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Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 RESEARCH Open Access On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points Nguyen Manh Hung1, Nguyen Thanh Anh1* and Phung Kim Chuc2 * Correspondence: thanhanh@hnue.edu.vn 1 Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam Full list of author information is available at the end of the article Abstract In this paper, we deal with the second initial boundary value problem for higher order hyperbolic systems in domains with conical points. We establish several results on the well-posedness and the regularity of solutions. 1 Introduction 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 [3-5] 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. 2 Notations and the formulation of the problem Let Ω be a bounded domain in ℝn, 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| Î G}, where G is a smooth domain on the unit sphere Sn-1. For each t, 0 l − then 2 2 l V2,α () ≡ Hαl () (2:3) with the norms being equivalent. Now we introduce the differential operator  Lu = L(x, t, D)u = (−1)|p| Dp (apq Dq u), |p|,|q|≤m where apq = apq (x, t) are the s × s matrices with the bounded complex-valued com∗ ponents in Q. We assume that apq = (−1)|p|+|q| a∗qp for all |p|, |q| ≤ m, where aqp is the transposed conjugate matrix to apq. This means the differential operator L is formally self-adjoint. We assume further that there exists a positive constant μ such that    2 ηp  apq (x, t)ηq ηp ≥ μ (2:4) |p|=|q|=m |p|=m for all hp Îℂs, |p| = m, and all (x, t) ∈ Q̄. 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 Nj = Nj (x, t, D), j = 1, 2,..., m on S such that integration equality   Luv̄ dx =   |p|,|q|≤m  apq Dq uDp v dx + m   j=1 ∂ Nj u ∂ j−1 v ds ∂ν j−1 (2:5) holds for all u, v ∈ C∞ () and for all t Î [0, ∞). The order of the operator Nj is 2m j for j = 1, 2,..., m. Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 4 of 18 In this paper, we consider the following problem: utt + Lu = f Nj u = on S, j = 1, . . . , m, 0 in Q, u|t=0 = ut |t=0 = 0 (2:6) (2:7) on . (2:8) A complex vector-valued function u Î Hm,1(Q, g) is called a generalized solution of problem (2.6)-(2.8) if and only if u|t = 0 = 0 and the equality     ut η̄t dx dt + apq Dq uDp η dx dt = f η̄ dx dt (2:9) Q |p|,|q|≤m Q Q holds for all h(x, t) Î Hm,1(Q) satisfying h(x, t) = 0 for all t ≥ T for some positive real number T. 3 The unique solvability and the regularity in time 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, ⎛ ⎞1   2 =⎝ |Dp u|2 dx⎠ ,  |p|=k  (u, v) =  uv̄ dx. For vector functions u and v defined in Q and τ > 0, we set  |u|k,Qτ = τ 0 Bt k (t, u, v) = |u(·, t)|2k, 1 dt 2 , |u|k,τ = |u(·, τ )|k, ,   ∂ k apq (·, t)Dq u(·, t)Dp v(·, t) dx, k ∂t  |p|,|q|≤m (u, v)τ = (u(·, τ ), v(·, τ )) , Bτt k (u, v) =  τ Bt k (t, u, v) dt. 0 Especially, we set B(t, u, v) = Bt 0 (t, u, v) and Bτ (u, v) = Bτt 0 (u, v). From the formally self-adjointness of the operator L, we see that B(τ , u, v) = B(τ , v, u). (3:1) 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, a, b are real-valued continuous on an interval [a, b], b is nonnegative and integrable on [a, b], a is nondecreasing satisfying  τ u(τ ) ≤ α(τ ) + β(t)u(t) dt for all a ≤ τ ≤ b. a Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 5 of 18 Then  τ u(τ ) ≤ α(τ ) exp  β(t) dt for all a ≤ τ ≤ b. (3:2) a From [[9], Th. 4.14], we have the following lemma. Lemma 3.2. For each positive real number ε and each integer j, 0 g0, if f Î L2(Q, s) for some nonnegative real number s, the problem (2.6)-(2.8) has a unique generalized solution u in the space Hm,1(Q, g + s) and  2 u2Hm,1 (Q,γ +σ ) ≤ C f L (Q,σ ) , (3:4) 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 m {ϕk }∞ k=1 is an orthogonal basis of H (Ω) which is orthonormal in L2(Ω). Put uN (x, t) = N  cN k (t)ϕk (x), k=1 N where (cN k (t))k=1 are the solution of the system of the following ordinary differential equations of second order: N (uN tt , ϕl )t + B(t, u , ϕl ) = (f , ϕl )t , l = 1, . . . , N, (3:5) with the initial conditions cN k (0) = 0, dcN k dt(0) = 0, k = 1, . . . , N. (3:6) dcN k (t) dt, 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 Let us multiply (3.5) by N N N N (uN tt , ut )t + B(t, u , ut ) = (f , ut )t . (3:7) Now adding this equality to its complex conjugate, then using (3.1) and the integration by parts, we obtain 2 N N τ N N N |uN t |0,τ + B(τ , u , u ) = Bt (u , u ) + 2Re(f , ut )Qτ . With noting that, for some positive real number r, ρ|uN |20,τ = 2Re ρ(uN , uN t )Qτ , (3:8) Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 6 of 18 we can rewrite (3.8) as follows 2 N N 2 τ N N |uN t |0,τ + B0 (τ , u , u ) + ρ|u|0,τ = Bt (u , u )   N apq Dq uN Dp uN dx + 2Re ρ(uN , uN − t )Qτ + 2 Re(f , ut )Qτ . |p|, |q| ≤ m |p| + |q| < 2m − 1 τ (3:9) By (2.4), the left-hand side of (3.9) is greater than  2 |uN (·, τ )|2 + μ|uN (·, τ )|2 + ρ u(·, τ ) . t 0, m, 0, 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   ∂apq   ∂apq q N p N D u D u dx dt + Dq uN Dp uN dx dt ≡ I1 + I2 . Qτ ∂t Qτ ∂t |p|=|q|=m |p|,|q|≤m |p|+|q|≤2m−1 Put μ1 = sup{|     ∂apq (x, t)| : p = q = m, (x, t) ∈ Q} ∂t and m =  1. |p|=m Then, by the Cauchy inequality, we have I1 ≤ μ 1  1 (|Dq uN |20,Qτ + |Dp uN |20,Qτ ) ≤ m μ1 |uN |2m,Qτ . 2 |p|=|q|=m By the Cauchy inequality and the interpolation inequality (3.3), for an arbitrary positive number ε1, we have I2 ≤ ε1 |uN |2m,Qτ + C1 |uN |20,Qτ , where C1 = C1(ε1) is a nonnegative constant independent of uN, f and τ. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number ε2 with ε2 <μ, it holds that II ≤ ε2 |uN (·, τ )|2m, + C2 |uN (·, τ )|20, , where C 2 = C 2 (ε 2) is a nonnegative constant independent of uN , f and τ. For the terms III and IV, by the Cauchy inequality, we have III ≤ (μ − ε2 )ρ 2 N 2 m μ1 + ε1 N 2 |u |0,Qτ + |ut |0,Qτ , m μ1 + ε1 μ − ε2 and 2 IV ≤ ε3 |uN t |0,Qτ + 1 2 |f | , ε3 0,Qτ Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 7 of 18 where ε3 > 0, arbitrary. Combining the above estimations we get from (3.9) that 2 N 2 N 2 N 2 |uN t (·, τ )|0, + (μ − ε2 )|u (·, τ )|m, + (ρ − C2 )|u (·, τ )|0, ≤ (m μ1 + ε1 )|u |m,Qτ     2 m μ1 + ε1 (μ − ε2 )ρ 1 2 2 + C1 + |uN |20,Qτ + + ε3 |uN |f | . t |0,Qτ + m μ1 + ε1 μ − ε2 ε3 0,Qτ (3:10) Now fix ε1, ε2 and consider the function C1 + g(ρ) = (μ − ε2 )ρ 2 m μ1 + ε1 ρ − C2 for ρ > C2 . We have C1 2 dg ρ − 2C2 ρ − A = dρ A(ρ − C2 )2 with A = (μ − ε2 )ρ 2 . m μ1 + ε1 We see that the function g has a unique minimum at  C1 ρ0 = ρ0 (ε1 , ε2 ) = C2 + C22 + . A We put γ0 = 1 2 inf max{ ε1 >0 0<ε2 <μ m μ1 + ε1 , g(ρ0 )}. μ − ε2 (3:11) Now we take real numbers g, g1 arbitrarily satisfying g0 C2(ε1, ε2)) and ε3 such that (μ − ε2 )ρ 2 m μ1 + ε1 < 2γ1 . ρ − C2 (ε1 , ε2 ) C1 (ε1 , ε2 ) + m μ1 + ε1 + ε3 < 2γ1 μ − ε2 and (3:12) From now to the end of the present proof, we fix such constants ε1, ε2, ε3 and r. Let |||uN (·, τ )2 ||| stand for the left-hand side of (3.10). It follows from (3.10) and (3.12) that  τ  τ N 2 2 |||u (·, τ )||| ≤ 2γ1 |||u(·, t)||| dt + C |f (·, t)|20, dt for all τ ≤ 0, (3:13) 0 0 1 . By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that ε3  τ (3:14) |||uN (·, τ )|||2 ≤ Ce2γ1 τ |f (·, t)|20, dt for allτ ≥ 0. where C = 0 We see that  τ  |f (·, t)|20, dt = e2σ τ 0 τ 0 |e−σ τ f (·, t)|20, dt ≤ e2σ τ  τ 0 |e−σ t f (·, t)|20, dt. Hence, it follows from (3.14) that  |||uN (·, τ )|||2 ≤ Ce2(γ1 +σ )τ τ 0  2 |e−σ t f (·, t)|20, dt ≤ Ce2(γ1 +σ )τ f L2 (Q,σ ) for τ ≤ 0. (3:15) Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 8 of 18 Now multiplying both sides of this inequality by e-2(g+s)τ, then integrating them with respect to τ from 0 to ∞, we arrive at  ∞  2 |||uN |||2Q,γ +σ := e−2(γ +σ )τ |||uN (·, τ )|||2 dτ ≤ C f L (Q,σ ) . (3:16) 2 0 It is clear that |||.|||Q,g+s is a norm in Hm,1(Q, g + s) which is equivalent to the norm .Hm,1 (Q,γ +σ ). Thus, it follows from (3.16) that  N 2  2 u  m,1 ≤ C f L2 (Q,σ ) . H (Q,γ +σ ) (3:17) From this inequality, by standard weakly convergent arguments (see, e.g., [[10], Ch. 7]), we can conclude that the sequence {uN }∞ N=1 possesses a subsequence convergent to a vector function u Î Hm,1(Q, g + s) 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 apq together with their derivatives with respect to t up to the order h are bounded on Q. Let g0 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 s (i) ftk ∈ L2 (Q, kγ0 + σ ), k ≤ h, (ii) ftk (x, 0) = 0, 0 ≤ k ≤ h − 1. Then for an arbitrary real number g satisfying g >g0 the generalized solution u in the space Hm,1(Q, g + s) of the problem (3.6)- (3.7) has derivatives with respect to t up to the order h with utk ∈ Hm,1 (Q, (k + 1)γ + σ ) for k = 0, 1,..., h and h  utk 2Hm,1 (Q,(k+1)γ +σ ) ≤ C k=0 h   2 ftk  , L2 (Q,kγ0 +σ ) (3:18) k=0 where C is a constant independent of u and f. Proof. From the assumptions on the regularities of the coefficients apq and of the N function f it follows that the solution (cN k (t))k=1 of the system (3.5), (3.6) has general- ized derivatives with respect to t up to the order h + 2. Now take an arbitrary real number g1 satisfying g0 0 (3:19) j=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 ·Hm (). 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 (uN t h+2 , ϕl )t + h    h k=0 k Bth−k (t, uN t k , ϕl ) = (ft h , ϕl )t , l = 1, . . . , N. (3:20) Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 9 of 18 From these equalities together with the initial (3.6) and the assumption (ii), we can show by induction on h that uN t k |t=0 = 0 for k = 0, . . . , h + 1. (3:21) Now multiplying both sides of (3.20) by dh+1 cN k dt h+1, then taking sum with respect to l from 1 to N, we get N (uN t h+2 , ut h+1 )t + h    h k=0 k N N Bth−k (t, uN t k , ut h+1 ) = (ft h , ut h+1 )t . (3:22) Adding the equality (3.22) to its complex conjugate, we have  h    ∂ ∂ N 2 h N N N , u ) − B h−k+1 (t, u , u ) = 2Re(fth , uN |uth+1 |0,t + Bth−k (t, uN k h k h t t t t t t h+1 )t . k ∂t ∂t k=0 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 2 |uN t h+1 |0,τ + N B(τ , uN t h , ut h ) τ =B N (uN t h , ut h ) + h−1    h k k=0 − h−1    h k=0 k N Bτth−k+1 (uN t k , ut h ) (3:23) N Bth−k (τ , uN t k , ut h ) 2Re(fth , uN t h+1 )Qτ . + This equality has the form (3.8) with uN replaced by uN and the last term of the th righthand side of (3.8) replaced by the following expression h−1    h k=0 k N Bτth−k+1 (uN t k , ut h ) − h−1    h k=0 k N N Bth−k (τ , uN t k , ut h ) + 2Re(ft h , ut h+1 )Qτ . Since the coefficients apq 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 | h−1    h k=0 k h−1   N   N N 2 N 2 u k (·, τ )2 Bth−k (τ , uN t k , ut h )| ≤ ε |ut h (·, τ )|m, + |ut h (·, τ )|0, + C t m, τ k=0 k   2  2 N 2 2(hγ1 +σ )τ ftj  ≤ ε |uN t h (·, τ )|m, + |ut h (·, τ )|0, + Ce L 2 (Q,jγ0 +σ ) j=0 | h−1    h k=0 k h−1   N 2  N N 2 N 2 u k  Bτth−k+1 (uN t k , ut h )| ≤ ε |ut h |m,Qτ + |ut h |0,Qτ + C t m,Q τ h−1   2 N 2 = ε |uN t h |m,Qτ + |ut h |0,Qτ + C k=0  2 N 2 ≤ ε |uN t h |m,Qτ + |ut h |0,Qτ + C k  j=0 k=0  τ 0  N  u k (·, t)2 dt t m,  2 ftj  L2 (Q,jγ0 +σ )  τ e2(hγ1 +σ )t dt 0 k   2  2 N 2 2(hγ1 +σ )τ ftj  + Ce | + |u | ≤ ε |uN h h t m,Qτ t 0,Qτ L 2 (Q,jγ0 +σ ) j=0 . , Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 10 of 18 and  2 |2Re(fth , uN ) | ≤ ε|uN |2 + C fth L2 (Q) t h+1 Qτ t h+1 0,Qτ  2 ≤ ε|uNh+1 |2 + Ce2(hγ1 +σ )τ fth  0,Qτ t L2 (Q,hγ0 +σ ) . 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)g+s)τ , then integrate them with respect to τ from 0 to ∞ to get k   N 2  2 u k  m,1 ftj  ≤ C , k = 0, . . . , h. t H (Q,(k+1)γ +σ ) L2 (Q,jγ0 +σ ) (3:24) j=0 From this inequality, by again standard weakly convergent arguments, we can con∞ possesses a subsequence convergent to a vector funcclude that the sequence {uN t k }N=1 tion u(k) Î Hm,1(Q, (k +1)g +s), moreover, u(k) is the kth generalized derivative in t of the generalized solution u of problem (2.6)-(2.8). The estimation (3.18) follows from (3.24) by passing the weak convergences. □ 4 The global regularity First, we introduce the operator pencil associated with the problem. See [11] for more detail. For convenience we rewrite the operators L(x, t, D), Nj (x, t, D) in the form L = L(x, t, ∂x ) =  ap (x, t) Dp |p|≤2m Nj = Nj (x, t, D) =  bjp (x, t) Dp , j = 1, . . . , m. |p|≤2m−j Let L0(x, t, D), N0j (x, t, D), be the principal homogeneous parts of L(x, t, D), Nj (x, t, D). It can be directly verified that the derivative Da can be written in the form Dα = r −|α| |α|  Pα,p (ω, Dω )(rDr )p , p=0 (ω, ∂ω) are differential operators of order ≤ |a| - p with smooth coeffi∂ ¯ , r = |x|, ω is an arbitrary local coordinate system on S n-1 , Dω = cients on  , ∂ω ∂ Dr = . Thus we can write L0(0, t, D) and N0j (0, t, D) in the form ∂r where Pa, p L0 (0, t, D) = r −2m L(ω, t, Dω , rDr ), N0,j (0, t, D) = r −2m+j Nj (ω, t, Dω , rDr ). The operator pencil associated with the problem is defined by U (λ, t) = (L(ω, t, Dω , λ), Nj (ω, t, Dω , λ)), λ ∈ C, t ∈ (0, +∞).