  A reaction-diffusion equation with power nonlinearity formulated either on
the half-line or on the finite interval with nonzero boundary conditions is
shown to be locally well-posed in the sense of Hadamard for data in Sobolev
spaces. The result is established via a contraction mapping argument, taking
advantage of a novel approach that utilizes the formula produced by the unified
transform method of Fokas for the forced linear heat equation to obtain linear
estimates analogous to those previously derived for the nonlinear
Schr\"odinger, Korteweg-de Vries and "good" Boussinesq equations. Thus, the
present work extends the recently introduced "unified transform method approach
to well-posedness" from dispersive equations to diffusive ones.
