The Korteweg-de Vries equation on an interval

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 051507 (2019) and may be found at https://doi.org/10.1063/1.5080366.The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega

Initial-boundary value problems for a reaction-diffusion equation

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 081509 (2019); doi: 10.1063/1.5118767 and may be found at https://aip.scitation.org/doi/10.1063/1.5118767.A novel approach that utilizes Fokas’s unified transform is employed for studying a reaction-diffusion equation with power nonlinearity formulated either on the half-line or on a finite interval with data in Sobolev spaces. This approach was recently introduced for initial-boundary value problems involving dispersive nonlinear equations such as the nonlinear Schrödinger and the Korteweg-de Vries equations. Thus, the present work extends the new approach from dispersive equations to diffusive ones, demonstrating the universality of the unified transform in the analysis of nonlinear evolution equations on domains with a boundary