Global Uniqueness and Hölder Stability for Recovering a Nonlinear Source Term in a Parabolic Equation ‡

Consider the semilinear parabolic equation −ut(x, t) + uxx + q(u) = f(x, t), with the initial condition u(x, 0) = u0(x), Dirichlet boundary conditions u(0, t) = φ0(t), u(1, t) = φ1(t) 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 ux(0, t) = ψ0(t), ux(1, t) = ψ1(t). 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). These results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.