ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

2006 ◽  
Vol 08 (02) ◽  
pp. 189-217 ◽  
Author(s):  
ROSA E. CARDIEL ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Pseudo Differential Operator ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

scholarly journals Benjamin-Ono Equation on a Half-Line

2010 ◽  
Vol 2010 ◽  
pp. 1-38 ◽  
Author(s):  
Nakao Hayashi ◽  
Elena I. Kaikina
Keyword(s):  
Boundary Value Problem ◽  
Large Time ◽  
Boundary Value ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Half Line ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Boundary Value ◽  
Time Existence

We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Landau–Ginzburg-type equations on the half-line in the critical case

2005 ◽  
Vol 135 (6) ◽  
pp. 1241-1262 ◽  
Author(s):  
Elena I. Kaikina ◽  
Hector F. Ruiz-Paredes
Keyword(s):  
Critical Case ◽  
Linear Part ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical case where β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

scholarly journals On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

2020 ◽  
pp. 547-558
Author(s):  
Alexander N. Polkovnikov
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Type Spaces ◽  
Bochner Space ◽  
Initial Boundary Value

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space

FRACTIONAL LANDAU–GINZBURG EQUATIONS ON A SEGMENT

2008 ◽  
Vol 10 (06) ◽  
pp. 1151-1181
Author(s):  
ELENA I. KAIKINA
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Existence Of Solutions ◽  
Asymptotic Representation Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.

scholarly journals The initial-boundary value problem for the Schrödinger–Korteweg–de Vries system on the half-line

2019 ◽  
Vol 21 (08) ◽  
pp. 1850066 ◽  
Author(s):  
Márcio Cavalcante ◽  
Adán J. Corcho
Keyword(s):  
Boundary Value Problem ◽  
Vries Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Half Line ◽  
Nonlinear Schrödinger ◽  
De Vries ◽  
Quadratic Nonlinearities ◽  
Initial Boundary Value

We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger–Korteweg–de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, one of these families being developed by Holmer to study the IBVP associated to the Korteweg–de Vries equation [The initial-boundary value problem for the Korteweg–de Vries equation, Comm. Partial Differential Equations 31 (2006) 1151–1190] and the other one was recently introduced by Cavalcante [The initial-boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential Integral Equations 30(7–8) (2017) 521–554] in the context of nonlinear Schrödinger with quadratic nonlinearities.

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