Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

[摘要] The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t=(k(x)u_x(x,t))_x + F(x,t)$ from the measured data at the final time $u(x,T)=\mu (x)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J(F) = \|\mu _T(x)- u(x,T)\|_{0}^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.

[学科分类] 应用数学

[关键词] inverse parabolic problem;unknown source;adjoint problem;Fréchet derivative;Lipschitz continuity