The existence of a solution to the dirichlet problem for second order hyperbolic equations in nonsmooth domains

Chia sẻ: Minh Minh | Ngày: | Loại File: PDF | Số trang:7

Thêm vào BST

Báo xấu

13
lượt xem 2
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

The purpose of this paper is to prove the existence of generalized solution of a boundary value problem for the second order hyperbolic equations without initial conditions in nonsmooth domains.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: The existence of a solution to the dirichlet problem for second order hyperbolic equations in nonsmooth domains

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2012, Vol. 57, No. 3, pp. 60-66 THE EXISTENCE OF A SOLUTION TO THE DIRICHLET PROBLEM FOR SECOND ORDER HYPERBOLIC EQUATIONS IN NONSMOOTH DOMAINS Nguyen Thi Hue Sao Do University, Hai Duong City E-mail: thaohue-117@yahoo.com.vn Abstract. The purpose of this paper is to prove the existence of general- ized solution of a boundary value problem for the second order hyperbolic equations without initial conditions in nonsmooth domains. Keywords: Dirichlet problem, hyperbolic equations, nonsmooth domains. 1. Introduction Let Ω be a nonsmooth domain in Rn (n ≥ 2). For h ∈ R, set Qh = Ω × (h, ∞), Sh = ∂Ω × (h, ∞), S = ∂Ω × R, Q = Ω × R. Let m, k be non-negative integers. Denote by H m (Ω), H ˚m (Ω) usual Sobolev spaces as in [1]. By the notation (., .) we mean the inner product in L2 (Ω). We denote by H −1 (Ω) the dual space of H˚1 (Ω). The pairing between H˚1 (Ω) and H −1 (Ω) is denoted by h., .i. Let X be a Banach space, γ = γ(t) be a real functions. We denote by L2 (a, +∞, γ; X), the space of functions f : (a, +∞) → X with the norm Z +∞ 21 kf kLp (a,+∞,γ;X) = kf (t)k2X e−γ(t)t dt < ∞. a Finally, we introduce the Sobolev space H∗1,1 (Qa , γ) which consists all functions u defined on Qa such that u ∈ L2 (a, +∞, γ; H ˚ 1(G)), ut ∈ L2 (a, +∞, γ; L2(Ω)) and utt ∈ L2 (a, +∞, γ; H −1(Ω)) with the norm kuk2H 1,1 (Qa ,γ) = kuk2L2 (a,+∞,γ;H 1 (Ω)) + kut k2L2 (a,+∞,γ;L2 (Ω)) + kutt k2L2 (a,+∞,γ;H −1 (Ω)) . ∗ To simplify notation, we set L2 (Q, γ0 ) = L2 (R, γ0; L2 (Ω)). Let n X n X L(x, t; D) = − Di (Aij (x, t)Dj ) + Bi (x, t)Di + C(x, t), i,j=1 i=1 be a second order partial differential operator, where Di = ∂xi , and Aij , Bi , C are bounded functions from C ∞ (Q). 60
The existence of solution of the Dirichlet problem for second order hyperbolic equations... We study the following problem: utt + L(x, t; D)u = f in Q, (1.1) u = 0 on S, (1.2) where f : Q → R is given. We assume that the operator L is uniformly strong elliptic, that is, there exists a constant µ0 > 0 such that n (1.3) X Aij (x, t)ξξ ≥ µ0 |ξ|2 i,j=1 for all ξ ∈ R and (x, t) ∈ Q. Let us introduce the following bilinear form Z X n n X B(u, v; t) = (Aij (x, t)Dj uDi v + Bi (x, t)Di uv + C(x, t)uv dx. Ω i,j=1 i=1 Then the following Green’s formula (L(x, t; D)u, v) = B(u, v; t) is valid for all u, v ∈ C0∞ (Ω) and a.e. t ∈ R. Definition 1.1. Let f ∈ L2 (Q, γ0 ), then a function u ∈ H∗1,1 (Q, γ) is called a generalized solution of problem (1.1) - (1.2) if and only if the equality hutt , vi + B(u, v; t) = (f, v), a.e. t ∈ R, (1.4) ˚1 (Ω). holds for all v ∈ H The problem with initial conditions was considered in [2,3] in which the solv- ability of the problem was proven. The boundary value problem without initial con- dition for parabolic equation has been investigated in [5,6,7]. In this work, we will prove the existence of generalized solutions of problem (1.1) - (1.2). Let us present the main results of this paper. Theorem 1.1. Suppose that the coefficients of the operator L satisfy sup{|Aij |, |Aijt|, |Bi |, |C| : i, j = 1, . . . , n; (x, t) ∈ Q} ≤ µ, µ = const. Then for each γ(t)t > γ0 (t)t, t ∈ R, problem (1.1)-(1.2) has a generalized solution u in the space H∗1,1 (Q, γ) and the following estimate holds kuk2H 1,1 (Q,γ) ≤ Ckf k2L2 (Q,γ0 ) (1.5) ∗ here C is a constant independent of u and f . 61
Nguyen Thi Hue 2. The proof of Theorem 1.1 To prove the theorem, we construct a family approximate solution uh of the solution u of problem (1.1)-(1.2). It is known that there is a smooth function χ(t) which is equal to 1 on [1, +∞), is equal to 0 on (−∞, 0] and assumes value in [0, 1] on [0; 1] (see [4, Th. 5.5] for more details). Moreover, we can suppose that all derivatives of χ(t) are bounded. Let h ∈ (−∞, 0] be an integer. Setting f h (x, t) = χ(t−h)f (x, t) then f if t ≥ h + 1 h f = 0 if t ≤ h. Moreover, if f ∈ L2 (Q, γ0 ), f h ∈ L2 (h, ∞, γ; L2(Ω)), f h ∈ L2 (Q; γ0 ) and kf h k2L2 (Qh ;γ0 ) = kf h k2L2 (Q;γ0 ) ≤ kf k2L2 (Q;γ0 ) . (2.1) Fixed f ∈ L2 (Q; γ0 ), we consider the following problem in the cylinder Qh : utt + L(x, t, D)u = f h (x, t) in Qh , (2.2) u = 0 on Sh , (2.3) u |t=h = 0, ut |t=h = 0 on Ω. (2.4) This is the initial boundary value problem for hyperbolic equations in cylinders Qh . A function uh ∈ H∗1,1 (Qh , γ) is called a generalized solution of the problem (2.2)-(2.4) iff uh (., h) = 0, uht (., h) = 0 and the equality huhtt , vi + B(uh , v; t) = (f h , v), holds for a.e. t ∈ (h, ∞) and all v ∈ H ˚1 (Ω). Lemma 2.1. For any h fixed, there exists a solution of the problem (2.2)-(2.4). Firstly, we will prove the existence by Galerkin’s approximating method. Let k=1 be an orthogonal basis of H (Ω) which is orthonormal in L2 (Ω). Put {ωk (x)}∞ ˚1 N X N u (x, t) = CkN (t)ωk (x) k=1 where CkN (t), k = 1, . . . , N, is the solution of the following ordinary differential system: (uN N h tt , ωk ) + B(u , ωk ; t) = (f , ωk ), k = 1, . . . , N, (2.1) with the initial conditions N CkN (h) = 0, Ckt (h) = 0, k = 1, . . . , N. (2.2) 62
The existence of solution of the Dirichlet problem for second order hyperbolic equations... Let us multiply (2.1) by Ckt N (t), then take the sum with respect to k from 1 to N to arrive at (uN N N N h N tt , ut ) + B(u , ut ; t) = (f , ut ). Since (uN tt , u N t ) = d dt ku t L2 (Ω) , we get N 2 k d kuN 2 N N h N (2.3) t kL2 (Ω) + 2B(u , ut ; t) = 2(f , ut ). dt By the Cauchy-Schwarz inequality, we have 2|(f h , uN h 2 N 2 t )| ≤ kf kL2 (Ω) + kut kL2 (Ω) . (2.4) Furthermore, we can write n Z X n Z X N B(u , uN t ; t) = Aij (x, t)Dj u N Di u N t dx+ Bi (x, t)Di uN uN N N t +C(x, t)u ut dx Ω i,j=1 Ω i=1 =: B1 + B2 . (2.5) It is easy to see n d 1 1Z X B1 = N N A[u , u , t] − Aijt Dj uN Di uN dx, (2.6) dt 2 2 i,j=1 Ω for the symmetric bilinear form n Z X N N A[u , u , t] = Aij Dj uN Di uN dx. Ω i,j=1 The equality (2.6) implies d 1 B1 ≥ A[uN , uN , t] − µkuN k2H 1 (Ω) , (2.7) dt 2 and we note also |B2 | ≤ µ kuN k2H 1 (Ω) + kuN k 2 t L2 (Ω) , (2.8) Combining estimates (2.3)-(2.8), we obtain d N 2 2 N 2 h 2 kut kL2 (Ω) + A[uN , uN , t] ≤ 2µ kuN k t L2 (Ω) + ku k H 1 (Ω) + kf kL2 (Ω) dt ≤ µ1 kuN k 2 t L2 (Ω) + A[u N , u N , t] + kf h k2L2 (Ω) (2.9) where we used (1.3), µ1 = max{2µ, µ2µ0 }. 63
Nguyen Thi Hue Now write 2 2 η(t) := kuN N N h t (., t)kL2 (Ω) + A[u , u , t]; ξ(t) := kf (., t)kL2 (Ω) , t ∈ [h, ∞). Then (2.9) implies η ′ (t) ≤ µ1 η(t) + ξ(t), for a.e. t ∈ [h, ∞). Thus the differential form of Gronwall-Belmann’s inequality yields the esti- mate Zt η(t) ≤ C eµ1 (t−s) ξ(s)ds, t ∈ [h, ∞). (2.10) h We obtain from (2.10) and (1.3) the following estimate Zt ku N (., t)k2L2 (Ω) + kuN k2H 1 (Ω) ≤C eµ1 (t−s) kf h k2L2 (Ω) ds ≤ Ceµ1 t kf h k2L2 (Q,γ0 ) , h where γ0 (t) ∈ [µ1 , +∞) for t < 0 and γ0 (t) ∈ [0, µ1) for t > 0. Now multiplying both sides of this inequality by e−γ(t)t , then integrating them with respect to t from h to ∞, we obtain kuN (., t)k2L2 (h,∞,γ,L2 (Ω)) + kuN k2L2 (h,∞,γ,H 1 (Ω)) ≤ Ckf h k2L2 (Q,γ0 ) , (2.11) where γ(t) ∈ [µ1 , +∞) for t > 0 and γ(t) ∈ [0, µ1 ) for t < 0. Fix any v ∈ H ˚1 (Ω), with kvk2 1 H (Ω) ≤ 1 and write v = v + v where v ∈ 1 2 1 ⊥ span{ωk }N k=1 and (v , ωk ) = 0, k = 1, . . . , N, (v ∈ span{ωk }k=1 ). We have kv kH 1 (Ω) ≤ 2 2 N 1 kvkH 1 (Ω) ≤ 1. Utilizing (2.1), we get tt , v ) + B(u , v ; t) = (f , v ) for a.e. t ∈ [h, +∞). 1 1 h 1 (uN N N From uN (x, t) = CkN (t)ωk , we can see that P k=1 1 h 1 1 (uN N N tt , v) = (utt , v ) = (f , v ) − B(u , v ; t). Consequently, h 2 N 2 |(uN tt , v)| ≤ C kf k L2 (Ω) + ku k H (Ω) . 1 Since this inequality holds for all v ∈ H˚1 (Ω), kvkH 1 (Ω) ≤ 1, the following inequality will be inferred kuN k 2 tt H (Ω) −1 ≤ C kf h 2 k L2 (Ω) + ku N 2 k H (Ω) . 1 (2.12) 64
The existence of solution of the Dirichlet problem for second order hyperbolic equations... Multiplying (2.12) by e−γ(t)t , then integrating them with respect to t from h to ∞, and by using (2.11), we obtain kuN 2 h 2 tt kL2 (h,∞,γ,H −1 (Ω)) ≤ Ckf kL2 (Q,γ0 ) . (2.13) Combining (2.11) and (2.13), we arrive at kuN k2H 1,1 (Q ≤ Ckf h k2L2 (Q,γ0 ) (2.14) ∗ h ,γ) where C is a absolute constant. From the inequality (2.14), by standard weakly convergent arguments, we can conclude that the sequence {uN }∞ N =1 possesses a subsequence convergent to a function u ∈ H∗ (Qh , γ), which is a generalized solution of problem (2.2)-(2.4). h 1,1 Let k be a integer less than h, denote uk a generalized solution of the problem (2.2)-(2.4) when we replaced h by k. We define uh in the cylinder Qk by setting uh (x, t) = 0 for k ≤ t ≤ h. Putting uhk = uh − uk , f hk = f h − f k , so uhk is the generalized solution of the following problem uhk tt + L(x, t, D)u hk = f hk (x, t) in Qk , (2.12) uhk = 0 on Sk , j = 1, ..., m, (2.13) t |t=k = 0 on Ω. uhk |t=k = 0, uhk (2.14) We have kuhk k2H 1,1 (Q ≤ Ckf h − f k k2L2 (Q,γ0 ) , ∗ k ,γ) and Zh+1 kf h − f k k2L2 (Q,γ0 ) = e−γ0 (t)t kf h − f k k2L2 (Ω) dt. k Zh+1 = e−γ0 (t)t |χ(t − h) − χ(t − k)|.kf k2L2 (Ω) dt k Zh+1 ≤2 e−γ0 (t)t kf k2L2 (Ω) dt. k Thus Zh+1 h ku − uk k2H 1,1 (Q ≤ 2C e−γ0 (t)t kf k2L2 (Ω) dt. (2.15) ∗ (k,∞) ,γ) k Rh Since f ∈ L2 (Q, γ0 ), e−γ0 (t)t kf k2L2 (Ω) dt → 0 when h, k → −∞. It follows that k h=0 is a Cauchy sequence. So {u } is convergent to u in H∗ (Qk , γ) (Consider {uh }−∞ h 1,1 65
Nguyen Thi Hue uh in the cylinder Q by setting uh (x, t) = 0 for all t < h). Because Zh+1 Zh kf h − f k2L2 (Q,γ0 ) = e−γ0 (t)t |χ(t − h) − 1|.kf k2L2(Ω) dt + e−γ0 (t)t kf k2L2 (Ω) dt, h −∞ so Zh+1 Zh h kf − f k2L2 (Q,γ0 ) ≤2 e−γ0 (t)t 2 kf kL2 (Ω) dt + e−γ0 (t)t kf k2L2 (Ω) dt, h −∞ {f h } is convergent to f in L2 (Q, γ0 ). Since uh is a generalized solution of the problem (2.2)-(2.4), we have huhtt , vi + B(uh , v; t) = (f h , v), holds for a.e. t ∈ (h, ∞) and all v ∈ H ˚1 (Ω). Sending h → −∞, we obtain (1.4). Thus u is a generalized solution of the problem (1.2)-(1.3). Using (2.14), letting N → ∞, we gain kuh k2H 1,1 (Q ,γ) ≤ Ckf h k2L2 (Q,γ0 ) . ∗ h Thus kuh k2H 1,1 (Q,γ) ≤ Ckf k2L2 (Q,γ0 ) . ∗ Sending h → −∞ we obtain (1.5). The proof of the Theorem is completed. REFERENCES [1] Evans LC, 1998. Partial Differential Equations. Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI. [2] N. M. Hung, V. T. Luong, 2009. Lp-Regularity of solutions to first initial- boundary value problem for hyperbolic equations in cusp domains. Electron. J. Differential Equations, Vol. 2009, No. 151, pp.1-18. [3] N. M. Hung, V. T. Luong, 2009. Regularity of the solution of the first initial-boundary value problem for hyperbolic equations in domains with cuspi- dal points on boundary. Boundary Value Problems, Vol. 2009, Art. ID 135730. doi:10.1155/2009/135730. [4] Micheal Renardy, Robert C. Rogers, 2004. An Introduction to Partial Differential Equations. Springer. [5] Yu. P. Krasovskii, 1992. An estimate of solutions of parabolic problems without an initial condition. Math. USSR Izvestiya, Vol. 39, No. 2. [6] N. M. Bokalo, 1990. Problem without initial conditions for nolinear parabolic equations, UDC 517. 95. [7] S. D. Ivasishen, 1983. Parabolic Boundary - Value Problems without initial con- ditions. UDC 917.946. 66