AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT Rakesh

Let Ω ⊂ R, n ≥ 2, be a bounded domain with smooth boundary. Consider utt −4xu+ q(x, t)u = 0 in Ω× [0, T ] u(x, 0) = 0, ut(x, 0) = 0 if x ∈ Ω u(x, t) = f(x, t) on ∂Ω× [0, T ] We show that if u and f are known on ∂Ω× [0, T ], for all f ∈ C∞ 0 (∂Ω× [0, T ]), then q(x, t) may be reconstructed on C = { (x, t) : x∈Ω, 0 < t < T, x− tω & x+ (T − t)ω 6∈ Ω ∀ ω ∈ R, |ω| = 1 } provided q is known at all points not in C. If u, f are known on ∂Ω× [0, T ], and u is known on t = T , for all f ∈ C∞ 0 (∂Ω× [0, T ]) then q(x, t) may be reconstructed on D = { (x, t) : x∈Ω, 0 < t < T, x− tω 6∈ Ω ∀ ω ∈ R, |ω| = 1} provided q is known at all points not in D. Let Ω ⊂ R be a bounded domain with smooth boundary, and q(x, t) a function on Ω. Suppose u(x, t) satisfies 2u+ q(x, t)u ≡ utt −4xu+ q(x, t)u = 0 in Ω× [0, T ] u(x, 0) = 0, ut(x, 0) = 0 if x ∈ Ω (1) u(x, t) = f(x, t) on ∂Ω× [0, T ] For a fixed q(x, t), (1) is a well posed initial boundary value problem, hence one may define the Dirichlet to Neumann map Λq : H (∂Ω× [0, T ]) −→ L(∂Ω× [0, T ]) f(x, t) 7−→ ∂u ∂n and the boundary to final data map Γq : H (∂Ω× [0, T ]) −→ H(Ω)× L(Ω) f(x, t) 7−→ ( u(x, T ) , ut(x, T ) ) We prove the following two results THEOREM A If n > 1, and q(x, t) is in C(Ω× [0, T ]), then knowing Λq we can reconstruct q(x, t) on the region C = { (x, t) : x∈Ω, 0 < t < T, x− tω & x+ (T − t)ω 6∈ Ω ∀ ω ∈ R, |ω| = 1 } provided q(x, t) is known at all points not in C. C is region III in the figure, and it is a subset of Ω× [0, T ] consisting of those points through which every line making an angle of 45 with the vertical meets the planes t = 0 and t = T outside Ω. THEOREM B If n > 1, and q(x, t) is in C(Ω× [0, T ]), then knowing Λq and Γq we can reconstruct q(x, t) on the region D = { (x, t) : x∈Ω, 0 < t < T, x− tω 6∈ Ω ∀ ω ∈ R, |ω| = 1 } provided q(x, t) is known at all points not in D. D is region III ∪ IV ∪ V in the figure, and is a subset of Ω× [0, T ] consisting of those points through which every line making an angle of 45 with the vertical meets the planes t = 0 outside Ω.