Conditional stability in determining source terms of semilinear parabolic partial differential equations

Abstract

We study an inverse source problem for a semilinear parabolic equation in a bounded domain, where the nonlinearity depends on the unknown function and its gradient through a quadratic reaction term and a Burgers-type convection term. From partial boundary observation of the time derivative and its spatial gradient on an open portion of the boundary, together with an interior snapshot of the solution at a fixed time, we aim to recover an unknown spatial source factor. The analysis combines (i) a paradifferential paralinearization in Besov spaces on a short time window, which converts the nonlinear model into a linear parabolic equation with small bounded coefficients, and (ii) a Carleman estimate for the time-differentiated equation, yielding conditional Holder stability. The approach extends the cut-off free Carleman method for linear inverse source problems to a nonlinear setting while keeping the observation geometry unchanged.