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 Hm(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 〈., .〉. 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 ‖f‖Lp(a,+∞,γ;X) = (∫ +∞ a ‖f(t)‖2Xe −γ(t)tdt ) 1 2 <∞. Finally, we introduce the Sobolev spaceH1,1∗ (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 ‖u‖2 H 1,1 ∗ (Qa,γ) = ‖u‖2L2(a,+∞,γ;H1(Ω)) + ‖ut‖ 2 L2(a,+∞,γ;L2(Ω)) + ‖utt‖ 2 L2(a,+∞,γ;H−1(Ω)) . To simplify notation, we set L2(Q, γ0) = L2(R, γ0;L2(Ω)). Let L(x, t;D) = − n∑ i,j=1 Di(Aij(x, t)Dj) + n∑ i=1 Bi(x, t)Di + C(x, t), 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∑ i,j=1 Aij(x, t)ξξ ≥ µ0|ξ| 2 (1.3) for all ξ ∈ R and (x, t) ∈ Q. Let us introduce the following bilinear form B(u, v; t) = ∫ Ω ( n∑ i,j=1 (Aij(x, t)DjuDiv + n∑ i=1 Bi(x, t)Diuv + C(x, t)uv ) dx. Then the following Green’s formula (L(x, t;D)u, v) = B(u, v; t) is valid for all u, v ∈ C∞0 (Ω) and a.e. t ∈ R. Definition 1.1. Let f ∈ L2(Q, γ0), then a function u ∈ H1,1∗ (Q, γ) is called a generalized solution of problem (1.1) - (1.2) if and only if the equality 〈utt, v〉+B(u, v; t) = (f, v), a.e. t ∈ R, (1.4) holds for all v ∈ H˚1(Ω). 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 H1,1∗ (Q, γ) and the following estimate holds ‖u‖2 H 1,1 ∗ (Q,γ) ≤ C‖f‖2L2(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 fh(x, t) = χ(t−h)f(x, t) then fh = { f if t ≥ h + 1 0 if t ≤ h. Moreover, if f ∈ L2(Q, γ0), fh ∈ L2(h,∞, γ;L2(Ω)), fh ∈ L2(Q; γ0) and ‖fh‖2L2(Qh;γ0) = ‖f h‖2L2(Q;γ0) ≤ ‖f‖ 2 L2(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 ∈ H1,1∗ (Qh, γ) is called a generalized solution of the problem (2.2)-(2.4) iff uh(., h) = 0, uht (., h) = 0 and the equality 〈uhtt, v〉+B(u h, v; t) = (fh, 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 {ωk(x)} ∞ k=1 be an orthogonal basis of H˚1(Ω) which is orthonormal in L2(Ω). Put uN(x, t) = N∑ k=1 CNk (t)ωk(x) where CNk (t), k = 1, . . . , N, is the solution of the following ordinary differential system: (uNtt , ωk) +B(u N , ωk; t) = (f h, ωk), k = 1, . . . , N, (2.1) with the initial conditions CNk (h) = 0, C N kt(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 CNkt(t), then take the sum with respect to k from 1 to N to arrive at (uNtt , u N t ) +B(u N , uNt ; t) = (f h, uNt ). Since (uNtt , uNt ) = ddt ( ‖uNt ‖ 2 L2(Ω) ) , we get d dt ( ‖uNt ‖ 2 L2(Ω) ) + 2B(uN , uNt ; t) = 2(f h, uNt ). (2.3) By the Cauchy-Schwarz inequality, we have 2|(fh, uNt )| ≤ ‖f h‖2L2(Ω) + ‖u N t ‖ 2 L2(Ω). (2.4) Furthermore, we can write B(uN , uNt ; t) = ∫ Ω n∑ i,j=1 Aij(x, t)Dju NDiu N t dx+ ∫ Ω n∑ i=1 Bi(x, t)Diu NuNt +C(x, t)u NuNt dx =: B1 +B2. (2.5) It is easy to see B1 = d dt (1 2 A[uN , uN , t] ) − 1 2 ∫ Ω n∑ i,j=1 AijtDju NDiu Ndx, (2.6) for the symmetric bilinear form A[uN , uN , t] = ∫ Ω n∑ i,j=1 AijDju NDiu Ndx. The equality (2.6) implies B1 ≥ d dt (1 2 A[uN , uN , t] ) − µ‖uN‖2H1(Ω), (2.7) and we note also |B2| ≤ µ ( ‖uN‖2H1(Ω) + ‖u N t ‖ 2 L2(Ω) ) , (2.8) Combining estimates (2.3)-(2.8), we obtain d dt ( ‖uNt ‖ 2 L2(Ω) + A[u N , uN , t] ) ≤ 2µ ( ‖uNt ‖ 2 L2(Ω) + ‖u N‖2H1(Ω) ) + ‖fh‖2L2(Ω) ≤ µ1 ( ‖uNt ‖ 2 L2(Ω) + A[u N , uN , t] ) + ‖fh‖2L2(Ω) (2.9) where we used (1.3), µ1 = max{2µ, 2µµ0 }. 63 Nguyen Thi Hue Now write η(t) := ‖uNt (., t)‖ 2 L2(Ω) + A[u N , uN , t]; ξ(t) := ‖fh(., t)‖2L2(Ω), t ∈ [h,∞). Then (2.9) implies η′(t) ≤ µ1η(t) + ξ(t), for a.e. t ∈ [h,∞). Thus the differential form of Gronwall-Belmann’s inequality yields the esti- mate η(t) ≤ C t∫ h eµ1(t−s)ξ(s)ds, t ∈ [h,∞). (2.10) We obtain from (2.10) and (1.3) the following estimate ‖uN(., t)‖2L2(Ω) + ‖u N‖2H1(Ω) ≤ C t∫ h eµ1(t−s)‖fh‖2L2(Ω)ds ≤ Ce µ1t‖fh‖2L2(Q,γ0), where γ0(t) ∈ [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 ‖uN(., t)‖2L2(h,∞,γ,L2(Ω)) + ‖u N‖2L2(h,∞,γ,H1(Ω)) ≤ C‖f h‖2L2(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 ‖v‖2 H1(Ω) ≤ 1 and write v = v 1 + v2 where v1 ∈ span{ωk}Nk=1 and (v2, ωk) = 0, k = 1, . . . , N, (v2 ∈ span{ωk}Nk=1 ⊥ ). We have ‖v1‖H1(Ω) ≤ ‖v‖H1(Ω) ≤ 1. Utilizing (2.1), we get (uNtt , v 1) +B(uN , v1; t) = (fh, v1) for a.e. t ∈ [h,+∞). From uN(x, t) = N∑ k=1 CNk (t)ωk, we can see that (uNtt , v) = (u N tt , v 1) = (fh, v1)− B(uN , v1; t). Consequently, |(uNtt , v)| ≤ C ( ‖fh‖2L2(Ω) + ‖u N‖2H1(Ω) ) . Since this inequality holds for all v ∈ H˚1(Ω), ‖v‖H1(Ω) ≤ 1, the following inequality will be inferred ‖uNtt ‖ 2 H−1(Ω) ≤ C ( ‖fh‖2L2(Ω) + ‖u N‖2H1(Ω) ) . (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 ‖uNtt ‖ 2 L2(h,∞,γ,H−1(Ω)) ≤ C‖fh‖2L2(Q,γ0). (2.13) Combining (2.11) and (2.13), we arrive at ‖uN‖2 H 1,1 ∗ (Qh,γ) ≤ C‖fh‖2L2(Q,γ0) (2.14) 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 uh ∈ H1,1∗ (Qh, γ), which is a generalized solution of problem (2.2)-(2.4). 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, fhk = fh − fk, so uhk is the generalized solution of the following problem uhktt + L(x, t,D)u hk = fhk(x, t) in Qk, (2.12) uhk = 0 on Sk, j = 1, ..., m, (2.13) uhk |t=k= 0, u hk t |t=k= 0 on Ω. (2.14) We have ‖uhk‖2 H 1,1 ∗ (Qk,γ) ≤ C‖fh − fk‖2L2(Q,γ0), and ‖fh − fk‖2L2(Q,γ0) = h+1∫ k e−γ0(t)t‖fh − fk‖2L2(Ω)dt. = h+1∫ k e−γ0(t)t|χ(t− h)− χ(t− k)|.‖f‖2L2(Ω)dt ≤ 2 h+1∫ k e−γ0(t)t‖f‖2L2(Ω)dt. Thus ‖uh − uk‖2 H 1,1 ∗ (Q(k,∞),γ) ≤ 2C h+1∫ k e−γ0(t)t‖f‖2L2(Ω)dt. (2.15) Since f ∈ L2(Q, γ0), h∫ k e−γ0(t)t‖f‖2L2(Ω)dt → 0 when h, k → −∞. It follows that {uh}−∞h=0 is a Cauchy sequence. So {uh} is convergent to u in H1,1∗ (Qk, γ) (Consider 65 Nguyen Thi Hue uh in the cylinder Q by setting uh(x, t) = 0 for all t < h). Because ‖fh − f‖2L2(Q,γ0) = h+1∫ h e−γ0(t)t|χ(t− h)− 1|.‖f‖2L2(Ω)dt+ h∫ −∞ e−γ0(t)t‖f‖2L2(Ω)dt, so ‖fh − f‖2L2(Q,γ0) ≤ 2 h+1∫ h e−γ0(t)t‖f‖2L2(Ω)dt+ h∫ −∞ e−γ0(t)t‖f‖2L2(Ω)dt, {fh} is convergent to f in L2(Q, γ0). 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