NONCLASSICAL PROBLEMS FOR THE SECOND ORDER HYPERBOLIC EQUATIONS

G. Avalishvili, D. Gordeziani

I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University

Nonclassical initial and boundary conditions often arise in mathematical modelling of various phenomena of physics, biology, ecology and other fields. Conditions of such type contain certain relations either between the values of an unknown function on the boundary of the domain and its internal values or between the values of an unknown function at the initial and some later moments of time. Consequently the nonclassical problems of the first type are called spatial nonlocal problems and the second ones − nonlocal problems in time.

First, a certain class of spatial nonlocal problems was investigated by A.V. Bitsadze and A.A. Samarskii [1]. Further, in [2,3] the problems of such a type were called Bitsadze-Samarskii nonlocal problems and iteration methods of resolution for the such type problems in the case of rather general elliptic and parabolic equations were suggested. Later, in [4-17] interesting generalizations of Bitsadze-Samarskii conditions were suggested. It should be pointed out, that the majority of workes devoted to the nonlocal problems studies elliptic and parabolic cases.

Let Ω ⊂ ℝ^{n} be a bounded domain with a piecewise smooth boundary Γ = ∂Ω. L^{2}(Ω) denotes the space of real functions square integrable in Ω in the Lebesgue sense. L^{∞}(Ω) is a space of measurable and almost everywhere bounded functions. W^{2,k}(Ω) = H^{k}(Ω) denotes the Sobolev space of order k, i.e. the subspace of L^{2}(Ω) of functions whose derivatives up to the order k in the sense of distributions are in L^{2}(Ω). W_{0}^{2,k}(Ω) = H_{0}^{k}(Ω) is the closure of the set D(Ω) of real infinitely differentiable functions with compact support in Ω in the space H^{k}(Ω). C^{0}(0, T; X) is a space of continuous functions of t with values in the Banach space X.

Suppose, that a_{ij} (i, j = \overline{1, n}) are prescribed real functions on Ω, which satisfy the following conditions

\begin{matrix}a, a_{ij} ∈ L^{∞}(Ω), \\ a(x) ≥ 0, a_{ij}(x) = a_{ji}(x), \\ ∑_{i,j=1}^{n} a_{ij}(x)ξ_{i}ξ_{j} ≥ α(ξ_{1}^{2} + ... + ξ_{n}^{2}), α > 0, ∀ξ = (ξ_{1}, ..., ξ_{n}) ∈ ℝ^{n}, \end{matrix}  (1)

almost everywhere on Ω. Let A : H_{0}^{1}(Ω) → L^{2}(Ω) be the linear operator

Au = − ∑_{i,j=1}^{n} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂u}{∂x_{j}} ) + a(x)u.

From conditions (1) it immediately follows, that the bilinear form

a(φ, ψ) = (Aφ, ψ) = ∑_{i,j=1}^{n} ∫_{Ω} a_{ij}(x)\frac{∂ψ}{∂x_{i}} \frac{∂φ}{∂x_{j}} dx + ∫_{Ω} a(x)φ(x)ψ(x)dx

is symmetric and coercive

a(φ, ψ) = a(ψ, φ) ∀φ, ψ ∈ H_{0}^{1}(Ω),
a(φ, φ) ≥ α∥φ∥_{H_{0}^{1}(Ω)}^{2}, α > 0,

and eigen functions of the operator A are dense in L^{2}(Ω).

Let us consider the nonlocal problem in time for the second order hyperbolic equation

\frac{∂^{2}u}{∂t^{2}} + Au = f(x, t), (x, t) ∈ Ω × (0, T),  (2)

\begin{matrix}u(x, 0) = ∑_{i=1}^{k} α_{i}u(x, T_{i}) + ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ρ_{ij}(τ)u(x, τ)dτ + u_{0}(x), \\ u_{t}(x, 0) = ∑_{i=1}^{k} β_{i}u_{t}(x, T_{i}) + ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ω_{ij}(τ)u_{t}(x, τ)dτ + u_{1}(x), \end{matrix} x ∈ Ω, (3)

u(x, t) = 0,  (x, t) ∈ ∂Ω × [0, T],  (4)

where ρ_{ij}, ω_{ij} are measurable, bounded functions, α_{i}, β_{i} are real constant numbers, T_{i} ∈ (0, T], u_{0} ∈ H_{0}^{1}(Ω), u_{1} ∈ L^{2}(Ω). We have to find the function u ∈ C^{0}(0, T; H_{0}^{1}(Ω)) with the generalized derivative u′ ∈ C^{0}(0, T; L^{2}(Ω)), which satisfies the following equation

\frac{d}{dt}(u′(·), v) + a(u(·), v) = (f(·), v), ∀v ∈ H_{0}^{1}(Ω),  (5)

in the sense of distributions on (0, T) and conditions (3) in the sense of spaces H_{0}^{1}(Ω) and L^{2}(Ω) respectively.

Under these conditions the following theorem is true.

Theorem. If there exists a positive real number q > 0, such that

\begin{matrix}| & (1 − ∑_{i=1}^{k} α_{i}cos(λ_{n}T_{i}) − ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ρ_{ij}(τ)cos(λ_{n}τ)dτ) & (1 − ∑_{i=1}^{k} β_{i}cos(λ_{n}T_{i})− \\ − ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ω_{ij}(τ)cos(λ_{n}τ)dτ) + (∑_{i=1}^{k} α_{i}sin(λ_{n}T_{i}) + ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ρ_{ij}(τ)sin(λ_{n}τ)dτ) \end{matrix}

(∑_{i=1}^{k} β_{i}sin(λ_{n}T_{i}) + ∑_{T_{i}<T_{j}} ∫_{T_{i}}^{T_{j}}ω_{ij}(τ)sin(λ_{n}τ)dτ)| > q, n = 1, 2, ... (6)

where {λ_{n}}_{n=1}^{∞} are eigen values of the operator A, u_{0} ∈ H_{0}^{1}(Ω), u_{1} ∈ L^{2}(Ω), f ∈ L^{2}(0, T; L^{2}(Ω)), then nonlocal problem (2)-(4) has a unique solution.

Let’s formulate one useful sufficient condition for the fulfilment of inequality (6).

Corollary. If in the nonlocal conditions (4) integral addends are omitted and coefficients α_{i}, β_{i} (i = \overline{1, m}) satisfy the following inequality

∑_{i=1}^{k}(|α_{i}| + |β_{i}|) < 1,

then the nonlocal problem has a unique solution.

Now we consider some particular cases of nonlocal problem (2)-(4) and show an essential difference between classical and above mentioned problems.

Let Ω = (0, l), A ≡ −\frac{∂^{2}}{∂x^{2}} , f ≡ 0 and conditions (3) take the form

\begin{matrix}u(x, 0) = ∑_{i=1}^{k} α_{i}u(x, T_{i}) + u_{0}(x), \\ u_{t}(x, 0) = ∑_{i=1}^{k} α_{i}u_{t}(x, T_{i}) + u_{1}(x), \end{matrix} x ∈ Ω.  (7)

Hence, we get nonlocal problem in time for a string oscillation equation, which

has a unique solution if ∑_{i=1}^{k} |α_{i}| < 1. In the degenerate case ∑_{i=1}^{k} |α_{i}| = 1 if

among the points {T_{i}}_{i=1}^{k} at least one is such, that quotient T_{i}/l is irrational, then the nonlocal problem does not have more than one solution. It must be pointed out, that for k = 1, |α_{1}| = 1 the condition of irrationality of T_{1}/l is a necessary and sufficient condition for the uniqueness of the solution of the nonlocal problem.

Similarly we may consider two-dimensional and multidimensional problem (2), (7) with homogeneous boundary conditions, when Ω = (0, l_{1}) × (0, l_{2}) ×

... × (0, l_{s}), A ≡ − ∑_{i=1}^{s} \frac{∂^{2}}{∂x_{i}^{2}} , f ≡ 0. Taking into account inequality (6), in the

case ∑_{i=1}^{k} |α_{i}| = 1 we obtain, that if for some T_{i} equation

(\frac{n_{1}}{l_{1}} )^{2} + (\frac{n_{2}}{l_{2}} )^{2} + ... + (\frac{n_{s}}{l_{s}} )^{2} = (\frac{p}{T_{i}} )^{2}

is unsolvable in integer numbers, i.e. for any integer n_{1}, n_{2}, ..., n_{s}, p ∈ ℤ quotients n_{1}/l_{1}, n_{2}/l_{2}, ..., n_{s}/l_{s}, p/T_{i} are not the so-called generalized Pythagorean numbers, then the nonlocal problem does not have more than one solution.

So, for the nonlocal problems in time the uniqueness and existence of the solution depend on algebraic properties of the quotients T_{1}/l, T_{2}/l, ..., T_{s}/l (in the one-dimensional case), that doesn’t take place for classical problems.

Volume 3, 1999 19

R e f e r e n c e s

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