Ill-posedness of degenerate dispersive equations
نویسندگان
چکیده
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 = (u)xxx + (u)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 H 2 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 H 2). Mathematics Subject Classification: 35G25, 35Q53, 34A34, 34A36 (Some figures may appear in colour only in the online journal)
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