In the paper we investigate a non-local contact-boundary value problem for a system of second order hyperbolic equations with discontinuous solutions. Under some conditions on input data, a priori estimate is obtained for the solution of this problem

Akhiev, S.S.

Electronic Journal of Qualitative Theory of Differential Equations

S. S. Akhiev

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Electronic journal of qualitative theory of differential equations

A priori estimate for discontinuous solutions of a second order linear hyperbolic problem

Seyidali S. Akhiev

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Akhiev S. S.

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2010, No. 23, 1-12; http://www.math.u-szeged.hu/ejqtde/A PRIORI ESTIMATE FORDISCONTINUOUS SOLUTIONS OF ASECOND ORDER LINEAR HYPERBOLICPROBLEMS. S. Akhiev(Azerbaijan State Pedagogical University)Abstract. In the paper we investigate a non-local contact-boundary value problemfor a system of second order hyperbolic equations with discontinuous solutions. Undersome conditions on input data a priori estimate is obtained for the solution of thisproblem.In the paper we consider the following hyperbolic system:(Lz) (t, x) ≡ ztx (t, x) + z (t, x)A0,0 (t, x) ++zt (t, x)A1,0 (t, x) + zx (t, x)A0,1 (t, x) =ϕ (t, x) ,(t, x) ∈ G = G0 ∪G1, G0 = (0, T )× (0, α) , G1 = (0, T )× (α, l) ,(1)where z (t, x) = (z1 (t, x) , ...., zn (t, x)) is the desired vector-function; Ai,j (t, x),i, j = 0, 1 are the given n× n-matrices on G; ϕ (t, x) is the given n-dimensionalvector-function on G; α is a fixed point from (0, l) .For the system (1) we give the following non-local contact boundary condi-tions(Lkz) (t) ≡ z (t, 0)β0,k (t) + z (t, α− 0)β1,k (t) + z (t, α+ 0)β2,k (t)++z (t, l)β3,k (t) + zt (t, 0) g0,k (t) + zt (t, α− 0) g1,k (t) + zt (t, α+ 0) g2,k (t)++zt (t, l) g3,k (t) = ϕk (t) , t ∈ (0, T ) , k = 1, 2 ; (2)(L3z) (x) ≡ zx (0, x) = ϕ3 (x) , x ∈ (0, l) ; (3)L0z ≡ z (0, 0) = ϕ0. (4)Here: βi,k (t), gi,k (t) , i = 0, 1, 2, 3; k = 1, 2 are the given n× n matrices on(0, T ); ϕk (t) , k = 1, 2 are the given n-dimensional vector-functions on (0, T ) ;ϕ3 (x) is the given n-dimensional vector-function on (0, l) ; ϕ0 is the given con-stant n-dimensional vector.EJQTDE, 2010 No. 23, p. 1We assume that the following conditions are satisfied:1) The matrices Ai,j (t, x) are measurable on G, A0,0 ∈ Lp,n×n (G) ; thereexit the functions A01,0 ∈ Lp (0, l) and A00,1 ∈ Lp (0, T ), such that ‖A1,0 (t, x)‖ ≤A01,0 (x), ‖A0,1 (t, x)‖ ≤ A00,1 (t) almost everywhere on G, where Lp,n×n (G) ,1 ≤ p ≤ ∞ is a Banach space of n × n matrices g = (gij) with elementsgij ∈ Lp (G) , wherein the norm is determined by the equality ‖g‖Lp,n×n(G) =∥∥g0∥∥Lp(G), moreover g0 = ‖g‖ ≡n∑i,j=1|gij | is the norm of the matrix g;2) βi,k ∈ Lp,n×n (0, T ) and gi,k ∈ L∞,n×n (0, T ) ;3) ϕ ∈ Lp,n (G) , ϕk ∈ Lp,n (0, T ), ϕ3 ∈ Lp,n (0, l), where Lp,n (G) , 1 ≤ p ≤∞, is a space of n-dimensional vector-functions ϕ = (ϕ1, ..., ϕn) with elementsfrom Lp (G); norm of ϕ ∈ Lp,n (G) is defined as ‖ϕ‖Lp,n(G) =∥∥ϕ0∥∥Lp(G)andϕ0(t, x) = ‖ϕ(t, x)‖ =n∑i=1|ϕi(t, x)| is norm of n-vectors ϕ(t, x) ∈ Rn for fixed(t, x) ∈ G. Rn is the space of all vectors ρ = (ρ1, . . . , ρn) with norm ‖ρ‖ =n∑i=1|ρi|.Non-local boundary value problems for integro-differential equations withcontinuous coefficients were studied in the paper [2].We‘ll consider the solution of problem (1)-(4) in the space Ŵp,n (G) , 1 ≤p ≤ ∞, [4] (p. 52) of all n-dimensional vector-functions z (t, x), which oneach domain Gk (k = 0, 1) belong to Wp,n (Gk) and are continuous at the point(0, α). Here Wp,n (Gk) is a space of all n-dimensional vector-functions z ∈Lp,n (Gk) , possessing generalized in S.L.Sobolev’s sense derivatives zt, zx andztx from Lp,n (Gk) , k = 0, 1. We‘ll define the norm in the space Ŵp,n (G) bythe equality [4] (p. 54)‖z‖cWp,n(G)=1∑k=0‖z‖Wp,n(Gk) ,where‖z‖Wp,n(Gk) = ‖z‖Lp,n(Gk) + ‖zt‖Lp,n(Gk) + ‖zx‖Lp,n(Gk) + ‖ztx‖Lp,n(Gk) .Since the operator Nz = (z (0, 0) , zt (t, 0) , zt (t, α+ 0) , zx (0, x) , ztx (t, x)),brings about isomorphism from Ŵp,n (G) to Q̂p,n =Rn×Lp,n (0, T )×Lp,n (0, T ) ×Lp,n (0, l) × Lp,n (G) , [1], we can reduce problem (1) - (4) to the followingoperator equationL̂z = ϕ̂,where L̂ = (L0, L1, L2, L3, L) , z ∈ Ŵp,n (G) is desired solution andϕ̂ = (ϕ0, ϕ1, ϕ2, ϕ3, ϕ) ∈ Q̂p,n is the given element. This equation is equiva-EJQTDE, 2010 No. 23, p. 2lent to the system of integro-algebraic equations with respect to elements of fivecomponentsb̂ = (b0, b1 (t) , b2 (t) , b3 (x) , b (t, x)) ≡≡ (z (0, 0) , zt (t, 0) , zt (t, a+ 0) , zx (0, x) , ztx (t, x))of the space Q̂p,n:b (t, x) +∫ t0∫ x0b (τ, ζ) q1 (ζ, x)A0,0 (t, x) dτdζ++∫ x0b (t, ζ) q1 (ζ, x)A1,0 (t, x) dζ +∫ t0b (τ, x)A0,1 (t, x) dτ++(∫ t0b1 (τ) θ (α− x) dτ +∫ t0b2 (τ) θ (x− α) dτ)A0,0 (t, x) ++ (b1 (t) θ (α− x) + b2 (t) θ (x− α))A1,0 (t, x) + b3 (x)A0,1 (t, x) ++∫ x0b3 (ζ)A0,0 (t, x) dζ + b0A0,0 (t, x) = ϕ(t, x), (t, x) ∈ G; (5)b1 (t) (g0,k (t) + g1,k (t)) + b2 (t) (g2,k (t) + g3,k (t))++∫ t0b1 (τ) (β0,k (t) + β1,k (t)) dτ++∫ t0b2 (τ) (β2,k (t) + β3,k (t)) dτ = ϕ0k (t) , t ∈ (0, T ) , k = 1, 2 ; (6)whereϕ0k(t) = ϕk(t) − b0(β0,k(t) + β1,k(t) + β2,k(t) + β3,k(t))−−∫ α0b3(ζ)(β1,k(t) + β2,k(t))dζ −∫ l0b3(ζ)β3,k(t)dζ − ϕk,b(t);ϕk,b(t) =∫ t0∫ α0b(τ, ζ)β1,k(t)dτdζ +∫ t0∫ lαb(τ, ζ)β3,k(t)dτdζ++∫ α0b(t, ζ)g1,k(t)dζ +∫ lαb(t, ζ)g3,k(t)dζ, t ∈ (0, T ); (6∗)b3 (x) = ϕ3 (x) , x ∈ (0, l) ; (7)b0 = ϕ0, (8)where θ (y) is one –dimensional Heaviside function on R = R1 and q1 (ζ, x) =θ (ζ − α) θ (x− α) + θ (α− x).EJQTDE, 2010 No. 23, p. 3If we succeed to estimate the components b0, b1 (t) , b2 (t) , b3 (x) , b (t, x) ofthe vector b̂, on the basis of [1] we get a priori estimate for the solution z ∈Ŵp,n (G) of problem (1)-(4)z(t, x) = b0 + θ (α− x)∫ T0b1(τ)θ (t− τ) dτ++θ (x− α)∫ T0b2(τ)θ (t− τ) dτ +∫ l0b3(ζ)θ (x− ζ) dζ++∫G∫θ (t− τ) θ (x− ζ) q1 (ζ, x) b(τ, ζ)dτdζ , (t, x) ∈ G. (9)The components b1 (t) , b2 (t) , b (t, x) are determined from the system ofequations (5),(6), since the components b0, b3 (x) are explicitly given by con-ditions (7), (8), therefore, it remains to estimate only b1 (t) , b2 (t) , b (t, x).It is obvious that by means of the matrix∆(t) =g0,1(t) + g1,1(t)g2,1(t) + g3,1(t)g0,2(t) + g1,2(t)g2,2(t) + g3,2(t)we can write the equality (6) in the compact form(b1(t), b2(t))∆(t) +∫ t0(b1(τ), b2(τ))B(t)dτ = (ϕ01(t), ϕ02(t)), t ∈ (0, T ), (10)whereB(t) =β0,1(t) + β1,1(t)β2,1(t) + β3,1(t)β0,2(t) + β1,2(t)β2,2(t) + β3,2(t)Assume that almost for all t ∈ (0, T ) the matrix ∆(t) is invertible and itholds‖∆(t)‖ ≤M1,∥∥∆−1(t)∥∥ ≤M1 (11)in the sense of almost everywhere on (0, T ). Then, from (10) we have(b1(t), b2(t)) +∫ t0(b1(τ), b2(τ))B1(t)dτ = (ϕ01(t), ϕ02(t))∆−1(t), t ∈ (0, T ), (12)whereB1(t) = B(t)∆−1(t).Passing in (12) to the vector norm we haveα(t) ≤∫ t0α(τ)l(t)dτ + S0(t), t ∈ (0, T ), (13)EJQTDE, 2010 No. 23, p. 4whereα(t) = ‖b1(t)‖ + ‖b2(t)‖ ,l(t) = ‖B1(t)‖ ≤M1 ‖B(t)‖ ,S0(t) =∥∥(ϕ01(t), ϕ02(t))∆−1(t)∥∥ ≤M1(∥∥ϕ01(t)∥∥+∥∥ϕ02(t)∥∥);here and below Mi are constants independent on ϕ̂ = (ϕ0, ϕ1, ϕ2, ϕ3, ϕ).Let the point τ ∈ (0, T ) be fixed and t ∈ (0, τ). Then integrating (13) withrespect to t on (0, τ) we getR(τ) ≤∫ τ0R(t)l(t)dt+ S1(τ), τ ∈ (0, T ), (14)whereS1(τ) =∫ τ0S0(t)dt,R(τ) =∫ τ0α(t)dt.We write the inequality (14) in the formR(τ) ≤ R1(τ) + S1(τ) + ε, (15)where ε > 0 is an arbitrary number andR1(τ) =∫ τ0R(t)l(t)dt.HenceR (τ)R1 (τ) + S1 (τ) + ε≤ 1, τ ∈ (0, T ).ThereforeR (τ) l (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, T ),orṘ1 (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, T ), (16)where the sign of point over some function of one argument means its firstderivative.The function S1(τ) is a monotonically increasing function.Therefore, if t isfixed and τ ∈ (0, t), then S1(τ) ≤ S1(t). Therefore from (16) we haveṘ1 (τ)R1 (τ) + S1 (t) + ε≤Ṙ1 (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, t),EJQTDE, 2010 No. 23, p. 5integrating it with respect to τ on (0, t) we getlnR1 (t) + S1 (t) + εR1 (0) + S1 (t) + ε≤∫ t0l (τ) dτorR1(t) + S1(t) + ε ≤ (S1(t) + ε)eRt0l(τ)dτ .Taking this into account in (15) we getR(t) ≤ S1(t)eRt0l(τ)dτ . (17)Writing (13) in the formα(t) ≤ R(t)l(t) + S0(t)and using (17) we getα(t) ≤ l(t)eRt0l(τ)dτ∫ t0S0(τ)dτ + S0(t) (18)Thus, we have provedLemma 1. If, for some non-negative functions α, l, S0 ∈ Lp(0, T ), the inequal-ity (13) holds, then the function α(t) also satisfies the condition (18).Taking into account the expression of the function l(t) in (18) we getα(t) ≤M2 ‖B(t)‖∫ t0S0(τ)dτ + S0(t), t ∈ (0, T ), (19)whereM2 = M1 exp(M1∫ T0‖B(τ)‖ dτ).Notice that from the conditions imposed on the matrix functions βi,k(t) itfollows that ‖B(·)‖ ∈ Lp(0, T ). Therefore M2 < +∞.Now by means of this lemma for the sum α(t) = ‖b1(t)‖ + ‖b2(t)‖ we havethe estimateα(t) = ‖b1(t)‖ + ‖b2(t)‖ ≤M2∫ t0S0(τ)dτ ‖B(t)‖ + S0(t), t ∈ (0, T ).Therefore, using the Hölder inequality, we obtain‖bk(t)‖ ≤ S0(t) +M2T1q∥∥S0∥∥Lp(0,T )‖B(t)‖ , t ∈ (0, T ), k = 1, 2,here and below q = p/(p− 1) denotes the number conjugate to p.EJQTDE, 2010 No. 23, p. 6Here, passing to the norm, by Minkowsky inequality we get‖bk‖Lp,n(0,T ) ≤∥∥S0∥∥Lp,(0,T )(1 +M2T1q ‖B‖Lp,2n×2n(0,T )) == M3∥∥S0∥∥Lp(0,T ), k = 1, 2;M3 = 1 +M2T1q ‖B‖Lp,2n×2n(0,T ).Obviously∥∥S0∥∥Lp(0,T )≤M1(∥∥ϕ01∥∥Lp,n(0,T )+∥∥ϕ02∥∥Lp,n(0,T )).Therefore‖bk‖Lp,n(0,T ) ≤M4(∥∥ϕ01∥∥Lp,n(0,T )+∥∥ϕ02∥∥Lp,n(0,T )), k = 1, 2, (20)where M4 = M3M1.Now, let’s estimate the norms∥∥ϕ0k∥∥Lp,n(0,T ), k = 1, 2. Obviously if the vectorz ∈ Ŵp,n (G) satisfies the conditions (1)-(4), then its independent elementsb̂ = (z(0, 0), zt(t, 0), zt(t, α+0), zx(0, x), ztx(t, x))= (b0, b1(t), b2(t), b3(x), b(t, x))satisfy the equalities (5)-(8) and therewith‖b0‖ ≤ ‖ϕ̂‖Q̂p,n‖b3‖Lp,n(0,l) ≤ ‖ϕ̂‖Q̂p,n (21)where‖ϕ̂‖Q̂p,n=∥∥∥L̂z∥∥∥Ŵp,n(G)=∥∥∥L̂(N−1(b̂))∥∥∥Q̂p,n== ‖ϕ0‖ + ‖ϕ1‖Lp,n(0,T ) + ‖ϕ2‖Lp,n(0,T ) ++ ‖ϕ3‖Lp,n(0,l) + ‖ϕ‖Lp,n(G) , z = N−1b̂.Therefore, from the expressions (6*) of the vectors ϕ0k(t) we have∥∥ϕ0k(t)∥∥ ≤ ‖ϕk(t)‖ + ‖b0‖ (‖β0,k(t)‖ + ‖β1,k(t)‖++ ‖β2,k(t)‖ + ‖β3,k(t)‖) + ‖b3‖Lp,n(0,l) α1q ××(‖β1,k(t)‖ + ‖β2,k(t)‖) + l1q ‖b3‖Lp,n(0,l) ‖β3,k(t)‖ + ‖ϕk,b(t)‖ , k = 1, 2.Hence by means of the Minkowsky inequality allowing for (20) we haveEJQTDE, 2010 No. 23, p. 7∥∥ϕ0k∥∥Lp,n(0,T )≤ ‖ϕk‖Lp,n(0,T ) + ‖b0‖3∑i=0‖βi,k‖Lp,n×n(0,T ) ++ ‖b3‖Lp,n(0,l) α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) ++ ‖b3‖Lp,n(0,l) l1q ‖β3,k‖Lp,n×n(0,T ) ++ ‖ϕk,b‖Lp,n(0,T ) ≤ ‖ϕ̂‖Q̂p,n (1 +3∑i=0‖βi,k‖Lp,n×n(0,T ) ++α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) + l1q ‖β3,k‖Lp,n×n(0,T )) + ‖ϕk,b‖Lp,n(0,T )or∥∥ϕ0k∥∥Lp,n(0,T )≤M5 ‖ϕ̂‖Q̂p,n + ‖ϕk,b‖Lp,n(0,T ) , (22)whereM5 = 1 +3∑i=0‖βi,k‖Lp,n×n(0,T ) + α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) ++l1q ‖β3,k‖Lp,n×n(0,T ) .Now let’s estimate the norm of the vector ϕk,b(t). Obviously‖ϕk,b(t)‖ ≤ (Tα)1q ‖β1,k‖Lp,n×n(0,T ) ‖b‖Lp,n(G) ++(T (l− α))1q ‖β3,k‖Lp,n×n(0,T ) ‖b‖Lp,n(G) ++α1q ‖b(t, ·)‖Lp,n(0,l)‖g1,k‖L∞,n×n(0,T ) ++(l − α)1q ‖b(t, ·)‖Lp,n(0,l)‖g3,k‖L∞,n×n(0,T ) .Therefore‖ϕk,b‖Lp,n(0,T ) ≤ ck ‖b‖Lp,n(G), k = 1, 2, (23)ck = (Tα)1q ‖β1,k‖Lp,n×n(0,T ) + (T (l− α))1q ‖β3,k‖Lp,n×n(0,T ) ++α1q ‖g1,k‖L∞,n×n(0,T ) + (l − α)1q ‖g3,k‖L∞,n×n(0,T ) . (24)EJQTDE, 2010 No. 23, p. 8Then using (23) we get from (22)∥∥ϕ0k∥∥Lp,n(0,T )≤M5 ‖ϕ̂‖Q̂p,n + ck ‖b‖Lp,n(G), k = 1, 2. (25)Now, equation (5) is written in the form(Ωb)(t, x) = ϕ0(t, x), (t, x) ∈ G, (26)where(Ωb)(t, x) = b(t, x) +∫ t0∫ x0b(τ, ζ)q1(ζ, x)A0,0(t, x)dτdζ++∫ x0b(t, ζ)q1(ζ, x)A1,0(t, x)dζ +∫ t0b(τ, x)A0,1(t, x)dτ, (27)ϕ0(t, x) = ϕ0,0(t, x) + ϕ0,1(t, x);ϕ0,0(t, x) = ϕ(t, x) − b3(x)A0,1(t, x)−−b0A0,0(t, x) −∫ x0b3(ζ)A0,0(t, x)dζ;ϕ0,1(t, x) = −(∫ t0b1(τ)θ(α − x)dτ +∫ t0b2(τ)θ(x − α)dτ)A0,0(t, x)−−(b1(t)θ(α − x) + b2(t)θ(x − α))A1,0(t, x).In the expression of the vector ϕ0,0(t, x) there are the vectors ϕ(t, x), b3(x), b0and the given matrices A0,1(t, x), A0,0(t, x). Above we have estimated the normsof the vectors ϕ(t, x), b3(x), b0 by ‖ϕ̂‖Q̂p,n . Therefore, from the expression ofthe vector ϕ0,0(t, x) by means of the Holder and Minkowsky inequalities we caneasily get the estimate∥∥ϕ0,0∥∥Lp,n(G)≤M6 ‖ϕ̂‖Q̂p,n , (28)where M6 > 0 is a suitable constant.Further, from the expression of the vector ϕ0,1(t, x) it is seen that∥∥ϕ0,1(t, x)∥∥ ≤ (T1q ‖b1‖Lp,n(0,T ) θ(α− x)++T1q ‖b2‖Lp,n(0,T ) θ(x− α))‖A0,0(t, x)‖++(‖b1‖Lp,n(0,T ) θ(α− x) + ‖b2‖Lp,n(0,T ) θ(x − α))A01,0(x).Hence we get∥∥ϕ0,1∥∥Lp,n(G)≤ (T1q ‖A0,0‖Lp,n×n(G) +EJQTDE, 2010 No. 23, p. 9+∥∥A01,0∥∥Lp(0,l))(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T )) == M7(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T )), (29)whereM7 = T1q ‖A0,0‖Lp,n×n(G) +∥∥A01,0∥∥Lp(0,l).The operator Ω, defined by the equality (27) acts in Lp,n(G), is boundedand has a bounded inverse in it [3]. Therefore from (26) we have‖b‖Lp,n(G)≤∥∥Ω−1∥∥∥∥ϕ0∥∥Lp,n(G).Hence by (28) and (29) we get‖b‖Lp,n(G)≤∥∥Ω−1∥∥ (M6 ‖ϕ̂‖Q̂p,n +M7(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ))). (30)Take into account, (25) in (20) and get‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) ≤ 2M4(2M5 ‖ϕ̂‖Q̂p,n + (c1 + c2) ‖b‖Lp,n(G)). (31)Hence substituting (31) into (30) we get‖b‖Lp,n(G)≤∥∥Ω−1∥∥(M8 ‖ϕ̂‖Q̂p,n + 2M4M7(c1 + c2) ‖b‖Lp,n(G)),where M8 = M6 + 4M4M5M7 > 0.If we assumeγ = 2M4M7∥∥Ω−1∥∥ (c1 + c2) < 1, (∗)then we can obtain‖b‖Lp,n(G)≤M9 ‖ϕ̂‖Q̂p,n , (32)with constantM9 = (1 − γ)−1∥∥Ω−1∥∥M8.Taking into account (32) in (31) we have‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) ≤ 2M4(2M5 +M9(c1 + c2)) ‖ϕ̂‖Q̂p,n. (33)Now, summing up the inequalities (21), (32), (33) for the totalityEJQTDE, 2010 No. 23, p. 10b̂ = (b0, b1(t), b2(t), b3(x), b(t, x)) we have the estimate∥∥∥b̂∥∥∥Q̂p,n= ‖b0‖ + ‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) + ‖b3‖Lp,n(0,l) ++ ‖b‖Lp,n(G)≤M10 ‖ϕ̂‖Q̂p,n = M10∥∥∥L̂z∥∥∥Q̂p,n,whereM10 = 2M4(2M5 +M9(c1 + c2)) > 0andL̂z = ϕ̂, b̂ = Nz.Using the last inequality we get‖z‖cWp,n(G)≤M11 ‖Nz‖bQp,n≤M11M10∥∥∥L̂z∥∥∥bQp,n,with suitable constant M11 > 0 independent on z. Hence the following theoremis true.Theorem 1. Let the matrix ∆(t) be invertible for almost all t ∈ (0, T ) andconditions (11) and (*) be fulfilled, where M4 and M7 are the constants definedabove by using the number M1, the constants ck (k = 1, 2) are given by theformula (24), and the operator Ω is given by the relation (27). Then, for everysolution z of problem (1)-(4), the a priori estimate ‖z‖cWp,n(G)≤ M∥∥∥L̂z∥∥∥bQp,nholds, where M > 0 is a positive constant independent on z.The operator L̂ is a linear and bounded operator from Ŵp,n (G) to Q̂p,n.Therefore, there exists a bounded, conjugated operator L̂∗ :(Q̂p,n)∗→(Ŵp,n (G))∗.Using general forms of linear bounded functional determined on Q̂p,n andŴp,n (G) we can prove that L̂∗ is a bounded vector operator of the form L̂∗ =(ω0, ω1, ω2, ω3, ω) acting in the space Q̂q,n, where 1/p+ 1/q = 1. Therefore, wecan consider the equation L̂∗f̂ = ψ̂ as a conjugated equation for problem(1)-(4),where f̂ is a desired solution, ψ̂ is an element from(Ŵp,n (G))∗. It follows fromTheorem 1 that the following theorem is true.Theorem 2. Let the conditions of Theorem 1 be satisfied. Then problem (1)-(4) may have at most one solution z ∈ Ŵp,n (G), and the conjugated equationL̂∗f̂ = ψ̂ for any right hand side ψ̂ ∈(Ŵp,n (G))∗has at least one solutionf̂ ∈ Q̂q,n.EJQTDE, 2010 No. 23, p. 11References[1] Akhiev S. S., On some problems of optimization of systems of linear hyper-bolic equation of second order with discontinuous solutions and non-localboundary conditions. Dep.in Az NII NTI 37 p., No 1860 Az from (22.07.92)No 11, (253). (Russian)[2] Nakhushev A. M., Boundary value problems for loaded integro-differentialequations of hyperbolic type and some their applications to the prognosis ofsoil moisture. Differentialniye Uravneniya, (1979),. Vol.15, No 1, p. 96-105.[3] Orucoglu K., Akhiev S. S., The Riemann function for the third order one-dimensional pseudoparabolic equation, Acta Appl. Math. 53(1998), No 3,p. 353-370.[4] Sobolev S. L., Some applications of functional analysis to mathematicalphysics. Novosibirsk (1962), 256 p. (Russian).(Received November 19, 2008)EJQTDE, 2010 No. 23, p. 12

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Electronic Journal of Qualitative Theory of Differential Equations2010, No. 23, 1-12; http://www.math.u-szeged.hu/ejqtde/A PRIORI ESTIMATE FORDISCONTINUOUS SOLUTIONS OF ASECOND ORDER LINEAR HYPERBOLICPROBLEMS. S. Akhiev(Azerbaijan State Pedagogical University)Abstract. In the paper we investigate a non-local contact-boundary value problemfor a system of second order hyperbolic equations with discontinuous solutions. Undersome conditions on input data a priori estimate is obtained for the solution of thisproblem.In the paper we consider the following hyperbolic system:(Lz) (t, x) ≡ ztx (t, x) + z (t, x)A0,0 (t, x) ++zt (t, x)A1,0 (t, x) + zx (t, x)A0,1 (t, x) =ϕ (t, x) ,(t, x) ∈ G = G0 ∪G1, G0 = (0, T )× (0, α) , G1 = (0, T )× (α, l) ,(1)where z (t, x) = (z1 (t, x) , ...., zn (t, x)) is the desired vector-function; Ai,j (t, x),i, j = 0, 1 are the given n× n-matrices on G; ϕ (t, x) is the given n-dimensionalvector-function on G; α is a fixed point from (0, l) .For the system (1) we give the following non-local contact boundary condi-tions(Lkz) (t) ≡ z (t, 0)β0,k (t) + z (t, α− 0)β1,k (t) + z (t, α+ 0)β2,k (t)++z (t, l)β3,k (t) + zt (t, 0) g0,k (t) + zt (t, α− 0) g1,k (t) + zt (t, α+ 0) g2,k (t)++zt (t, l) g3,k (t) = ϕk (t) , t ∈ (0, T ) , k = 1, 2 ; (2)(L3z) (x) ≡ zx (0, x) = ϕ3 (x) , x ∈ (0, l) ; (3)L0z ≡ z (0, 0) = ϕ0. (4)Here: βi,k (t), gi,k (t) , i = 0, 1, 2, 3; k = 1, 2 are the given n× n matrices on(0, T ); ϕk (t) , k = 1, 2 are the given n-dimensional vector-functions on (0, T ) ;ϕ3 (x) is the given n-dimensional vector-function on (0, l) ; ϕ0 is the given con-stant n-dimensional vector.EJQTDE, 2010 No. 23, p. 1We assume that the following conditions are satisfied:1) The matrices Ai,j (t, x) are measurable on G, A0,0 ∈ Lp,n×n (G) ; thereexit the functions A01,0 ∈ Lp (0, l) and A00,1 ∈ Lp (0, T ), such that ‖A1,0 (t, x)‖ ≤A01,0 (x), ‖A0,1 (t, x)‖ ≤ A00,1 (t) almost everywhere on G, where Lp,n×n (G) ,1 ≤ p ≤ ∞ is a Banach space of n × n matrices g = (gij) with elementsgij ∈ Lp (G) , wherein the norm is determined by the equality ‖g‖Lp,n×n(G) =∥∥g0∥∥Lp(G), moreover g0 = ‖g‖ ≡n∑i,j=1|gij | is the norm of the matrix g;2) βi,k ∈ Lp,n×n (0, T ) and gi,k ∈ L∞,n×n (0, T ) ;3) ϕ ∈ Lp,n (G) , ϕk ∈ Lp,n (0, T ), ϕ3 ∈ Lp,n (0, l), where Lp,n (G) , 1 ≤ p ≤∞, is a space of n-dimensional vector-functions ϕ = (ϕ1, ..., ϕn) with elementsfrom Lp (G); norm of ϕ ∈ Lp,n (G) is defined as ‖ϕ‖Lp,n(G) =∥∥ϕ0∥∥Lp(G)andϕ0(t, x) = ‖ϕ(t, x)‖ =n∑i=1|ϕi(t, x)| is norm of n-vectors ϕ(t, x) ∈ Rn for fixed(t, x) ∈ G. Rn is the space of all vectors ρ = (ρ1, . . . , ρn) with norm ‖ρ‖ =n∑i=1|ρi|.Non-local boundary value problems for integro-differential equations withcontinuous coefficients were studied in the paper [2].We‘ll consider the solution of problem (1)-(4) in the space Ŵp,n (G) , 1 ≤p ≤ ∞, [4] (p. 52) of all n-dimensional vector-functions z (t, x), which oneach domain Gk (k = 0, 1) belong to Wp,n (Gk) and are continuous at the point(0, α). Here Wp,n (Gk) is a space of all n-dimensional vector-functions z ∈Lp,n (Gk) , possessing generalized in S.L.Sobolev’s sense derivatives zt, zx andztx from Lp,n (Gk) , k = 0, 1. We‘ll define the norm in the space Ŵp,n (G) bythe equality [4] (p. 54)‖z‖cWp,n(G) =1∑k=0‖z‖Wp,n(Gk) ,where‖z‖Wp,n(Gk) = ‖z‖Lp,n(Gk) + ‖zt‖Lp,n(Gk) + ‖zx‖Lp,n(Gk) + ‖ztx‖Lp,n(Gk) .Since the operator Nz = (z (0, 0) , zt (t, 0) , zt (t, α+ 0) , zx (0, x) , ztx (t, x)),brings about isomorphism from Ŵp,n (G) to Q̂p,n=Rn×Lp,n (0, T )×Lp,n (0, T )×Lp,n (0, l) × Lp,n (G) , [1], we can reduce problem (1) - (4) to the followingoperator equationL̂z = ϕ̂,where L̂ = (L0, L1, L2, L3, L) , z ∈ Ŵp,n (G) is desired solution andϕ̂ = (ϕ0, ϕ1, ϕ2, ϕ3, ϕ) ∈ Q̂p,n is the given element. This equation is equiva-EJQTDE, 2010 No. 23, p. 2lent to the system of integro-algebraic equations with respect to elements of fivecomponentsb̂ = (b0, b1 (t) , b2 (t) , b3 (x) , b (t, x)) ≡≡ (z (0, 0) , zt (t, 0) , zt (t, a+ 0) , zx (0, x) , ztx (t, x))of the space Q̂p,n:b (t, x) +∫ t0∫ x0b (τ, ζ) q1 (ζ, x)A0,0 (t, x) dτdζ++∫ x0b (t, ζ) q1 (ζ, x)A1,0 (t, x) dζ +∫ t0b (τ, x)A0,1 (t, x) dτ++(∫ t0b1 (τ) θ (α− x) dτ +∫ t0b2 (τ) θ (x− α) dτ)A0,0 (t, x) ++ (b1 (t) θ (α− x) + b2 (t) θ (x− α))A1,0 (t, x) + b3 (x)A0,1 (t, x) ++∫ x0b3 (ζ)A0,0 (t, x) dζ + b0A0,0 (t, x) = ϕ(t, x), (t, x) ∈ G; (5)b1 (t) (g0,k (t) + g1,k (t)) + b2 (t) (g2,k (t) + g3,k (t))++∫ t0b1 (τ) (β0,k (t) + β1,k (t)) dτ++∫ t0b2 (τ) (β2,k (t) + β3,k (t)) dτ = ϕ0k (t) , t ∈ (0, T ) , k = 1, 2 ; (6)whereϕ0k(t) = ϕk(t)− b0(β0,k(t) + β1,k(t) + β2,k(t) + β3,k(t))−−∫ α0b3(ζ)(β1,k(t) + β2,k(t))dζ −∫ l0b3(ζ)β3,k(t)dζ − ϕk,b(t);ϕk,b(t) =∫ t0∫ α0b(τ, ζ)β1,k(t)dτdζ +∫ t0∫ lαb(τ, ζ)β3,k(t)dτdζ++∫ α0b(t, ζ)g1,k(t)dζ +∫ lαb(t, ζ)g3,k(t)dζ, t ∈ (0, T ); (6∗)b3 (x) = ϕ3 (x) , x ∈ (0, l) ; (7)b0 = ϕ0, (8)where θ (y) is one –dimensional Heaviside function on R = R1 and q1 (ζ, x) =θ (ζ − α) θ (x− α) + θ (α− x).EJQTDE, 2010 No. 23, p. 3If we succeed to estimate the components b0, b1 (t) , b2 (t) , b3 (x) , b (t, x) ofthe vector b̂, on the basis of [1] we get a priori estimate for the solution z ∈Ŵp,n (G) of problem (1)-(4)z(t, x) = b0 + θ (α− x)∫ T0b1(τ)θ (t− τ) dτ++θ (x− α)∫ T0b2(τ)θ (t− τ) dτ +∫ l0b3(ζ)θ (x− ζ) dζ++∫G∫θ (t− τ) θ (x− ζ) q1 (ζ, x) b(τ, ζ)dτdζ , (t, x) ∈ G. (9)The components b1 (t) , b2 (t) , b (t, x) are determined from the system ofequations (5),(6), since the components b0, b3 (x) are explicitly given by con-ditions (7), (8), therefore, it remains to estimate only b1 (t) , b2 (t) , b (t, x).It is obvious that by means of the matrix∆(t) = g0,1(t) + g1,1(t)g2,1(t) + g3,1(t)g0,2(t) + g1,2(t)g2,2(t) + g3,2(t)we can write the equality (6) in the compact form(b1(t), b2(t))∆(t) +∫ t0(b1(τ), b2(τ))B(t)dτ = (ϕ01(t), ϕ02(t)), t ∈ (0, T ), (10)whereB(t) = β0,1(t) + β1,1(t)β2,1(t) + β3,1(t)β0,2(t) + β1,2(t)β2,2(t) + β3,2(t)Assume that almost for all t ∈ (0, T ) the matrix ∆(t) is invertible and itholds‖∆(t)‖ ≤M1,∥∥∆−1(t)∥∥ ≤M1 (11)in the sense of almost everywhere on (0, T ). Then, from (10) we have(b1(t), b2(t)) +∫ t0(b1(τ), b2(τ))B1(t)dτ = (ϕ01(t), ϕ02(t))∆−1(t), t ∈ (0, T ), (12)whereB1(t) = B(t)∆−1(t).Passing in (12) to the vector norm we haveα(t) ≤∫ t0α(τ)l(t)dτ + S0(t), t ∈ (0, T ), (13)EJQTDE, 2010 No. 23, p. 4whereα(t) = ‖b1(t)‖+ ‖b2(t)‖ ,l(t) = ‖B1(t)‖ ≤M1 ‖B(t)‖ ,S0(t) =∥∥(ϕ01(t), ϕ02(t))∆−1(t)∥∥ ≤M1(∥∥ϕ01(t)∥∥+ ∥∥ϕ02(t)∥∥);here and below Mi are constants independent on ϕˆ = (ϕ0, ϕ1, ϕ2, ϕ3, ϕ).Let the point τ ∈ (0, T ) be fixed and t ∈ (0, τ). Then integrating (13) withrespect to t on (0, τ) we getR(τ) ≤∫ τ0R(t)l(t)dt+ S1(τ), τ ∈ (0, T ), (14)whereS1(τ) =∫ τ0S0(t)dt,R(τ) =∫ τ0α(t)dt.We write the inequality (14) in the formR(τ) ≤ R1(τ) + S1(τ) + ε, (15)where ε > 0 is an arbitrary number andR1(τ) =∫ τ0R(t)l(t)dt.HenceR (τ)R1 (τ) + S1 (τ) + ε≤ 1, τ ∈ (0, T ).ThereforeR (τ) l (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, T ),orR˙1 (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, T ), (16)where the sign of point over some function of one argument means its firstderivative.The function S1(τ) is a monotonically increasing function.Therefore, if t isfixed and τ ∈ (0, t), then S1(τ) ≤ S1(t). Therefore from (16) we haveR˙1 (τ)R1 (τ) + S1 (t) + ε≤R˙1 (τ)R1 (τ) + S1 (τ) + ε≤ l(τ), τ ∈ (0, t),EJQTDE, 2010 No. 23, p. 5integrating it with respect to τ on (0, t) we getlnR1 (t) + S1 (t) + εR1 (0) + S1 (t) + ε≤∫ t0l (τ) dτorR1(t) + S1(t) + ε ≤ (S1(t) + ε)eRt0l(τ)dτ .Taking this into account in (15) we getR(t) ≤ S1(t)eRt0l(τ)dτ . (17)Writing (13) in the formα(t) ≤ R(t)l(t) + S0(t)and using (17) we getα(t) ≤ l(t)eRt0l(τ)dτ∫ t0S0(τ)dτ + S0(t) (18)Thus, we have provedLemma 1. If, for some non-negative functions α, l, S0 ∈ Lp(0, T ), the inequal-ity (13) holds, then the function α(t) also satisfies the condition (18).Taking into account the expression of the function l(t) in (18) we getα(t) ≤M2 ‖B(t)‖∫ t0S0(τ)dτ + S0(t), t ∈ (0, T ), (19)whereM2 =M1 exp(M1∫ T0‖B(τ)‖ dτ).Notice that from the conditions imposed on the matrix functions βi,k(t) itfollows that ‖B(·)‖ ∈ Lp(0, T ). Therefore M2 < +∞.Now by means of this lemma for the sum α(t) = ‖b1(t)‖ + ‖b2(t)‖ we havethe estimateα(t) = ‖b1(t)‖ + ‖b2(t)‖ ≤M2∫ t0S0(τ)dτ ‖B(t)‖ + S0(t), t ∈ (0, T ).Therefore, using the Ho¨lder inequality, we obtain‖bk(t)‖ ≤ S0(t) +M2T1q∥∥S0∥∥Lp(0,T )‖B(t)‖ , t ∈ (0, T ), k = 1, 2,here and below q = p/(p− 1) denotes the number conjugate to p.EJQTDE, 2010 No. 23, p. 6Here, passing to the norm, by Minkowsky inequality we get‖bk‖Lp,n(0,T ) ≤∥∥S0∥∥Lp,(0,T )(1 +M2T1q ‖B‖Lp,2n×2n(0,T )) ==M3∥∥S0∥∥Lp(0,T ), k = 1, 2;M3 = 1 +M2T1q ‖B‖Lp,2n×2n(0,T ).Obviously∥∥S0∥∥Lp(0,T )≤M1(∥∥ϕ01∥∥Lp,n(0,T ) + ∥∥ϕ02∥∥Lp,n(0,T )).Therefore‖bk‖Lp,n(0,T ) ≤M4(∥∥ϕ01∥∥Lp,n(0,T ) + ∥∥ϕ02∥∥Lp,n(0,T )), k = 1, 2, (20)where M4 =M3M1.Now, let’s estimate the norms∥∥ϕ0k∥∥Lp,n(0,T ), k = 1, 2. Obviously if the vectorz ∈ Ŵp,n (G) satisfies the conditions (1)-(4), then its independent elementsbˆ = (z(0, 0), zt(t, 0), zt(t, α+0), zx(0, x), ztx(t, x))= (b0, b1(t), b2(t), b3(x), b(t, x))satisfy the equalities (5)-(8) and therewith‖b0‖ ≤ ‖ϕˆ‖Qˆp,n‖b3‖Lp,n(0,l) ≤ ‖ϕˆ‖Qˆp,n (21)where‖ϕˆ‖Qˆp,n=∥∥∥Lˆz∥∥∥Wˆp,n(G)=∥∥∥Lˆ(N−1(bˆ))∥∥∥Qˆp,n== ‖ϕ0‖+ ‖ϕ1‖Lp,n(0,T ) + ‖ϕ2‖Lp,n(0,T ) ++ ‖ϕ3‖Lp,n(0,l) + ‖ϕ‖Lp,n(G) , z = N−1bˆ.Therefore, from the expressions (6*) of the vectors ϕ0k(t) we have∥∥ϕ0k(t)∥∥ ≤ ‖ϕk(t)‖+ ‖b0‖ (‖β0,k(t)‖ + ‖β1,k(t)‖++ ‖β2,k(t)‖+ ‖β3,k(t)‖) + ‖b3‖Lp,n(0,l) α1q××(‖β1,k(t)‖+ ‖β2,k(t)‖) + l1q ‖b3‖Lp,n(0,l) ‖β3,k(t)‖ + ‖ϕk,b(t)‖ , k = 1, 2.Hence by means of the Minkowsky inequality allowing for (20) we haveEJQTDE, 2010 No. 23, p. 7∥∥ϕ0k∥∥Lp,n(0,T ) ≤ ‖ϕk‖Lp,n(0,T ) + ‖b0‖ 3∑i=0‖βi,k‖Lp,n×n(0,T ) ++ ‖b3‖Lp,n(0,l) α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) ++ ‖b3‖Lp,n(0,l) l1q ‖β3,k‖Lp,n×n(0,T ) ++ ‖ϕk,b‖Lp,n(0,T ) ≤ ‖ϕˆ‖Qˆp,n (1 +3∑i=0‖βi,k‖Lp,n×n(0,T ) ++α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) + l1q ‖β3,k‖Lp,n×n(0,T )) + ‖ϕk,b‖Lp,n(0,T )or∥∥ϕ0k∥∥Lp,n(0,T ) ≤M5 ‖ϕˆ‖Qˆp,n + ‖ϕk,b‖Lp,n(0,T ) , (22)whereM5 = 1 +3∑i=0‖βi,k‖Lp,n×n(0,T ) + α1q2∑i=1‖βi,k‖Lp,n×n(0,T ) ++l1q ‖β3,k‖Lp,n×n(0,T ) .Now let’s estimate the norm of the vector ϕk,b(t). Obviously‖ϕk,b(t)‖ ≤ (Tα)1q ‖β1,k‖Lp,n×n(0,T ) ‖b‖Lp,n(G) ++(T (l− α))1q ‖β3,k‖Lp,n×n(0,T ) ‖b‖Lp,n(G) ++α1q ‖b(t, ·)‖Lp,n(0,l)‖g1,k‖L∞,n×n(0,T ) ++(l − α)1q ‖b(t, ·)‖Lp,n(0,l)‖g3,k‖L∞,n×n(0,T ) .Therefore‖ϕk,b‖Lp,n(0,T ) ≤ ck ‖b‖Lp,n(G), k = 1, 2, (23)ck = (Tα)1q ‖β1,k‖Lp,n×n(0,T ) + (T (l− α))1q ‖β3,k‖Lp,n×n(0,T ) ++α1q ‖g1,k‖L∞,n×n(0,T ) + (l − α)1q ‖g3,k‖L∞,n×n(0,T ) . (24)EJQTDE, 2010 No. 23, p. 8Then using (23) we get from (22)∥∥ϕ0k∥∥Lp,n(0,T ) ≤M5 ‖ϕˆ‖Qˆp,n + ck ‖b‖Lp,n(G) , k = 1, 2. (25)Now, equation (5) is written in the form(Ωb)(t, x) = ϕ0(t, x), (t, x) ∈ G, (26)where(Ωb)(t, x) = b(t, x) +∫ t0∫ x0b(τ, ζ)q1(ζ, x)A0,0(t, x)dτdζ++∫ x0b(t, ζ)q1(ζ, x)A1,0(t, x)dζ +∫ t0b(τ, x)A0,1(t, x)dτ, (27)ϕ0(t, x) = ϕ0,0(t, x) + ϕ0,1(t, x);ϕ0,0(t, x) = ϕ(t, x) − b3(x)A0,1(t, x)−−b0A0,0(t, x) −∫ x0b3(ζ)A0,0(t, x)dζ;ϕ0,1(t, x) = −(∫ t0b1(τ)θ(α − x)dτ +∫ t0b2(τ)θ(x − α)dτ)A0,0(t, x)−−(b1(t)θ(α − x) + b2(t)θ(x − α))A1,0(t, x).In the expression of the vector ϕ0,0(t, x) there are the vectors ϕ(t, x), b3(x), b0and the given matrices A0,1(t, x), A0,0(t, x). Above we have estimated the normsof the vectors ϕ(t, x), b3(x), b0 by ‖ϕˆ‖Qˆp,n . Therefore, from the expression ofthe vector ϕ0,0(t, x) by means of the Holder and Minkowsky inequalities we caneasily get the estimate∥∥ϕ0,0∥∥Lp,n(G)≤M6 ‖ϕˆ‖Qˆp,n , (28)where M6 > 0 is a suitable constant.Further, from the expression of the vector ϕ0,1(t, x) it is seen that∥∥ϕ0,1(t, x)∥∥ ≤ (T 1q ‖b1‖Lp,n(0,T ) θ(α− x)++T1q ‖b2‖Lp,n(0,T ) θ(x− α))‖A0,0(t, x)‖++(‖b1‖Lp,n(0,T ) θ(α− x) + ‖b2‖Lp,n(0,T ) θ(x − α))A01,0(x).Hence we get∥∥ϕ0,1∥∥Lp,n(G)≤ (T1q ‖A0,0‖Lp,n×n(G) +EJQTDE, 2010 No. 23, p. 9+∥∥A01,0∥∥Lp(0,l))(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T )) ==M7(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T )), (29)whereM7 = T1q ‖A0,0‖Lp,n×n(G) +∥∥A01,0∥∥Lp(0,l) .The operator Ω, defined by the equality (27) acts in Lp,n(G), is boundedand has a bounded inverse in it [3]. Therefore from (26) we have‖b‖Lp,n(G)≤∥∥Ω−1∥∥∥∥ϕ0∥∥Lp,n(G).Hence by (28) and (29) we get‖b‖Lp,n(G)≤∥∥Ω−1∥∥ (M6 ‖ϕˆ‖Qˆp,n +M7(‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ))). (30)Take into account, (25) in (20) and get‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) ≤ 2M4(2M5 ‖ϕˆ‖Qˆp,n + (c1 + c2) ‖b‖Lp,n(G)). (31)Hence substituting (31) into (30) we get‖b‖Lp,n(G)≤∥∥Ω−1∥∥(M8 ‖ϕˆ‖Qˆp,n + 2M4M7(c1 + c2) ‖b‖Lp,n(G)) ,where M8 =M6 + 4M4M5M7 > 0.If we assumeγ = 2M4M7∥∥Ω−1∥∥ (c1 + c2) < 1, (∗)then we can obtain‖b‖Lp,n(G)≤M9 ‖ϕˆ‖Qˆp,n , (32)with constantM9 = (1− γ)−1∥∥Ω−1∥∥M8.Taking into account (32) in (31) we have‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) ≤ 2M4(2M5 +M9(c1 + c2)) ‖ϕˆ‖Qˆp,n. (33)Now, summing up the inequalities (21), (32), (33) for the totalityEJQTDE, 2010 No. 23, p. 10bˆ = (b0, b1(t), b2(t), b3(x), b(t, x)) we have the estimate∥∥∥bˆ∥∥∥Qˆp,n= ‖b0‖+ ‖b1‖Lp,n(0,T ) + ‖b2‖Lp,n(0,T ) + ‖b3‖Lp,n(0,l) ++ ‖b‖Lp,n(G)≤M10 ‖ϕˆ‖Qˆp,n =M10∥∥∥Lˆz∥∥∥Qˆp,n,whereM10 = 2M4(2M5 +M9(c1 + c2)) > 0andL̂z = ϕ̂, b̂ = Nz.Using the last inequality we get‖z‖cWp,n(G) ≤M11 ‖Nz‖ bQp,n ≤M11M10∥∥∥L̂z∥∥∥ bQp,n ,with suitable constant M11 > 0 independent on z. Hence the following theoremis true.Theorem 1. Let the matrix ∆(t) be invertible for almost all t ∈ (0, T ) andconditions (11) and (*) be fulfilled, where M4 and M7 are the constants definedabove by using the number M1, the constants ck (k = 1, 2) are given by theformula (24), and the operator Ω is given by the relation (27). Then, for everysolution z of problem (1)-(4), the a priori estimate ‖z‖cWp,n(G) ≤ M∥∥∥L̂z∥∥∥ bQp,nholds, where M > 0 is a positive constant independent on z.The operator L̂ is a linear and bounded operator from Ŵp,n (G) to Q̂p,n.Therefore, there exists a bounded, conjugated operator L̂∗ :(Q̂p,n)∗→(Ŵp,n (G))∗.Using general forms of linear bounded functional determined on Q̂p,n andŴp,n (G) we can prove that L̂∗ is a bounded vector operator of the form L̂∗ =(ω0, ω1, ω2, ω3, ω) acting in the space Q̂q,n, where 1/p+ 1/q = 1. Therefore, wecan consider the equation L̂∗f̂ = ψ̂ as a conjugated equation for problem(1)-(4),where f̂ is a desired solution, ψ̂ is an element from(Ŵp,n (G))∗. It follows fromTheorem 1 that the following theorem is true.Theorem 2. Let the conditions of Theorem 1 be satisfied. Then problem (1)-(4) may have at most one solution z ∈ Ŵp,n (G), and the conjugated equationL̂∗f̂ = ψ̂ for any right hand side ψ̂ ∈(Ŵp,n (G))∗has at least one solutionf̂ ∈ Q̂q,n.EJQTDE, 2010 No. 23, p. 11References[1] Akhiev S. S., On some problems of optimization of systems of linear hyper-bolic equation of second order with discontinuous solutions and non-localboundary conditions. Dep.in Az NII NTI 37 p., No 1860 Az from (22.07.92)No 11, (253). (Russian)[2] Nakhushev A. M., Boundary value problems for loaded integro-differentialequations of hyperbolic type and some their applications to the prognosis ofsoil moisture. Differentialniye Uravneniya, (1979),. Vol.15, No 1, p. 96-105.[3] Orucoglu K., Akhiev S. S., The Riemann function for the third order one-dimensional pseudoparabolic equation, Acta Appl. Math. 53(1998), No 3,p. 353-370.[4] Sobolev S. L., Some applications of functional analysis to mathematicalphysics. Novosibirsk (1962), 256 p. (Russian).(Received November 19, 2008)EJQTDE, 2010 No. 23, p. 12

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A priori estimate for discontinuous solutions of a second order linear hyperbolic problem

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