A PRIORI ESTIMATE FOR DISCONTINUOUS SOLUTIONS OF A SECOND ORDER LINEAR HYPERBOLIC PROBLEM

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.


S. S. Akhiev (Azerbaijan State Pedagogical University)
Abstract.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.
We'll consider the solution of problem ( 1)-( 4) in the space W p,n (G) , 1 ≤ p ≤ ∞, [4] (p. 52) of all n-dimensional vector-functions z (t, x), which on each domain G k (k = 0, 1) belong to W p,n (G k ) and are continuous at the point (0, α).Here W p,n (G k ) is a space of all n-dimensional vector-functions z ∈ L p,n (G k ) , possessing generalized in S.L.Sobolev's sense derivatives z t , z x and z tx from L p,n (G k ) , k = 0, 1. We'll define the norm in the space W p,n (G) by the equality [4] (p. 54) [1], we can reduce problem (1) -(4) to the following operator equation Lz = ϕ, where EJQTDE, 2010 No. 23, p. 3 If we succeed to estimate the components b 0 , b 1 (t) , b 2 (t) , b 3 (x) , b (t, x) of the vector b, on the basis of [1] we get a priori estimate for the solution z ∈ W p,n (G) of problem ( 1)-( 4) The components b 1 (t) , b 2 (t) , b (t, x) are determined from the system of equations ( 5),(6), since the components b 0 , b 3 (x) are explicitly given by conditions (7), (8), therefore, it remains to estimate only b It is obvious that by means of the matrix we can write the equality (6) in the compact form where Assume that almost for all t ∈ (0, T ) the matrix ∆(t) is invertible and it holds in the sense of almost everywhere on (0, T ).Then, from (10) we have where Passing in (12) to the vector norm we have where ); here and below M i are constants independent on φ = (ϕ 0 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ).
Let the point τ ∈ (0, T ) be fixed and t ∈ (0, τ ).Then integrating (13) with respect to t on (0, τ ) we get where We write the inequality (14) in the form where ε > 0 is an arbitrary number and where the sign of point over some function of one argument means its first derivative.
The function S 1 (τ ) is a monotonically increasing function.Therefore, if t is fixed and τ ∈ (0, t), then S 1 (τ ) ≤ S 1 (t).Therefore from (16) we have EJQTDE, 2010 No. 23, p. 5 integrating it with respect to τ on (0, t) we get ln Taking this into account in (15) we get Writing (13) in the form and using (17) we get Thus, we have proved Lemma 1. If, for some non-negative functions α, l, S 0 ∈ L p (0, T ), the inequality (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 get where Notice that from the conditions imposed on the matrix functions β i,k (t) it follows that B(•) ∈ L p (0, T ).Therefore M 2 < +∞.
Now by means of this lemma for the sum α(t) = b 1 (t) + b 2 (t) we have the estimate Therefore, using the Hölder inequality, we obtain here and below q = p/(p − 1) denotes the number conjugate to p.
The operator Ω, defined by the equality (27) acts in L p,n (G), is bounded and has a bounded inverse in it [3].Therefore from (26) we have Hence by ( 28) and (29) we get Take into account, (25) in (20) and get Hence substituting (31) into (30) we get where with constant Taking into account (32) in (31) we have , with suitable constant M 11 > 0 independent on z.Hence the following theorem is true.
Theorem 1.Let the matrix ∆ (t) be invertible for almost all t ∈ (0, T ) and conditions (11) and (*) be fulfilled, where M 4 and M 7 are the constants defined above by using the number M 1 , the constants c k (k = 1, 2) are given by the formula (24), and the operator Ω is given by the relation (27).Then, for every solution z of problem (1)-( 4), the a priori estimate z holds, where M > 0 is a positive constant independent on z.
The operator L is a linear and bounded operator from W p,n (G) to Q p,n .
Using general forms of linear bounded functional determined on Q p,n and W p,n (G) we can prove that L * is a bounded vector operator of the form L * = (ω 0 , ω 1 , ω 2 , ω 3 , ω) acting in the space Q q,n , where 1/p + 1/q = 1.Therefore, we can consider the equation L * f = ψ as a conjugated equation for problem(1)-( 4), where f is a desired solution, ψ is an element from W p,n (G) * . It follows from Theorem 1 that the following theorem is true.

c
Wp,n(G) ≤ M Lz b Qp,n

Theorem 2 .
Let the conditions of Theorem 1 be satisfied.Then problem (1)-(4) may have at most one solution z ∈ W p,n (G), and the conjugated equation L * f = ψ for any right hand side ψ ∈ W p,n (G) * has at least one solution f ∈ Q q,n .EJQTDE, 2010 No. 23, p. 11