American Institute of Mathematical Sciences

November  2016, 15(6): 2179-2202. doi: 10.3934/cpaa.2016033

Sharp well-posedness for the Chen-Lee equation

 1 Departamento de Matemáticas, Universidad Nacional de Colombia, AK 30 45-03, 111321, Bogotá, Colombia 2 Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

Received  December 2015 Revised  May 2016 Published  September 2016

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(\mathbb{R})$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.
Citation: Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033
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References:
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