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 ap riorion 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 H¨ older stability of the functions u and q with respect to errors in the Neumann data ψ0 ,ψ 1, the initial condition u0 and the ap rioriknowledge 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.

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