Two dimensional determination of source terms in linear parabolic equation from the final overdetermination

A case of steady-case heat flow through a plane wall, which can be formulated as ut(x,y,t)−div(k(x,y)∇u(x,y,t))=F(x,y,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{t}(x,y,t)- \operatorname{div} (k(x,y) \nabla u(x,y,t)) = F(x,y,t)$\end{document} with Robin boundary condition −k(1,y)ux(1,y,t)=ν1[u(1,y,t)−T0(t)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-k(1,y)u_{x}(1,y,t)= \nu_{1} [u(1,y,t)-T_{0}(t)]$\end{document}, −k(x,1)uy(x,1,t)=ν2[u(x,1,t)−T1(t)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-k(x,1)u_{y}(x,1,t)= \nu_{2} [u(x,1,t)-T_{1}(t)]$\end{document}, where ω:={F(x,y,t);T0(t);T1(t)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\omega:=\{F(x,y,t);T_{0}(t);T_{1}(t)\}$\end{document} is to be determined, from the measured final data μT(x,y)=u(x,y,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu_{T}(x,y)=u(x,y,T)$\end{document} is investigated. It is proved that the Fréchet gradient of the cost functional J(ω)=∥μT(x,y)−u(x,y,T;ω)∥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J(\omega )=\|\mu_{T}(x,y)-u(x,y,T;\omega)\|^{2}$\end{document} can be found via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove the existence of a quasi-solution of the inverse problem. A steepest descent method with line search, which produces a monotone iteration scheme based on the gradient, is formulated. Some convergence results are given.