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Existence of solution to Korteweg‐de Vries equation in domains that can be transformed into rectangles
Author(s) -
Benia Yassine,
Sadallah BoubakerKhaled
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4773
Subject(s) - mathematics , korteweg–de vries equation , uniqueness , domain (mathematical analysis) , mathematical analysis , lebesgue integration , space (punctuation) , boundary value problem , boundary (topology) , derivative (finance) , pure mathematics , nonlinear system , linguistics , philosophy , physics , quantum mechanics , financial economics , economics
We study the Korteweg‐de Vries equation ∂ t v ( t , y ) + a ( t ) v ( t , y ) ∂ y v ( t , y ) + b ( t ) ∂ y 3 v ( t , y ) = g ( t , y ) subject to boundary condition in nonrectangular domain Ω = { ( t , y ) ∈ R 2 ; 0 < t < T , y ∈ I t } , where I t = { y ∈ R ; φ 1 ( t ) < y < φ 2 ( t ) , t ∈ ( 0 , T ) } , with some assumptions on functions ( φ i ( t )) 1≤ i ≤2 and the coefficients of equation. The right‐hand side and its derivative with respect to t are in the Lebesgue space L 2 (Ω). Our goal is to establish the existence, the uniqueness, and the regularity of the solution.