BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS WITH OPERATOR COEFFICIENTS

Let T be some bounded simply connected region in R2 with ∂ T = u0003̄1 ∩ u0003̄2. We seek a function u(x, t), ((x, t) ∈ T ) with values in a Hilbert space H which satisfies the equation ALu(x, t) = Bu(x, t)+f (x, t,u,ut ), (x, t) ∈ T , where A(x, t), B(x, t) are families of linear operators (possibly unbounded) with everywhere dense domain D (D does not depend on (x, t)) in H and Lu(x, t) = utt +a11uxx +a1ut +a2ux . The values u(x, t); ∂u(x, t)/∂n are given in 1. This problem is not in general well posed in the sense of Hadamard. We give theorems of uniqueness and stability of the solution of the above problem.

The Cauchy problem is the problem to find the solution of (1.3) which satisfies condition (1.5) with f, g ∈ D(A) ∩ D(B).
The Cauchy problem (1.3), (1.4), and (1.5) is not in general well posed in the sense of Hadamard.This type of problems for differential-operator equations were studied by Kreȋn [3], Levine [5], Buchgeim [1], and others.We will prove theorems of uniqueness and stability of the solution of the Cauchy problem using Lavrent'ev's method [2,4].

Uniqueness
Theorem 2.1.Let A = 1, Lu ≡ u ≡ u xx + u tt , and let B be a selfadjoint constant operator.Suppose w(x, t) satisfies is defined, and where c 1 , c 2 are positive constants.Then there exist constants i ≥ 0, i = 1, 2, such that with 2 ds (2.13) and obtain Re α t, s, u, u t +ε(v), u t ds ( By deducing this formula we use (2.8) and integration by parts.Integrating (2.14) from t 0 till t we get the following: where (2.16) Substituting (2.15) in (2.12) we get (2.17) where k 1 , k 2 , and k 3 are nonnegative constants which depend on T and the constants c 1 and c 2 .Then for ϕ (t) we get the following: (2.22) We consider now the function ψ(t) = ln[ϕ(t)+γ ], using the Cauchy inequality, and (2.22) we transform the second derivative or where p, q are nonnegative constants that depend on T and the constants c 1 , c 2 .Theorem 2.1 is proved.
Remark 2.2.It is known from the theory of ordinary differential equations if the function ψ(t) satisfies the inequality (2.7), then it satisfies the following inequality: where ψ 0 (t) is a solution of the differential equation From inequality (2.30) it is easy to see that the following corollary follows.

Corollary 2.4. The solution of the Cauchy problem for (1.3) is stable in the space
(2.33)

Stability
Let A be a constant selfadjoint operator and (Au, u) > 0 for all u = 0, (Au, u) = 0 if and only if u = 0. Let B(x, t) be a selfadjoint operator for every (x, y) ∈ D and with then where c 1 , c 2 are constants.
Theorem 3.1.Let the coefficients a 11 , a 1 , and a 2 satisfy condition (1.4).If the solution of (3.6) is equal to zero on 1 and satisfies the inequality where and , p, q are constants that depend on the coefficients T and c i , i = 1, 2.