On the support of solutions to the generalized KdV equation (original) (raw)
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Communications in Partial Differential Equations, 2014
We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u 0 ∈ H 3/4 + (R) whose restriction belongs to H l ((b, ∞)) for some l ∈ Z + and b ∈ R we prove that the restriction of the corresponding solution u(·,t) belongs to H l ((β , ∞)) for any β ∈ R and any t ∈ (0, T ). Thus, this type of regularity propagates with infinite speed to its left as time evolves.
On the regularity of solutions to the kkk-generalized Korteweg-de Vries equation
Proceedings of the American Mathematical Society, 2016
This work is concerned with special regularity properties of solutions to the k-generalized Korteweg-de Vries equation. In [4] it was established that if the initial datun u 0 ∈ H l ((b, ∞)) for some l ∈ Z + and b ∈ R, then the corresponding solution u(•, t) belongs to H l ((β, ∞)) for any β ∈ R and any t ∈ (0, T). Our goal here is to extend this result to the case where l > 3/4.
The Korteweg-de Vries equation on an interval
Journal of Mathematical Physics
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.
Korteweg-de Vries Equation in Bounded Domains
Boletim da Sociedade Paranaense de Matemática, 2009
Introduction 30 2 Notations and results 31 3 Solvability of the problem (2.1)-(2.3) 32 4 Solvability of the KdV equation 33 5 Stability 36 * The author was partially supported by a grant from CNPq-Brazil.
On the boundary‐value problem for the Korteweg–de Vries equation
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2003
We consider the initial-boundary value problem on the half-line for the Korteweg-de Vries equation ut + uux + uxxx = 0, t > 0, x > 0, u(x, 0) = u 0 (x), x > 0, u (0, t) = 0, t > 0.