Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

Forcella, Luigi; Fujiwara, Kazumasa; Georgiev, Vladimir; Ozawa, Tohru

doi:10.1007/978-3-030-10937-0_6

Cited by 1 publication

(1 citation statement)

References 26 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theoretical analysis of Cauchy problem for the fractional Ginzburg-Landau equation has also well developed. Among them, [10,22,30,39,40,43]. Due to the nonlocality of fractional operators, the study of a boundary value problem even in the case of a half-line is difficult.…”

Section: Previous Resultsmentioning

confidence: 99%

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

Carreño-Diaz,

Kaikina

2024

Nonlinearity

View full text Add to dashboard Cite

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.

show abstract