In the space of variables (x, t) ∈ ℝ^{n+1}, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ⊂ ℝ^{n+1} whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τϕ(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ϕ(x, t) to be the function ϕ(x, t) = s^{2}(x, x^{0}) − pt^{2}, where s(x, x^{0}) is the distance between points x and x^{0} in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x^{0} can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k_{0} ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k_{0} and sup_{x∈Ω}s^{2}(x, x^{0}) satisfies some smallness condition.

Keywords

Carleman estimate Cauchy problem stability uniqueness