Cauchy Problem for for some high order generalization of Korteweg - de Vries equation

2011/05/31 by Z. A. Sobirov, Sobirov, Z. A., S. Abdinazarov +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1105.6149

arxiv created 2011/05/31 · arxiv updated 2011/06/01

Abstract

In this work we study Cauchy problem for a high-order differential equation (∂ u(y,x))/(∂ y)+P((∂)/(∂ x))u(y,x)=γ(∂)/(∂ x)(u²(y,x))+F(y,x). We prove that the problem is well-posed both for linear (γ=0) and nonlinear equations on the class of rapidly decaying Schwartz functions. Furthermore, for the case when the initial condition is given on L₂(R¹) we prove the existence of the unique solution on the space L_∞(0,y₀; L₂(R¹))\bigcap L₂(0,y₀; H^n-1(R¹))\bigcap L₂(0,y₀;Hⁿ(-r, r)), where r is an arbitrary positive number. It is also shown that the solution continuously depends on the initial conditions.

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