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Inverse parabolic problems with the final overdetermination. (English) Zbl 0729.35146

Let Ω be a bounded (smooth) domain in n and let I be either interval (-,T) or (0,T) (T>0). The author considers the following identification problems P σ (σ=0,1): determine a pair of functions u: Ω×I and f: Ω ¯ such that

(1)D t u(x,t)+Au(x,t)=α(x,t)f(x)+F(x,t),(x,t)Q:=Ω×I,
(2)B σ u(x,t)=g(x,t),(x,t)Γ:=Ω×I,(3)u(x,0)=h 0 (x),xΩ,(4)u(x,T)=h T (x),xΩ·

Here A=a jk (x,t)D j D k +a j (x,t)D j +a(x,t) (D j =/x j , j=1,···,n) is a uniformly elliptic operator with negative definite principal part, while B 0 u=u and B 1 u=u/+bu with (x,t)·N(x)>ϵ for some positive constant ϵ, N(x) denoting the outward normal at a point (x,t)Γ. Finally, α,F,g,h 0 ,h T are prescribed functions in suitable (anisotropic) Hölder spaces.

Under suitable hypotheses on A,B and α, the author shows that problem (1)-(4) is Fredholm with zero index. Then, by restriction of the classes of admissible unknowns f and weights α, the author proves the uniqueness (and, consequently, the existence and the stability) of the solution to problem (1)-(4). Two basic assumptions for this are the positivity and the strict monotonicity of α with respect to time (α(x,t)>0, D t α(x,t)>0(x,t)Q). In connection with such hypotheses, the author shows that, when D t α fails to be strictly positive in Q ¯, there exist positive C -weight functions α for which problem (1)-(4) admits nontrivial solutions.

A further identification problem is studied in the case where α=F=0 and the coefficient a in A is supposed to be unknown and to depend only on x. Also for such a problem existence and uniqueness results are proved.

The paper concludes with the proof of a stability result for the solution to problem (1)-(4), when Q is a non-cylindrical domain.


MSC:
35R30Inverse problems for PDE
35K20Second order parabolic equations, initial boundary value problems