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In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition u(1,t)=f(t)Open image in new window in the quasi-linear parabolic equation ut(x,t)=(k(u(x,t))ux(x,t))xOpen image in new window, with Dirichlet boundary conditions u(0,t)=ψ0Open image in new window, u(1,t)=f(t)Open image in new window, by making use of the over measured data u(x0,t)=ψ1Open image in new window and ux(x0,t)=ψ2Open image in new window separately. The purpose of this study is to identify the unknown boundary condition u(1,t)Open image in new window at x=1Open image in new window by using the over measured data u(x0,t)=ψ1Open image in new window and ux(x0,t)=ψ2Open image in new window. First, by using over measured data as a boundary condition, we define the problem on ΩT0={(x,t)∈R2:0<x<x0,0<t≤T}Open image in new window, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval [0,1]Open image in new window, we reach the result. The main point here is the unique extensions of the solutions on [0,x0]Open image in new window to the closed interval [0,1]Open image in new window which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition u(1,t)Open image in new window at x=1Open image in new window.

Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method