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 Hung 1 , Nguyen Thanh Anh 1* and Phung Kim Chuc 2 * 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 singulari- ties have been thoroughly investi gated (see, e.g, [1,2] and the ex tensive bibliography in this book). We are concerned w ith initial boundary value problems for hyperbolic equations and systems in domains with co nical points. These problems with t he 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 sys - tems with the coefficients independent of time in domains with conical points was stu- died 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 exis- tence 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 ℝ n , n ≥ 2, with the boundary ∂Ω. We suppose that ∂Ω is an infinitely differentiable surface everywhere exce pt the origin, in a neighborhood of which Ω coincides with the cone K ={x : x/|x| Î G}, where G is a smooth doma in on the unit sphere S n-1 . For each t,0<t ≤∞,denoteQ t = Ω ×(0,t),Ω t = Ω ×{t}. Especially, we set Q = Q ∞ , Γ = ∂Ω\{0}, S = Γ × [0, +∞). Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 © 2011 Hung et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in a ny medium, provided the original work is properly cited. For each multi-index p =(p 1 , , p n ) Î N n , we use notations |p|=p 1 + + p n , D p = ∂ |p| ∂x p 1 1 ∂x p n n . For a complex-valued vector function u =(u 1 , , u s ) defined on Q, we denote D p u =  D p u1 , , D p us  , u t j = ∂ j u ∂t j =( ∂ j u 1 ∂t j , , ∂ j u s ∂t j ), |u| =( s  j =1 |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 () = ⎛ ⎝    |p|≤l |D p u| 2 dx ⎞ ⎠ 1 2 < ∞ . H l− 1 2 (  ) - the space of traces of vector functions from H l (Ω)onΓ with the norm  u  H l− 1 2 () = inf   v  H l () : v ∈ H l (), v|  = u  . H l,0 (Q, g)(g Î 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 −2γ t dxdt ⎞ ⎠ 1 2 < ∞ . Especially, we set L 2 (Q, g)=H 0,0 (Q, g). H l,1 (Q, g)(g Î 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 −2γ t dxdt ⎞ ⎠ 1 2 < ∞ . V l 2 , α ( ) - the closure of C ∞ 0 (\{0} ) with respect to the norm  u  V l 2,α () = ⎛ ⎝  |p|≤l   r 2(α+|p|−l) |D p u| 2 dx ⎞ ⎠ 1 2 , where r = |x| =   n k=1 x 2 k  1 2 . H l α ()(α ∈ R ) -the weighted Sobolev space of vector functions u defined in Ω with the norm  u  H l α () = ⎛ ⎝  |p|≤l   r 2α |D p u| 2 dx ⎞ ⎠ 1 2 . Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 2 of 18 If l ≥ 1, then V l− 1 2 α (  ) , H l− 1 2 α (  ) denote the spaces consisting of traces of functions from respective spaces V l 2 , α (), H l α ( ) on the boundary Γ with the respective norms  u  V l− 1 2 α () = inf   v  V l 2,α () : v ∈ V l 2,α (), v|  = u  ,  u  H l− 1 2 α (  ) = inf   v  H l α () : v ∈ H l α (), v|  = u  . H l,1 α (Q, γ )(α, γ ∈ R ) - the weighted Sobolev space of vector functions u defined in Q with the norm  u  H l,1 α (Q,γ ) = ⎛ ⎝  Q ⎛ ⎝  |p|≤l r 2α |D p u| 2 + |u t | 2 ⎞ ⎠ e −2γ t dx dt ⎞ ⎠ 1 2 < ∞ . From the definitions it follows the continuous imbeddings V l 2 , α () ⊂ H l α () (2:1) and V l+k 2 , α+k () ⊂ V l 2,α ( ) (2:2) for arbitrary nonnegative integers l, k and real number a. It is also well known (see [[2], Th. 7.1.1]) that if α < − n 2 or α > l − n 2 then V l 2 , α () ≡ H l α () (2:3) with the norms being equivalent. Now we introduce the differential operator Lu = L(x, t, D)u =  | p |,| q |≤m (−1) |p| D p (a pq D q u) , where a pq = a pq (x, t)arethes × s matrices with the bounded complex-valued com- ponents in Q . We assume that a pq =(−1) |p|+|q| a ∗ qp 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 pq (x, t)η q η p ≥ μ  | p |=m   η p   2 (2:4) for all h p Îℂ 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 boundar y operators N j = N j (x, t, D), j =1,2, ,m on S such that integration equality   Lu ¯ vdx=  |p|,|q|≤m   a pq D q uD p vdx+ m  j =1  ∂ N j u ∂ j−1 v ∂ν j−1 d s (2:5) holds for all u, v ∈ C ∞ (  ) and for all t Î [0, ∞). The order of the operator N j 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 3 of 18 In this paper, we consider the following problem: u tt + Lu = f in Q , (2:6) N j u =0onS, j =1, , m , (2:7) u | t=0 = u t | t=0 =0 on . (2:8) A complex vector-valued function u Î H m,1 (Q, g)iscalledageneralizedsolutionof problem (2.6)-(2.8) if and only if u| t =0 = 0 and the equality  Q u t ¯η t dx dt +  | p |,| q |≤m  Q a pq D q uD p η dx dt =  Q f ¯η dx d t (2:9) holds for all h(x, t) Î H m,1 (Q) satisfying h(x, t) = 0 for all t ≥ T for some positive real number T. 3 Th e 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, = ⎛ ⎝    |p|=k |D p u| 2 dx ⎞ ⎠ 1 2 ,(u, v)  =   u ¯ vdx . For vector functions u and v defined in Q and τ > 0, we set |u| k,Q τ =   τ 0 |u(·, t)| 2 k, dt  1 2 , |u| k, τ = |u(·, τ )| k, ,(u, v)  τ =(u(·, τ ), v(·, τ ))  , B t k (t, u, v)=  | p |,| q |≤m   ∂ k a pq ∂t k (·, t)D q u(·, t)D p v(·, t ) dx, B τ t k (u, v)=  τ 0 B t k (t, u, v) dt . Especially, we set B(t , u, v)=B t 0 (t , u, v)andB τ (u, v)=B τ t 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 followi ng Gronwall-Bellman and interpolation inequalities as two fundamental tools to establish the theorems on the unique existence and the regu- larity in time. Lemma 3.1 ([8], Lemma 3.1) Assume u, a, b are real-valued continuous on an inter- val [a, b], b is nonnegative and integrable on [a, b], a is nondecreasing satisfying u (τ ) ≤ α(τ )+  τ a β(t)u(t) dt for all a ≤ τ ≤ b . Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 4 of 18 Then u (τ ) ≤ α(τ ) exp   τ a β(t) dt  for all a ≤ τ ≤ b . (3:2) From [[9], Th. 4.14], we have the following lemma. Lemma 3.2. Fo r each positive real number ε and each integer j,0<j <m, there exists apositiverealnumberC= C (Ω, m, ε) which is dependent on only Ω, mandε such that the inequality |u| 2 j , ≤ ε|u| 2 m, + C|u| 2 0,  (3:3) holds for all u Î 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 withrespecttotareboundedon Q . Then there exists a positive rea l number g 0 such that for each g >g 0 , if f Î L 2 (Q, s) for some nonnegative real number s, the problem (2.6)-(2.8) has a unique generalized solution u in the space H m,1 (Q, g + s) and  u  2 H m,1 (Q,γ +σ ) ≤ C   f   2 L 2 (Q,σ ) , (3:4) 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 exist ence 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)= N  k =1 c N k (t ) ϕ k (x) , where (c N k (t )) N k = 1 are the solution of the system of the following ordinary differential equations of second order: (u N tt , ϕ l )  t + B(t, u N , ϕ l )=(f , ϕ l )  t , l =1, , N , (3:5) with the initial conditions c N k (0) = 0, dc N k dt(0) = 0, k =1, , N . (3:6) Let us multiply (3.5) by dc N k (t) dt , take the sum with respect to l from 1 to N, and inte- grate the obtained equality with respect to t from 0 to τ (0 <τ < ∞) to receive (u N tt , u N t )  t + B(t, u N , u N t )=(f , u N t )  t . (3:7) Now adding this equality to its complex conjugate, then using (3.1) and the integra- tion by parts, we obtain | u N t | 2 0, τ + B(τ , u N , u N )=B τ t (u N , u N )+2Re(f , u N t ) Q τ . (3:8) With noting that, for some positive real number r, ρ|u N | 2 0, τ =2Reρ(u N , u N t ) Q τ , Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 5 of 18 we can rewrite (3.8) as follows |u N t | 2 0, τ + B 0 (τ ,u N , u N )+ρ|u| 2 0, τ = B τ t (u N , u N ) −  |p|, |q|≤m | p | + | q | < 2m − 1   τ a pq D q u N D p u N dx +2Reρ(u N , u N t ) Q τ +2Re(f , u N t ) Q τ . (3:9) By (2.4), the left-hand side of (3.9) is greater than | u N t (·, τ )| 2 0, + μ|u N (·, τ )| 2 m, + ρ   u(·, τ )   2 0 ,  . We denote by I, II, III, IV the terms from the first, second, third, and forth, respec- tively, of the righ t-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 pq ∂t D q u N D p u N dx dt +  |p|,|q|≤m | p |+| q |≤2m−1  Q τ ∂a pq ∂t D q u N D p u N dx dt ≡ I 1 + I 2 . Put μ 1 =sup{| ∂ a pq ∂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 | 2 0,Q τ + |D p u N | 2 0,Q τ ) ≤ m  μ 1 |u N | 2 m,Q τ . By the Cauchy inequality and the interpolation inequality (3.3), fo r an arbitra ry posi- tive number ε 1 , we have I 2 ≤ ε 1 |u N | 2 m, Q τ + C 1 |u N | 2 0, Q τ , where C 1 = C 1 (ε 1 ) is a nonnegative constant independent of u N , f and τ.Nowusing again t he Cauchy and interpolation inequalities, for an arbitrary positive number ε 2 with ε 2 <μ, it holds that II ≤ ε 2 |u N (·, τ )| 2 m ,  + C 2 |u N (·, τ )| 2 0 ,  , 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 III ≤ (μ − ε 2 )ρ 2 m  μ 1 + ε 1 |u N | 2 0,Q τ + m  μ 1 + ε 1 μ − ε 2 |u N t | 2 0,Q τ , and IV ≤ ε 3 |u N t | 2 0,Q τ + 1 ε 3 |f | 2 0,Q τ , Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 6 of 18 where ε 3 > 0, arbitrary. Combining the above estimations we get from (3.9) that |u N t (·, τ )| 2 0, +(μ − ε 2 )|u N (·, τ )| 2 m, +(ρ − C 2 )|u N (·, τ )| 2 0, ≤ (m  μ 1 + ε 1 )|u N | 2 m,Q τ +  C 1 + (μ − ε 2 )ρ 2 m  μ 1 + ε 1  |u N | 2 0,Q τ +  m  μ 1 + ε 1 μ − ε 2 + ε 3  |u N t | 2 0,Q τ + 1 ε 3 |f | 2 0,Q τ . (3:10) 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 dg dρ = ρ 2 − 2C 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 )} . (3:11) Nowwetakerealnumbersg, g 1 arbitrarily satisfying g 0 <g 1 <g. Then there are posi- tive real numbers ε 1 , ε 2 ,(ε 2 <μ), r (r >C 2 (ε 1 , ε 2 )) and ε 3 such that m  μ 1 + ε 1 μ − ε 2 + ε 3 < 2γ 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 ) < 2γ 1 . (3:12) From now to the end of the present proof, we fix such constants ε 1 , ε 2 , ε 3 and r. Let |||u N (·, τ ) 2  || | stand for the left-hand side of (3.10). It follows from (3.10) and (3.12) that | ||u N (·, τ )||| 2  ≤ 2γ 1  τ 0 |||u(·, t)||| 2  dt + C  τ 0 |f (·, t)| 2 0, dt for all τ ≤ 0 , (3:13) where C = 1 ε 3 . By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that |||u N (·, τ )||| 2  ≤ Ce 2γ 1 τ  τ 0 |f (·, t)| 2 0, dt for allτ ≥ 0 . (3:14) We see that  τ 0 |f (·, t)| 2 0, dt = e 2στ  τ 0 |e −στ f (·, t)| 2 0, dt ≤ e 2στ  τ 0 |e −σ t f (·, t)| 2 0, dt . Hence, it follows from (3.14) that |||u N (·, τ )||| 2  ≤ Ce 2(γ 1 +σ )τ  τ 0 |e −σ t f (·, t)| 2 0, dt ≤ Ce 2(γ 1 +σ )τ   f   2 L 2 (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 7 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 | ||u N ||| 2 Q,γ +σ :=  ∞ 0 e −2(γ +σ )τ |||u N (·, τ )||| 2  dτ ≤ C   f   2 L 2 (Q,σ ) . (3:16) It is clear that |||.||| Q,g+s is a norm in H m,1 (Q, g + s) which is equivalent to the norm  .  H m,1 ( Q,γ +σ ) . Thus, it follows from (3.16) that   u N   2 H m,1 ( Q,γ +σ ) ≤ C   f   2 L 2 ( Q,σ ) . (3:17) 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 Î H m,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 a pq together with their derivatives with respect to t up to the order h are bounded on Q . Let g 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 s (i) f t k ∈ L 2 ( Q, kγ 0 + σ ) , k ≤ h , (ii) f t k ( x,0 ) =0,0≤ k ≤ h − 1 . Then for an arbitrary real number g satis fying g >g 0 the generalize d solution u in the space H m,1 (Q, g + s) of the problem (3.6)- (3.7) has deriva tives 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 h  k = 0  u t k  2 H m,1 (Q,(k+1)γ +σ) ≤ C h  k = 0   f t k   2 L 2 (Q,kγ 0 +σ ) , (3:18) 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 N k (t )) N 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 g 1 satisfying g 0 <g 1 <g. We will prove by induction that   u N t k (·, τ )   2 H m () ≤ Ce 2 ( (k+1)γ 1 +σ ) τ k  j =0   f t j   2 L 2 (Q,jγ 0 +σ ) for τ> 0 (3:19) 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 (  ) . 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 N t h+2 , ϕ l )  t + h  k = 0  h k  B t h−k (t, u N t k , ϕ l )=(f t h , ϕ l )  t , l =1, , N . (3:20) Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 8 of 18 From these equalities together with the initial (3.6) and the assumption (ii), we can show by induction on h that u N t k | t=0 =0 fork =0, , h +1 . (3:21) Now multiplying both sides of (3.20) by d h+1 c N k dt h+ 1 , then taking sum with respect to l from 1 to N, we get (u N t h+2 , u N t h+1 )  t + h  k = 0  h k  B t h−k (t, u N t k , u N t h+1 )=(f t h , u N t h+1 )  t . (3:22) Adding the equality (3.22) to its complex conjugate, we have ∂ ∂t |u N t h+1 | 2 0, t + h  k = 0  h k  ∂ ∂t B t h−k (t, u N t k , u N t h ) − B t h−k+1 (t, u N t k , u N t h )  =2Re(f t h , u N t h+1 )  t . Integrating both sides of t his 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 N t h+1 | 2 0, τ + B(τ , u N t h , u N t h )=B τ (u N t h , u N t h )+ h −1  k=0  h k  B τ t h−k+1 (u N t k , u N t h ) − h−1  k = 0  h k  B t h−k (τ , u N t k , u N t h )+2Re(f t h , u N t h+1 ) Q τ . (3:23) This equality has the form (3.8) with u N replaced by u N t h and the last term of the righthand side of (3.8) replaced by the following expression h −1  k = 0  h k  B τ t h−k+1 (u N t k , u N t h ) − h −1  k = 0  h k  B t h−k (τ , u N t k , u N t h )+2Re(f t h ,u N t h+1 ) 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 | h −1  k=0  h k  B t h−k (τ , u N t k , u N t h )|≤ε  |u N t h (·, τ )| 2 m, + |u N t h (·, τ )| 2 0,  + C h −1  k=0   u N t k (·, τ )   2 m, τ ≤ ε  |u N t h (·, τ )| 2 m, + |u N t h (·, τ )| 2 0,  + Ce 2(hγ 1 +σ)τ k  j =0   f t j   2 L 2 (Q,jγ 0 +σ ) , | h−1  k=0  h k  B τ t h−k+1 (u N t k , u N t h )|≤ε  |u N t h | 2 m,Q τ + |u N t h | 2 0,Q τ  + C h−1  k=0   u N t k   2 m,Q τ = ε  |u N t h | 2 m,Q τ + |u N t h | 2 0,Q τ  + C h−1  k=0  τ 0   u N t k (·, t)   2 m, dt ≤ ε  |u N t h | 2 m,Q τ + |u N t h | 2 0,Q τ  + C k  j=0   f t j   2 L 2 (Q,jγ 0 +σ )  τ 0 e 2(hγ 1 +σ)t dt ≤ ε  |u N t h | 2 m,Q τ + |u N t h | 2 0,Q τ  + Ce 2(hγ 1 +σ)τ k  j =0   f t j   2 L 2 (Q,jγ 0 +σ ) . Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 9 of 18 and |2Re(f t h ,u N t h+1 ) Q τ |≤ε|u N t h+1 | 2 0,Q τ + C   f t h   2 L 2 (Q) ≤ ε|u N t h+1 | 2 0,Q τ + Ce 2(hγ 1 +σ)τ   f t h   2 L 2 ( Q,hγ 0 +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). Nowwemultiplybothsidesof(3.19)bye -2((k+1)g+s)τ , then integrate them with respect to τ from 0 to ∞ to get   u N t k   2 H m,1 (Q,(k+1)γ +σ) ≤ C k  j =0   f t j   2 L 2 (Q,jγ 0 +σ ) , k =0, , h . (3:24) From this inequality, by again standard weakly convergent arguments, we can con- clude that the sequence {u N t k } ∞ N= 1 possesses a subsequence convergent to a vector func- tion u (k) Î H m,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 Th e global regularity First, we introduce the operator pencil associa ted with the problem. See [11] for more detail. For convenience we rewrite the operators L(x, t, D), N j (x, t, D) in the form L = L(x, t, ∂ x )=  | p |≤2m a p (x, t) D p N j = N j (x, t, D)=  | p |≤2m− j b jp (x, t) D p , j =1, , m . Let L 0 (x, t, D), N 0j (x, t, D), be the principal homogeneous parts of L(x, t, D), N j (x, t, D). It can be directly verified that the derivative D a can be written in the form D α = r −|α| |α|  p =0 P α,p (ω, D ω )(rD r ) p , where P a,p (ω , ∂ ω ) are differential operators of order ≤ |a|-p with smooth coeffi- cients on ¯  , r =|x|, ω is an arbitrary local coordinate system on S n-1 , D ω = ∂ ∂ω , D r = ∂ ∂r . Thus we can write L 0 (0, t, D) and N 0j (0, t, D) in the form L 0 ( 0, t, D ) = r −2m L ( ω, t, D ω , rD r ), N 0, j (0, t, D)=r −2m+j N j (ω, t, D ω , rD r ) . The operator pencil associated with the problem is defined by U (λ, t)=(L(ω, t, D ω , λ), N j (ω, t, D ω , λ)), λ ∈ C, t ∈ (0, +∞) . Hung et al. Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17 Page 10 of 18 [...]... strongly hyperbolic systems near a conical point at the boundary of the domain Math Sb 190(7), 103–126 (1999) (in Russian) Trans in Sb Math 190 (78), 1035-1058 (1999) 4 Hung, NM, Kim, BT: On the solvability of the first mixed problem for strongly hyperbolic system in infinite nonsmooth cylinders Taiwanese J Math 12(9), 2601–2618 (2008) 5 Hung, NM, Yao, JC: Cauchy-Dirichlet problem for second- order hyperbolic. .. the second initial boundary value problem for hyperbolic systems in domains with conical points Boundary Value Problems 2011 2011:17 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article... Boundary value problems for partial differential equations in nonsmooth domains Usp Math Nauka 38(2), 1–86 (1983) 2 Kozlov, VA, Maz’ya, VG, Rossmann, J: Elliptic Boundary Problems in Domains With Point Singularities, Mathematical Surveys and Monographs vol 85 American Mathematical Society, Providence, Rhode Island (1997) 3 Hung, NM: Asymptotic behaviour of solutions of the first boundary- value problem for. .. Hale, J: Theory of Functional Differential Equations Springer, New York (1977) 9 Adams, RA: Sobolev Spaces Academic Press, London (1975) 10 Evans, LC: Partial Differential Equations, Graduate Studies in Mathematics vol 19 American Mathematical Society, Providence, Rhode Island (1998) 11 Hung, NM, Anh, NT: The initial- boundary value problems for parabolic equations in domains with conical points In: Albert,... http://www.boundaryvalueproblems.com/content/2011/1/17 Page 18 of 18 13 Kozlov, VA, Maz’ya, VG, Rossmann, J: Spectral Problems Associated With Corner Singularities Of Solutions to Elliptic Equations In Mathematical Surveys and Monographs, vol 85,American Mathematical Society, Providence, Rhode Island (2001) doi:10.1186/1687-2770-2011-17 Cite this article as: Hung et al.: On the regularity of the solution for the second initial boundary. .. second- order hyperbolic equations in cylinder with non-smooth base Nonlinear Anal TMA 70, 741–756 (2009) doi:10.1016/j.na.2008.01.007 6 Kokotov, A, Plamenevskii, BA: On the asymptotic on solutions to the Neumann problem for hyperbolic systems in domain with conical point St Petersburg Math J 16(3), 477–506 (2005) 7 Renardy, M, Rogers, RC: An Introduction to Partial Differential Equations Springer, New York, 2... Proof Without generality we assume that the domain Ω coincides with the cone K in the unit ball Set Ω0 = {x Î Ω: |x| ≥ 2-1}, k = {x|x ∈ , 2−k ≤ |x| ≤ 2−k+1 }, k = 1, 2, , and Γk = ∂Ω ∩ ∂Ωk, k = 0, 1 According to well known results on the regularity of solutions of elliptic boundary problems in smooth domains (see, e.g., [12]), we have Hung et al Boundary Value Problems 2011, 2011:17 http://www.boundaryvalueproblems.com/content/2011/1/17... spectrum of the operator U (λ0 , t) for each t Î (0, ∞), is an enumerable set of eigenvalues (see [[2], Th 5.2.1]) Now let us give the main theorem of this section: Theorem 4.1 Suppose that all the assumptions of Theorem 3.4 hold for a given positive integer h Assume further that the strip m−ε− n n ≤ Reλ ≤ 2m − α − 2 2 (4:1) does not contain any eigenvalue of U (λ, t) for all t Î (0, +∞) and for some... the arguments 2m,1 above to conclude that uth−1 ∈ Hα (Q, (h + 1)γ + σ ) with the estimate (4.2) for k = h -1 The proof is completed 5 An example In this section we apply the previous results to the Cauchy-Neumann problem for the classical wave equation We consider the following problem: utt − u= f ∂u = ∂ν 0 on S, u|t=0 = ut |t=0 = in Q, (5:1) (5:2) 0 on , (5:3) where Δ is the Laplace operator For problem. .. is a constant independent of u, f1 and t Now multiplying both sides of (4.27) with e-2(2g+s)t, then integrating with respect to t from 0 to ∞ and using estimates from Theorem 3.4, we obtain u 2 2m,1 Hα (Q,2γ +σ ) ≤C f 2 , L2 (Q,σ ) (4:28) where C is a constant independent of u and f Hence, the theorem is valid for h = 1 Assume that the theorem is true for some nonnegative h - 2 We will prove it for h . article as: Hung et al.: On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points. Boundary Value Problems 2011 2011:17. Submit. 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 Hung 1 , Nguyen. near the conical points of the solutions are established. The Neumann boundary problem for general second- order hyperbolic sys - tems with the coefficients independent of time in domains with conical