Abstract
In this paper we provide numerical and analytical evidence that some degenerate dispersive partial differential equations are ill-posed. Specifically we study the K (2, 2) equation ut = (u2)xxx + (u2)x and the ‘degenerate Airy’ equation ut = 2uuxxx. For K (2, 2) our results are computational in nature: we conduct a series of numerical simulations which demonstrate that data which is very small in H2 can be of unit size at a fixed time which is independent of the data's size. For the degenerate Airy equation, our results are fully rigorous: we prove the existence of a compactly supported self-similar solution which, when combined with certain scaling invariances, implies ill-posedness (also in H2).
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