Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)ux(x,t))x, with Dirichlet boundary conditions u(0,t)=ψ0, u(1,t)=ψ1. Main goal of this study is to investigate the distinguishability of the input–output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k′(0) and k′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically, by an integral representation. Hence the input–output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.