Identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation by semigroup approach with mixed boundary conditions

In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k ( u ( x, t )) in the quasi‐linear parabolic equation u t ( x, t )=( k ( u ( x, t )) u x ( x, t )) x , with Dirichlet boundary conditions u x (0, t )=ψ 0 , u (1, t )=ψ 1 . The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : → C 1 [0, T ], Ψ[·] : → C 1 [0, T ] using semigroup theory. In this article, it is shown that if the null space of semigroups T ( t ) and S ( t ) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd.