Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

Abstract

Consider the semilinear parabolic equation with the initial condition Dirichlet boundary conditions and a sufficiently regular source term q(⋅), which is assumed to be known a priori on the range of u0(x). We investigate the inverse problem of determining the function q(⋅) outside this range from measurements of the Neumann boundary data Via the method of Carleman estimates, we derive global uniqueness of a solution (u, q) to this inverse problem and Hölder stability of the functions u and q with respect to errors in the Neumann data ψ0, ψ1, the initial condition u0 and the a priori knowledge of the function q (on the range of u0). The results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.