Wave and scattering operators for the nonlinear Klein-Gordon equation on a quarter-plane

Ballesteros, Miguel; Iniesta, Diego; Naumkin, Ivan; Peña, Clemente

doi:10.1016/j.jde.2022.03.009

Cited by 3 publications

(1 citation statement)

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“…For example, in the papers [27][28][29]36] considered IBV problems on D for the nonlinear Schrödinger equation with different boundary data types. In the paper [2], the global well-posedness of the Klein-Gordon equation on D with the Dirichlet type of boundary data was obtained. As far as we know, there are no results on the IBV problem for the fractional Ginzburg-Landau equation in the two-dimensional case.…”

Section: Previous Resultsmentioning

confidence: 99%

Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

Carreño-Diaz,

Kaikina

2024

Nonlinearity

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We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane { u t ( t , x ) − ∇ β u ( t , x ) = | u ( t , x ) | σ u ( t , x ) , x ∈ D , t > 0 u ( 0 , x ) = u 0 ( x ) , x ∈ D , u | x i = 0 = h j ( t , x j ) , x j > 0 , j = 1 , 2 , t > 0 , where D = { x 1 > 0 , x 2 > 0 } , β ∈ ( 3 2 , 2 ) , σ > 0 and ∇ β is a fractional Laplacian defined as ∇ β u = ∑ j 1 Γ ( 2 − β ) ∫ 0 x j u y j y j ( x j − y j ) β − 1 d y . We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the L 2 − based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.

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