Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 23, 1-12; http://www.math.u-szeged.hu/ejqtde/

A PRIORI ESTIMATE FOR DISCONTINUOUS SOLUTIONS OF A

SECOND ORDER LINEAR HYPERBOLIC 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.

In the paper we consider the following hyperbolic system:

\begin{matrix}(Lz) (t, x) ≡ z_{tx} (t, x) + z (t, x) A_{0,0} (t, x) + \\ +z_{t} (t, x) A_{1,0} (t, x) + z_{x} (t, x) A_{0,1} (t, x) =φ (t, x) , \\ (t, x) ∈ G = G_{0} ∪ G_{1}, G_{0} = (0, T) × (0, α) , G_{1} = (0, T) × (α, l) , \end{matrix} (1)

where z (t, x) = (z_{1} (t, x) , ...., z_{n} (t, x)) is the desired vector-function; A_{i,j} (t, x), i, j = 0, 1 are the given n × n-matrices on G; φ (t, x) is the given n-dimensional vector-function on G; α is a fixed point from (0, l) .

For the system (1) we give the following non-local contact boundary conditions

(L_{k}z) (t) ≡ z (t, 0) β_{0,k} (t) + z (t, α − 0) β_{1,k} (t) + z (t, α + 0) β_{2,k} (t) +

+z (t, l) β_{3,k} (t) + z_{t} (t, 0) g_{0,k} (t) + z_{t} (t, α − 0) g_{1,k} (t) + z_{t} (t, α + 0) g_{2,k} (t) +

+z_{t} (t, l) g_{3,k} (t) = φ_{k} (t) , t ∈ (0, T) , k = 1, 2 ;  (2)
(L_{3}z) (x) ≡ z_{x} (0, x) = φ_{3} (x) , x ∈ (0, l) ;  (3)
L_{0}z ≡ z (0, 0) = φ_{0}.  (4)

Here: β_{i,k} (t), g_{i,k} (t) , i = 0, 1, 2, 3; k = 1, 2 are the given n × n matrices on (0, T); φ_{k} (t) , k = 1, 2 are the given n-dimensional vector-functions on (0, T) ; φ_{3} (x) is the given n-dimensional vector-function on (0, l) ; φ_{0} is the given constant n-dimensional vector.

We assume that the following conditions are satisfied:

1) The matrices A_{i,j} (t, x) are measurable on G, A_{0,0} ∈ L_{p,n×n} (G) ; there exit the functions A_{1,0}^{0} ∈ L_{p} (0, l) and A_{0,1}^{0} ∈ L_{p} (0, T), such that ∥A_{1,0} (t, x)∥ ≤ A_{1,0}^{0} (x), ∥A_{0,1} (t, x)∥ ≤ A_{0,1}^{0} (t) almost everywhere on G, where L_{p,n×n} (G) , 1 ≤ p ≤ ∞ is a Banach space of n × n matrices g = (g_{ij}) with elements g_{ij} ∈ L_{p} (G) , wherein the norm is determined by the equality ∥g∥_{L_{p,n×n}(G)} =

∥g^{0}∥_{L_{p}(G)} , moreover g^{0} = ∥g∥ ≡ ∑_{i,j=1}^{n} |g_{ij}| is the norm of the matrix g;

2) β_{i,k} ∈ L_{p,n×n} (0, T) and g_{i,k} ∈ L_{∞,n×n} (0, T) ;

3) φ ∈ L_{p,n} (G) , φ_{k} ∈ L_{p,n} (0, T), φ_{3} ∈ L_{p,n} (0, l), where L_{p,n} (G) , 1 ≤ p ≤ ∞, is a space of n-dimensional vector-functions φ = (φ_{1}, ..., φ_{n}) with elements from L_{p} (G); norm of φ ∈ L_{p,n} (G) is defined as ∥φ∥_{L_{p,n}(G)} = ∥φ^{0}∥_{L_{p}(G)} and

φ^{0}(t, x) = ∥φ(t, x)∥ = ∑_{i=1}^{n} |φ_{i}(t, x)| is norm of n-vectors φ(t, x) ∈ R^{n} for fixed (t, x) ∈ G. R^{n} is the space of all vectors ρ = (ρ_{1}, …, ρ_{n}) with norm ∥ρ∥ =

∑_{i=1}^{n} |ρ_{i}|.

Non-local boundary value problems for integro-differential equations with continuous coefficients were studied in the paper [2].

We‘ll consider the solution of problem (1)-(4) in the space Ŵ_{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 Ŵ_{p,n} (G) by the equality [4] (p. 54)

∥z∥_{Ŵ_{p,n}(G)} = ∑_{k=0}^{1} ∥z∥_{W_{p,n}(G_{k})} ,

where

∥z∥_{W_{p,n}(G_{k})} = ∥z∥_{L_{p,n}(G_{k})} + ∥z_{t}∥_{L_{p,n}(G_{k})} + ∥z_{x}∥_{L_{p,n}(G_{k})} + ∥z_{tx}∥_{L_{p,n}(G_{k})} .

Since the operator Nz = (z (0, 0) , z_{t} (t, 0) , z_{t} (t, α + 0) , z_{x} (0, x) , z_{tx} (t, x)), brings about isomorphism from Ŵ_{p,n} (G) to Q̂_{p,n} =R^{n}×L_{p,n} (0, T)×L_{p,n} (0, T) × L_{p,n} (0, l) × L_{p,n} (G) , [1], we can reduce problem (1) - (4) to the following operator equation

L̂z = φ̂,

where L̂ = (L_{0}, L_{1}, L_{2}, L_{3}, L) , z ∈ Ŵ_{p,n} (G) is desired solution and φ̂ = (φ_{0}, φ_{1}, φ_{2}, φ_{3}, φ) ∈ Q̂_{p,n} is the given element. This equation is equivalent to the system of integro-algebraic equations with respect to elements of five components

b̂ = (b_{0}, b_{1} (t) , b_{2} (t) , b_{3} (x) , b (t, x)) ≡
≡ (z (0, 0) , z_{t} (t, 0) , z_{t} (t, a + 0) , z_{x} (0, x) , z_{tx} (t, x))

of the space Q̂_{p,n}:

b (t, x) + ∫_{0}^{t} ∫_{0}^{x} b (τ, ζ) q_{1} (ζ, x) A_{0,0} (t, x) dτdζ+
+ ∫_{0}^{x} b (t, ζ) q_{1} (ζ, x) A_{1,0} (t, x) dζ + ∫_{0}^{t} b (τ, x) A_{0,1} (t, x) dτ+
+ (∫_{0}^{t} b_{1} (τ) θ (α − x) dτ + ∫_{0}^{t} b_{2} (τ) θ (x − α) dτ) A_{0,0} (t, x) +
+ (b_{1} (t) θ (α − x) + b_{2} (t) θ (x − α)) A_{1,0} (t, x) + b_{3} (x) A_{0,1} (t, x) +
+ ∫_{0}^{x} b_{3} (ζ) A_{0,0} (t, x) dζ + b_{0}A_{0,0} (t, x) = φ(t, x), (t, x) ∈ G;  (5)
b_{1} (t) (g_{0,k} (t) + g_{1,k} (t)) + b_{2} (t) (g_{2,k} (t) + g_{3,k} (t)) +
+ ∫_{0}^{t} b_{1} (τ) (β_{0,k} (t) + β_{1,k} (t)) dτ+
+ ∫_{0}^{t} b_{2} (τ) (β_{2,k} (t) + β_{3,k} (t)) dτ = φ _{k}^{0} (t) , t ∈ (0, T) , k = 1, 2 ; (6)

where

φ_{k}^{0}(t) = φ_{k}(t) − b_{0}(β_{0,k}(t) + β_{1,k}(t) + β_{2,k}(t) + β_{3,k}(t))−
− ∫_{0}^{α} b_{3}(ζ)(β_{1,k}(t) + β_{2,k}(t))dζ − ∫_{0}^{l} b_{3}(ζ)β_{3,k}(t)dζ − φ_{k,b}(t);
φ_{k,b}(t) = ∫_{0}^{t} ∫_{0}^{α} b(τ, ζ)β_{1,k}(t)dτdζ + ∫_{0}^{t} ∫_{α}^{l} b(τ, ζ)β_{3,k}(t)dτdζ+
+ ∫_{0}^{α} b(t, ζ)g_{1,k}(t)dζ + ∫_{α}^{l} b(t, ζ)g_{3,k}(t)dζ, t ∈ (0, T);  (6^{∗})
b_{3} (x) = φ_{3} (x) , x ∈ (0, l) ;  (7)
b_{0} = φ_{0},  (8)

where θ (y) is one –dimensional Heaviside function on R = R^{1} and q_{1} (ζ, x) = θ (ζ − α) θ (x − α) + θ (α − x).

If we succeed to estimate the components b_{0}, b_{1} (t) , b_{2} (t) , b_{3} (x) , b (t, x) of the vector b̂, on the basis of [1] we get a priori estimate for the solution z ∈ Ŵ_{p,n} (G) of problem (1)-(4)

z(t, x) = b_{0} + θ (α − x) ∫_{0}^{T} b_{1}(τ)θ (t − τ) dτ+
+θ (x − α) ∫_{0}^{T} b_{2}(τ)θ (t − τ) dτ + ∫_{0}^{l} b_{3}(ζ)θ (x − ζ) dζ+
+ ∫_{G} ∫ θ (t − τ) θ (x − ζ) q_{1} (ζ, x) b(τ, ζ)dτdζ , (t, x) ∈ G.  (9)

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_{1} (t) , b_{2} (t) , b (t, x).

It is obvious that by means of the matrix

Δ(t) = ( \begin{matrix}g_{0,1}(t) + g_{1,1}(t) \\ g_{2,1}(t) + g_{3,1}(t) \end{matrix} \begin{matrix}g_{0,2}(t) + g_{1,2}(t) \\ g_{2,2}(t) + g_{3,2}(t) \end{matrix} )

we can write the equality (6) in the compact form

(b_{1}(t), b_{2}(t))Δ(t) + ∫_{0}^{t}(b_{1}(τ), b_{2}(τ))B(t)dτ = (φ_{1}^{0}(t), φ_{2}^{0}(t)), t ∈ (0, T), (10)

where

B(t) = ( \begin{matrix}β_{0,1}(t) + β_{1,1}(t) \\ β_{2,1}(t) + β_{3,1}(t) \end{matrix} \begin{matrix}β_{0,2}(t) + β_{1,2}(t) \\ β_{2,2}(t) + β_{3,2}(t) \end{matrix} )

Assume that almost for all t ∈ (0, T) the matrix Δ(t) is invertible and it holds

∥Δ(t)∥ ≤ M_{1}, ∥Δ^{−1}(t)∥ ≤ M_{1}  (11)

in the sense of almost everywhere on (0, T). Then, from (10) we have

(b_{1}(t), b_{2}(t)) + ∫_{0}^{t}(b_{1}(τ), b_{2}(τ))B_{1}(t)dτ = (φ_{1}^{0}(t), φ_{2}^{0}(t))Δ^{−1}(t), t ∈ (0, T), (12)

where

B_{1}(t) = B(t)Δ^{−1}(t).

Passing in (12) to the vector norm we have

α(t) ≤ ∫_{0}^{t} α(τ)l(t)dτ + S^{0}(t), t ∈ (0, T),  (13)

where

α(t) = ∥b_{1}(t)∥ + ∥b_{2}(t)∥ ,
l(t) = ∥B_{1}(t)∥ ≤ M_{1} ∥B(t)∥ ,
S^{0}(t) = ∥(φ_{1}^{0}(t), φ_{2}^{0}(t))Δ^{−1}(t)∥ ≤ M_{1}(∥φ_{1}^{0}(t)∥ + ∥φ_{2}^{0}(t)∥);

here and below M_{i} are constants independent on φ̂ = (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

R(τ) ≤ ∫_{0}^{τ} R(t)l(t)dt + S^{1}(τ), τ ∈ (0, T),  (14)

where

S^{1}(τ) = ∫_{0}^{τ} S^{0}(t)dt,
R(τ) = ∫_{0}^{τ} α(t)dt.

We write the inequality (14) in the form

R(τ) ≤ R^{1}(τ) + S^{1}(τ) + ε,  (15)

where ε > 0 is an arbitrary number and

R^{1}(τ) = ∫_{0}^{τ} R(t)l(t)dt.

Hence

\frac{R (τ)}{R^{1} (τ) + S^{1} (τ) + ε} ≤ 1, τ ∈ (0, T).

Therefore

\frac{R (τ) l (τ)}{R^{1} (τ) + S^{1} (τ) + ε} ≤ l(τ), τ ∈ (0, T),

or

\frac{Ṙ^{1} (τ)}{R^{1} (τ) + S^{1} (τ) + ε} ≤ l(τ), τ ∈ (0, T),  (16)

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

\frac{Ṙ^{1} (τ)}{R^{1} (τ) + S^{1} (t) + ε} ≤ \frac{Ṙ^{1} (τ)}{R^{1} (τ) + S^{1} (τ) + ε} ≤ l(τ), τ ∈ (0, t),

integrating it with respect to τ on (0, t) we get

ln \frac{R^{1} (t) + S^{1} (t) + ε}{R^{1} (0) + S^{1} (t) + ε} ≤ ∫_{0}^{t} l (τ) dτ

or

R^{1}(t) + S^{1}(t) + ε ≤ (S^{1}(t) + ε)e^{∫_{0}^{t} l(τ)dτ} .

Taking this into account in (15) we get

R(t) ≤ S^{1}(t)e^{∫_{0}^{t} l(τ)dτ} .  (17)

Writing (13) in the form

α(t) ≤ R(t)l(t) + S^{0}(t)

and using (17) we get

α(t) ≤ l(t)e^{∫_{0}^{t} l(τ)dτ} ∫_{0}^{t} S^{0}(τ)dτ + S^{0}(t)  (18)

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

α(t) ≤ M_{2} ∥B(t)∥ ∫_{0}^{t} S^{0}(τ)dτ + S^{0}(t), t ∈ (0, T),  (19)

where

M_{2} = M_{1} exp (M_{1} ∫_{0}^{T} ∥B(τ)∥ dτ) .

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

α(t) = ∥b_{1}(t)∥ + ∥b_{2}(t)∥ ≤ M_{2} ∫_{0}^{t} S^{0}(τ)dτ ∥B(t)∥ + S^{0}(t), t ∈ (0, T).

Therefore, using the Hölder inequality, we obtain

∥b_{k}(t)∥ ≤ S^{0}(t) + M_{2}T^{\frac{1}{q}} ∥S^{0}∥_{L_{p}(0,T)} ∥B(t)∥ , t ∈ (0, T), k = 1, 2,

here and below q = p/(p − 1) denotes the number conjugate to p.

Here, passing to the norm, by Minkowsky inequality we get

∥b_{k}∥_{L_{p,n}(0,T)} ≤ ∥S^{0}∥_{L_{p,}(0,T)} (1 + M_{2}T^{\frac{1}{q}} ∥B∥_{L_{p,2n×2n}(0,T)}) =
= M_{3} ∥S^{0}∥_{L_{p}(0,T)} , k = 1, 2;
M_{3} = 1 + M_{2}T^{\frac{1}{q}} ∥B∥_{L_{p,2n×2n}(0,T)} .

Obviously

∥S^{0}∥_{L_{p}(0,T)} ≤ M_{1}(∥φ_{1}^{0}∥_{L_{p,n}(0,T)} + ∥φ_{2}^{0}∥_{L_{p,n}(0,T)}).

Therefore

∥b_{k}∥_{L_{p,n}(0,T)} ≤ M_{4}(∥φ_{1}^{0}∥_{L_{p,n}(0,T)} + ∥φ_{2}^{0}∥_{L_{p,n}(0,T)}), k = 1, 2,  (20)

where M_{4} = M_{3}M_{1}.

Now, let’s estimate the norms ∥φ_{k}^{0}∥_{L_{p,n}(0,T)}, k = 1, 2. Obviously if the vector z ∈ Ŵ_{p,n} (G) satisfies the conditions (1)-(4), then its independent elements b̂ = (z(0, 0), z_{t}(t, 0), z_{t}(t, α+0), z_{x}(0, x), z_{tx}(t, x))= (b_{0}, b_{1}(t), b_{2}(t), b_{3}(x), b(t, x)) satisfy the equalities (5)-(8) and therewith

∥b_{0}∥ ≤ \left∥{ \atop φ̂}\right∥Q̂_{p,n}

∥b_{3}∥_{L_{p,n}(0,l)} ≤ \left∥{ \atop φ̂}\right∥Q̂_{p,n}  (21)

where

\left∥{ \atop φ̂}\right∥Q̂_{p,n} = ∥L̂z∥_{Ŵ_{p,n}(G)} = ∥L̂(N^{−1}(b̂))∥Q̂_{p,n} =

= ∥φ_{0}∥ + ∥φ_{1}∥_{L_{p,n}(0,T)} + ∥φ_{2}∥_{L_{p,n}(0,T)} +

+ ∥φ_{3}∥_{L_{p,n}(0,l)} + ∥φ∥_{L_{p,n}(G)} , z = N^{−1}b̂.

Therefore, from the expressions (6*) of the vectors φ_{k}^{0}(t) we have

∥φ_{k}^{0}(t)∥ ≤ ∥φ_{k}(t)∥ + ∥b_{0}∥ (∥β_{0,k}(t)∥ + ∥β_{1,k}(t)∥ +
+ ∥β_{2,k}(t)∥ + ∥β_{3,k}(t)∥) + ∥b_{3}∥_{L_{p,n}(0,l)} α^{\frac{1}{q}} ×

×(∥β_{1,k}(t)∥ + ∥β_{2,k}(t)∥) + l^{\frac{1}{q}} ∥b_{3}∥_{L_{p,n}(0,l)} ∥β_{3,k}(t)∥ + ∥φ_{k,b}(t)∥ , k = 1, 2.

Hence by means of the Minkowsky inequality allowing for (20) we have

∥φ_{k}^{0}∥_{L_{p,n}(0,T)} ≤ ∥φ_{k}∥_{L_{p,n}(0,T)} + ∥b_{0}∥ ∑_{i=0}^{3} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} +

+ ∥b_{3}∥_{L_{p,n}(0,l)} α^{\frac{1}{q}} ∑_{i=1}^{2} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} +

+ ∥b_{3}∥_{L_{p,n}(0,l)} l^{\frac{1}{q}} ∥β_{3,k}∥_{L_{p,n×n}(0,T)} +

+ ∥φ_{k,b}∥_{L_{p,n}(0,T)} ≤ \left∥{ \atop φ̂}\right∥Q̂_{p,n} (1 + ∑_{i=0}^{3} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} +

+α^{\frac{1}{q}} ∑_{i=1}^{2} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} + l^{\frac{1}{q}} ∥β_{3,k}∥_{L_{p,n×n}(0,T)}) + ∥φ_{k,b}∥_{L_{p,n}(0,T)}

or

∥φ_{k}^{0}∥_{L_{p,n}(0,T)} ≤ M_{5} \left∥{ \atop φ̂}\right∥Q̂_{p,n} + ∥φ_{k,b}∥_{L_{p,n}(0,T)} ,  (22)

where

M_{5} = 1 + ∑_{i=0}^{3} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} + α^{\frac{1}{q}} ∑_{i=1}^{2} ∥β_{i,k}∥_{L_{p,n×n}(0,T)} +

+l^{\frac{1}{q}} ∥β_{3,k}∥_{L_{p,n×n}(0,T)} .

Now let’s estimate the norm of the vector φ_{k,b}(t). Obviously

∥φ_{k,b}(t)∥ ≤ (T α)^{\frac{1}{q}} ∥β_{1,k}∥_{L_{p,n×n}(0,T)} ∥b∥_{L_{p,n}(G)} +

+(T (l − α))^{\frac{1}{q}} ∥β_{3,k}∥_{L_{p,n×n}(0,T)} ∥b∥_{L_{p,n}(G)} +
+α^{\frac{1}{q}} ∥b(t, ·)∥_{L_{p,n}(0,l)} ∥g_{1,k}∥_{L_{∞,n×n}(0,T)} +
+(l − α)^{\frac{1}{q}} ∥b(t, ·)∥_{L_{p,n}(0,l)} ∥g_{3,k}∥_{L_{∞,n×n}(0,T)} .

Therefore

∥φ_{k,b}∥_{L_{p,n}(0,T)} ≤ c^{k} ∥b∥_{L_{p,n}(G)} , k = 1, 2,  (23)
c^{k} = (T α)^{\frac{1}{q}} ∥β_{1,k}∥_{L_{p,n×n}(0,T)} + (T (l − α))^{\frac{1}{q}} ∥β_{3,k}∥_{L_{p,n×n}(0,T)} +
+α^{\frac{1}{q}} ∥g_{1,k}∥_{L_{∞,n×n}(0,T)} + (l − α)^{\frac{1}{q}} ∥g_{3,k}∥_{L_{∞,n×n}(0,T)} .  (24)

Then using (23) we get from (22)

∥φ_{k}^{0}∥_{L_{p,n}(0,T)} ≤ M_{5} \left∥{ \atop φ̂}\right∥Q̂_{p,n} + c^{k} ∥b∥_{L_{p,n}(G)} , k = 1, 2.  (25)

Now, equation (5) is written in the form

(Ωb)(t, x) = φ^{0}(t, x), (t, x) ∈ G,  (26)

where

(Ωb)(t, x) = b(t, x) + ∫_{0}^{t} ∫_{0}^{x} b(τ, ζ)q_{1}(ζ, x)A_{0,0}(t, x)dτdζ+

+ ∫_{0}^{x} b(t, ζ)q_{1}(ζ, x)A_{1,0}(t, x)dζ + ∫_{0}^{t} b(τ, x)A_{0,1}(t, x)dτ,  (27)
φ^{0}(t, x) = φ^{0,0}(t, x) + φ^{0,1}(t, x);
φ^{0,0}(t, x) = φ(t, x) − b_{3}(x)A_{0,1}(t, x)−
−b_{0}A_{0,0}(t, x) − ∫_{0}^{x} b_{3}(ζ)A_{0,0}(t, x)dζ;

φ^{0,1}(t, x) = − (∫_{0}^{t} b_{1}(τ)θ(α − x)dτ + ∫_{0}^{t} b_{2}(τ)θ(x − α)dτ) A_{0,0}(t, x)−

−(b_{1}(t)θ(α − x) + b_{2}(t)θ(x − α))A_{1,0}(t, x).

In the expression of the vector φ^{0,0}(t, x) there are the vectors φ(t, x), b_{3}(x), b_{0} and the given matrices A_{0,1}(t, x), A_{0,0}(t, x). Above we have estimated the norms of the vectors φ(t, x), b_{3}(x), b_{0} by \left∥{ \atop φ̂}\right∥Q̂_{p,n}. Therefore, from the expression of the vector φ^{0,0}(t, x) by means of the Holder and Minkowsky inequalities we can easily get the estimate

∥φ^{0,0}∥_{L_{p,n}(G)} ≤ M_{6} \left∥{ \atop φ̂}\right∥Q̂_{p,n} ,  (28)

where M_{6} > 0 is a suitable constant.

Further, from the expression of the vector φ^{0,1}(t, x) it is seen that

∥φ^{0,1}(t, x)∥ ≤ (T^{\frac{1}{q}} ∥b_{1}∥_{L_{p,n}(0,T)} θ(α − x)+
+T^{\frac{1}{q}} ∥b_{2}∥_{L_{p,n}(0,T)} θ(x − α))∥A_{0,0}(t, x)∥+
+(∥b_{1}∥_{L_{p,n}(0,T)} θ(α − x) + ∥b_{2}∥_{L_{p,n}(0,T)} θ(x − α))A_{1,0}^{0}(x).

Hence we get

∥φ^{0,1}∥_{L_{p,n}(G)} ≤ (T^{\frac{1}{q}} ∥A_{0,0}∥_{L_{p,n×n}(G)} +

+ ∥A_{1,0}^{0}∥_{L_{p}(0,l)})(∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)}) =
= M_{7}(∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)}),  (29)

where

M_{7} = T^{\frac{1}{q}} ∥A_{0,0}∥_{L_{p,n×n}(G)} + ∥A_{1,0}^{0}∥_{L_{p}(0,l)} .

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

∥b∥_{L_{p,n}(G)} ≤ ∥Ω^{−1}∥ ∥φ^{0}∥_{L_{p,n}(G)} .

Hence by (28) and (29) we get

∥b∥_{L_{p,n}(G)} ≤ ∥Ω^{−1}∥ (M_{6} \left∥{ \atop φ̂}\right∥Q̂_{p,n} + M_{7}(∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)})). (30)

Take into account, (25) in (20) and get

∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)} ≤ 2M_{4}(2M_{5} \left∥{ \atop φ̂}\right∥Q̂_{p,n} + (c^{1} + c^{2}) ∥b∥_{L_{p,n}(G)}). (31)

Hence substituting (31) into (30) we get

∥b∥_{L_{p,n}(G)} ≤ ∥Ω^{−1}∥ (M_{8} \left∥{ \atop φ̂}\right∥Q̂_{p,n} + 2M_{4}M_{7}(c^{1} + c^{2}) ∥b∥_{L_{p,n}(G)}) ,

where M_{8} = M_{6} + 4M_{4}M_{5}M_{7} > 0.

If we assume

γ = 2M_{4}M_{7} ∥Ω^{−1}∥ (c^{1} + c^{2}) < 1,  (∗)

then we can obtain

∥b∥_{L_{p,n}(G)} ≤ M_{9} \left∥{ \atop φ̂}\right∥Q̂_{p,n} ,  (32)

with constant

M_{9} = (1 − γ)^{−1} ∥Ω^{−1}∥ M_{8}.

Taking into account (32) in (31) we have

∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)} ≤ 2M_{4}(2M_{5} + M_{9}(c^{1} + c^{2})) \left∥{ \atop φ̂}\right∥Q̂_{p,n} .  (33)

Now, summing up the inequalities (21), (32), (33) for the totality b̂ = (b_{0}, b_{1}(t), b_{2}(t), b_{3}(x), b(t, x)) we have the estimate

∥b̂∥Q̂_{p,n} = ∥b_{0}∥ + ∥b_{1}∥_{L_{p,n}(0,T)} + ∥b_{2}∥_{L_{p,n}(0,T)} + ∥b_{3}∥_{L_{p,n}(0,l)} +

+ ∥b∥_{L_{p,n}(G)} ≤ M_{10} \left∥{ \atop φ̂}\right∥Q̂_{p,n} = M_{10} ∥L̂z∥Q̂_{p,n} ,

where

M_{10} = 2M_{4}(2M_{5} + M_{9}(c^{1} + c^{2})) > 0

and

L̂z = φ̂, b̂ = Nz.

Using the last inequality we get

∥z∥_{Ŵ_{p,n}(G)} ≤ M_{11} ∥Nz∥Q̂_{p,n} ≤ M_{11}M_{10} ∥L̂z∥Q̂_{p,n} ,

with suitable constant M_{11} > 0 independent on z. Hence the following theorem is true.

Theorem 1. Let the matrix Δ (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∥_{Ŵ_{p,n}(G)} ≤ M ∥L̂z∥Q̂_{p,n} holds, where M > 0 is a positive constant independent on z.

The operator L̂ is a linear and bounded operator from Ŵ_{p,n} (G) to Q̂_{p,n}. Therefore, there exists a bounded, conjugated operator L̂^{∗} :\binom{}{Q_{p,n}}^{∗}→(Ŵ_{p,n} (G))^{∗}. Using general forms of linear bounded functional determined on Q̂_{p,n} and Ŵ_{p,n} (G) we can prove that L̂^{∗} is a bounded vector operator of the form L̂^{∗} = (ω_{0}, ω_{1}, ω_{2}, ω_{3}, ω) acting in the space Q̂_{q,n}, where 1/p + 1/q = 1. Therefore, we can 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 (Ŵ_{p,n} (G))^{∗}. It follows from Theorem 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 ∈ Ŵ_{p,n} (G), and the conjugated equation L̂^{∗}f̂ = ψ̂ for any right hand side ψ̂ ∈ (Ŵ_{p,n} (G))^{∗} has at least one solution f̂ ∈ Q̂_{q,n}.

References

[1] Akhiev S. S., On some problems of optimization of systems of linear hyperbolic equation of second order with discontinuous solutions and non-local boundary 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-differential equations of hyperbolic type and some their applications to the prognosis of soil moisture. Differentialniye Uravneniya, (1979),. Vol.15, No 1, p. 96-105.

[3] Orucoglu K., Akhiev S. S., The Riemann function for the third order onedimensional pseudoparabolic equation, Acta Appl. Math. 53(1998), No 3, p. 353-370.

[4] Sobolev S. L., Some applications of functional analysis to mathematical physics. Novosibirsk (1962), 256 p. (Russian).

(Received November 19, 2008)