Well-posed boundary value problems for integrable evolution equations on a finite interval

Pelloni, B. (2002) Well-posed boundary value problems for integrable evolution equations on a finite interval. Theoretical and Mathematical Physics, 133 (2). pp. 1598-1606. ISSN 0040-5779 doi: 10.1023/A:1021163230712

Abstract/Summary

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.