Identifying unknown time delay and spatially varying coefficients in a reaction-diffusion equation from boundary measurements

Abstract

This work investigates an inverse problem for a general class of linear reaction-diffusion systems incorporating multiple delayed contributions, namely retarded diffusion, retarded time derivatives, and retarded source terms. The objective is to simultaneously recover the unknown time lag $τ>0$ and spatially heterogeneous coefficients from boundary flux measurements alone. The recovery strategy exploits the singular temporal behavior generated by an incompatibility between the prescribed initial history and the boundary data. In contrast to prior inverse problems for delay equations, which assume $τ$ known and are confined to ODE or abstract settings, our approach operates in a parabolic PDE framework with spatially varying coefficients. Once $τ$ is identified, we establish a Lipschitz stability estimate via a Carleman inequality, in the case of time-independent coefficients $p=p(x)$, $q=q(x)$. This is the first result to simultaneously recover an unknown delay and spatially dependent coefficients in a delayed parabolic PDE from boundary data.