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.

A reaction-diffusion equation with power nonlinearity formulated either on the halfline 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 Schrödinger, 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. -boundary value problems, wellposedness in Sobolev spaces on the half-line and the finite interval, L 2 -boundedness of Laplace transform, unified transform method of Fokas. F. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p . Indiana Univ. Math.

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The well-posedness of the Neumann and Robin problems for the Korteweg–de Vries equation is studied with data in Sobolev spaces. Using the Fokas unified transform method, the corresponding linear problems with forcing are solved and solution estimates are derived. Then, using these, an iteration map is defined, and it is proved to be a contraction in appropriate solution spaces after the needed bilinear estimates are derived.

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