Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

Abstract This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown 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 (0, t )=ψ 0 , u (1, t )=ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[⋅]: → C 1 [0, T ], Ψ[⋅]:→ C 1 [0, T ] via semigroup theory. In this paper, it is shown 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. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f ( t ):= k ( u (0, t )) u x (0, t ) or/and h ( t ):= k ( u (1, t )) u x (1, t ), the values k (ψ 0 ) and k (ψ 1 ) of the unknown diffusion coefficient k ( u ( x , t )) at ( x , t )=(0,0) and ( x , t )=(1,0), respectively, can be determined explicitly. In addition to these, the values k u (ψ 0 ) and k u (ψ 1 ) of the unknown coefficient k ( u ( x , t )) at ( x , t )=(0,0) and ( x , t )=(1,0), 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[⋅]:→ C 1 [0, T ], Ψ[⋅]:→ C 1 [0, T ] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd.