ | p | = | q | = m 1 2 ( | D q u N | 0 , Q τ 2 + | D p u N | 0 , Q τ 2 ) ≤ m ′ μ 1 | u N | m , Q τ 2 .

By the Cauchy inequality and the interpolation inequality (3.3), for an arbitrary positive number ε ₁, we have

I 2 ≤ ε 1 | u N | m , Q τ 2 + C 1 | u N | 0 , Q τ 2 ,

where C ₁ = C ₁(ε ₁) is a nonnegative constant independent of u^N , f and τ. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number ε ₂ with ε ₂ < μ, it holds that

I I ≤ ε 2 | u N ( ⋅ , τ ) | m , Ω 2 + C 2 | u N ( ⋅ , τ ) | 0 , Ω 2 ,

where C ₂ = C ₂(ε ₂) is a nonnegative constant independent of u^N , f and τ. For the terms III and IV, by the Cauchy inequality, we have

I I I ≤ ( μ - ε 2 ) ρ 2 m ′ μ 1 + ε 1 | u N | 0 , Q τ 2 + m ′ μ 1 + ε 1 μ - ε 2 | u t N | 0 , Q τ 2 ,

and

I V ≤ ε 3 | u t N | 0 , Q τ 2 + 1 ε 3 | f | 0 , Q τ 2 ,

where ε ₃ > 0, arbitrary. Combining the above estimations we get from (3.9) that

| u t N ( ⋅ , τ ) | 0 , Ω 2 + ( μ - ε 2 ) | u N ( ⋅ , τ ) | m , Ω 2 + ( ρ - C 2 ) | u N ( ⋅ , τ ) | 0 , Ω 2 ≤ ( m ′ μ 1 + ε 1 ) | u N | m , Q τ 2 + C 1 + ( μ - ε 2 ) ρ 2 m ′ μ 1 + ε 1 | u N | 0 , Q τ 2 + m ′ μ 1 + ε 1 μ - ε 2 + ε 3 | u t N | 0 , Q τ 2 + 1 ε 3 | f | 0 , Q τ 2 .

Now fix ε ₁, ε ₂ and consider the function

g ( ρ ) = C 1 + ( μ - ε 2 ) ρ 2 m ′ μ 1 + ε 1 ρ - C 2 for ρ > C 2 .

We have

d g d ρ = ρ 2 - 2 C 2 ρ - C 1 A A ( ρ - C 2 ) 2 with A = ( μ - ε 2 ) ρ 2 m ′ μ 1 + ε 1 .

We see that the function g has a unique minimum at

ρ 0 = ρ 0 ( ε 1 , ε 2 ) = C 2 + C 2 2 + C 1 A .

We put

γ 0 = 1 2     inf ε 1 > 0 0 < ε 2 < μ   max { m ′ μ 1 + ε 1 μ − ε 2 , g ( ρ 0 ) } .

Now we take real numbers γ, γ ₁ arbitrarily satisfying γ ₀ < γ ₁ < γ. Then there are positive real numbers ε ₁, ε ₂, (ε ₂ < μ), ρ (ρ > C ₂(ε ₁, ε ₂)) and ε ₃ such that

m ′ μ 1 + ε 1 μ - ε 2 + ε 3 < 2 γ 1 and C 1 ( ε 1 , ε 2 ) + ( μ - ε 2 ) ρ 2 m ′ μ 1 + ε 1 ρ - C 2 ( ε 1 , ε 2 ) < 2 γ 1 .

From now to the end of the present proof, we fix such constants ε ₁, ε ₂, ε ₃ and ρ. Let | | | u N ( ⋅ , τ ) Ω 2 | | | stand for the left-hand side of (3.10). It follows from (3.10) and (3.12) that

| | | u N ( ⋅ , τ ) | | | Ω 2 ≤ 2 γ 1 ∫ 0 τ | | | u ( ⋅ , t ) | | | Ω 2 d t + C ∫ 0 τ | f ( ⋅ , t ) | 0 , Ω 2 d t for all τ ≤ 0 ,

where C = 1 ε 3 . By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that

| | | u N ( ⋅ , τ ) | | | Ω 2 ≤ C e 2 γ 1 τ ∫ 0 τ | f ( ⋅ , t ) | 0 , Ω 2 d t for all τ ≥ 0 .

We see that

∫ 0 τ | f ( ⋅ , t ) | 0 , Ω 2 d t = e 2 σ τ ∫ 0 τ | e - σ τ f ( ⋅ , t ) | 0 , Ω 2 d t ≤ e 2 σ τ ∫ 0 τ | e - σ t f ( ⋅ , t ) | 0 , Ω 2 d t .

Hence, it follows from (3.14) that

| | | u N ( ⋅ , τ ) | | | Ω 2 ≤ C e 2 ( γ 1 + σ ) τ ∫ 0 τ | e - σ t f ( ⋅ , t ) | 0 , Ω 2 d t ≤ C e 2 ( γ 1 + σ ) τ f L 2 ( Q , σ ) 2 for τ ≤ 0 .

Now multiplying both sides of this inequality by e ^-2(γ+σ)τ , then integrating them with respect to τ from 0 to ∞, we arrive at

| | | u N | | | Q , γ + σ 2 : = ∫ 0 ∞ e - 2 ( γ + σ ) τ | | | u N ( ⋅ , τ ) | | | Ω 2 d τ ≤ C f L 2 ( Q , σ ) 2 .

It is clear that |||.||| _Q,γ+σ is a norm in H ^m,1(Q, γ + σ) which is equivalent to the norm . H m , 1 ( Q , γ + σ ) . Thus, it follows from (3.16) that

u N H m , 1 ( Q , γ + σ ) 2 ≤ C f L 2 ( Q , σ ) 2 .

From this inequality, by standard weakly convergent arguments (see, e.g., [ 10 , Ch. 7]), we can conclude that the sequence { u N } N = 1 ∞ possesses a subsequence convergent to a vector function u ∈ H ^m,1(Q, γ + σ) which is a generalized solution of problem (2.6)-(2.8). Moreover, it follows from (3.17) that the inequality (3.4) holds.   □

Theorem 3.4. Let h be a nonnegative integer. Assume that all the coefficients a_pq together with their derivatives with respect to t up to the order h are bounded on Q ¯ . Let γ ₀ be the number as in Theorem 3.3 which was defined by formula (3.11). Let the vector function f satisfy the following conditions for some nonnegative real number σ

(i) f t k ∈ L 2 ( Q , k γ 0 + σ ) , k ≤ h ,

(ii) f t k ( x , 0 ) = 0 , 0 ≤ k ≤ h - 1 .

Then for an arbitrary real number γ satisfying γ > γ ₀ the generalized solution u in the space H ^m,1(Q, γ + σ) of the problem (3.6)- (3.7) has derivatives with respect to t up to the order h with u t k ∈ H m , 1 ( Q , ( k + 1 ) γ + σ ) for k = 0, 1,..., h and

∑ k = 0 h u t k H m , 1 ( Q , ( k + 1 ) γ + σ ) 2 ≤ C ∑ k = 0 h f t k L 2 ( Q , k γ 0 + σ ) 2 ,

where C is a constant independent of u and f.

Proof. From the assumptions on the regularities of the coefficients a_pq and of the function f it follows that the solution ( c k N ( t ) ) k = 1 N of the system (3.5), (3.6) has generalized derivatives with respect to t up to the order h + 2. Now take an arbitrary real number γ ₁ satisfying γ ₀ < γ ₁ < γ. We will prove by induction that

u t k N ( ⋅ , τ ) H m ( Ω ) 2 ≤ C e 2 ( k + 1 ) γ 1 + σ τ ∑ j = 0 k f t j L 2 ( Q , j γ 0 + σ ) 2 for τ > 0

and for k = 0,..., h, where the constant C is independent of N, f and τ. From (3.15) it follows that (3.19) holds for k = 0 since the norm |||·||| is equivalent to the norm ⋅ H m ( Ω ) . Assuming by induction that (3.19) holds for k = h - 1, we will show it to be true for k = h. To this end we differentiate h times both sides of (3.5) with respect to t to receive the following equality

( u t h + 2 N , φ l ) Ω t + ∑ k = 0 h ( h k ) B t h − k ( t , u t k N , φ l ) = ( f t h , φ l ) Ω t , l = 1 , … , N .

From these equalities together with the initial (3.6) and the assumption (ii), we can show by induction on h that

u t k N | t = 0 = 0 for k = 0 , … , h + 1 .

Now multiplying both sides of (3.20) by d h + 1 c k N ¯ d t h + 1 , then taking sum with respect to l from 1 to N, we get

( u t h + 2 N , u t h + 1 N ) Ω t + ∑ k = 0 h ( h k ) B t h − k ( t , u t k N , u t h + 1 N ) = ( f t h , u t h + 1 N ) Ω t .

Adding the equality (3.22) to its complex conjugate, we have

∂ ∂ t | u t h + 1 N | 0 , Ω t 2 + ∑ k = 0 h h k ∂ ∂ t B t h - k ( t , u t k N , u t h N ) - B t h - k + 1 ( t , u t k N , u t h N ) = 2 Re ( f t h , u t h + 1 N ) Ω t .

Integrating both sides of this equality with respect to t from 0 to a positive real τ with using the integration by parts and (3.21), we arrive at

| u t h + 1 N | 0 , Ω τ 2 + B ( τ , u t h N , u t h N ) = B τ ( u t h N , u t h N ) + ∑ k = 0 h - 1 h k B t h - k + 1 τ ( u t k N , u t h N ) - ∑ k = 0 h - 1 h k B t h - k ( τ , u t k N , u t h N ) + 2 Re ( f t h , u t h + 1 N ) Q τ .

This equality has the form (3.8) with u^N replaced by u t h N and the last term of the righthand side of (3.8) replaced by the following expression

∑ k = 0 h - 1 h k B t h - k + 1 τ ( u t k N , u t h N ) - ∑ k = 0 h - 1 h k B t h - k ( τ , u t k N , u t h N ) + 2 Re ( f t h , u t h + 1 N ) Q τ .

Since the coefficients a_pq together with their derivatives with respect to t up to the order h are bounded, by the Cauchy and interpolation inequalities and the induction assumption, we see that

| ∑ k = 0 h - 1 h k B t h - k ( τ , u t k N , u t h N ) | ≤ ε | u t h N ( ⋅ , τ ) | m , Ω 2 + | u t h N ( ⋅ , τ ) | 0 , Ω 2 + C ∑ k = 0 h - 1 u t k N ( ⋅ , τ ) m , Ω τ 2 ≤ ε | u t h N ( ⋅ , τ ) | m , Ω 2 + | u t h N ( ⋅ , τ ) | 0 , Ω 2 + C e 2 h γ 1 + σ τ ∑ j = 0 k f t j L 2 ( Q , j γ 0 + σ ) 2 ,

| ∑ k = 0 h - 1 h k B t h - k + 1 τ ( u t k N , u t h N ) | ≤ ε | u t h N | m , Q τ 2 + | u t h N | 0 , Q τ 2 + C ∑ k = 0 h - 1 u t k N m , Q τ 2 = ε | u t h N | m , Q τ 2 + | u t h N | 0 , Q τ 2 + C ∑ k = 0 h - 1 ∫ 0 τ u t k N ( ⋅ , t ) m , Ω 2 d t ≤ ε | u t h N | m , Q τ 2 + | u t h N | 0 , Q τ 2 + C ∑ j = 0 k f t j L 2 ( Q , j γ 0 + σ ) 2 ∫ 0 τ e 2 h γ 1 + σ t d t ≤ ε | u t h N | m , Q τ 2 + | u t h N | 0 , Q τ 2 + C e 2 h γ 1 + σ τ ∑ j = 0 k f t j L 2 ( Q , j γ 0 + σ ) 2 .

and

| 2 Re ( f t h , u t h + 1 N ) Q τ | ≤ ε | u t h + 1 N | 0 , Q τ 2 + C f t h L 2 ( Q ) 2 ≤ ε | u t h + 1 N | 0 , Q τ 2 + C e 2 h γ 1 + σ τ f t h L 2 ( Q , h γ 0 + σ ) 2 .

Thus, repeating the arguments which were used to get (3.15) from (3.8), we can obtain (3.19) for k = h from (3.23).

Now we multiply both sides of (3.19) by e ^{-2((k+1)γ+σ)τ} , then integrate them with respect to τ from 0 to ∞ to get

u t k N H m , 1 ( Q , ( k + 1 ) γ + σ ) 2 ≤ C ∑ j = 0 k f t j L 2 ( Q , j γ 0 + σ ) 2 , k = 0 , … , h .