An inverse problem for a heat equation with piecewise-constant thermal conductivity

Abstract

The governing equation is ut=(a(x)ux)x, 0≤x≤1, t>0, u(x,0)=0, u(0,t)=0, a(1)u′(1,t)=f(t). The extra data are u(1,t)=g(t). It is assumed that a(x) is a piecewise-constant function and f≢0. It is proved that the function a(x) is uniquely defined by the above data. No restrictions on the number of discontinuity points of a(x) and on their locations are made. The number of discontinuity points is finite, but this number can be arbitrarily large. If a(x)∊C2[0,1], then a uniqueness theorem has been established earlier for multidimensional problem, x∊Rn,n>1 [see A. G. Ramm, Multidimensional inverse problems and completeness of the products of solutions to PDE, J. Math. Anal. Appl., 134, 211 (1988)] for the stationary problem with infinitely many boundary data. The novel point in this work is the treatment of the discontinuous piecewise-constant function a(x) and the proof of Property C for a pair of the operators {ℓ1,ℓ2}, where ℓj≔−(d2/dx2)+k2qj2(x), j=1,2, and qj2(x)>0 are piecewise-constant functions, and for the pair {L1,L2}, where Lju≔−[aj(x)u′(x)]′+λu, j=1,2, and aj(x)>0 are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations [see A. G. Ramm, Inverse Problems (Springer, New York, 2005)].