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.

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.

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 Critical convective-type equations on a half-line

2006 ◽  
Vol 2006 ◽  
pp. 1-24
Author(s):  
Elena I. Kaikina
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behavior ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equationsut+ℕ(u,ux)+(an∂xn+am∂xm)u=0,(x,t)∈ℝ+×ℝ+,u(x,0)=u0(x),x∈ℝ+,∂xj−1u(0,t)=0forj=1,…,m/2, where the constantsan,am∈ℝ,n,mare integers, the nonlinear termℕ(u,ux)depends on the unknown functionuand its derivativeuxand satisfies the estimate|ℕ(u,v)|≤C|u|ρ|v|σwithσ≥0,ρ≥1, such that((n+2)/2n)(σ+ρ−1)=1,ρ≥1,σ∈[0,m). Also we suppose that∫ℝ+xn/2ℕdx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

The wave front set of the solution of a simple initial-boundary value problem with glancing rays. II

1977 ◽  
Vol 81 (1) ◽  
pp. 97-120 ◽  
Author(s):  
F. G. Friedlander ◽  
R. B. Melrose
Keyword(s):  
Boundary Value Problem ◽  
Wave Front ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Principal Result ◽  
Wave Front Set ◽  
Precise Meaning ◽  
Image Position ◽  
Initial Boundary Value

This paper is a sequel to an earlier paper in these Proceedings by one of us ((5); this will be referred to as [I]). The question considered there was that of determining the wave front set of the solution of the boundary value problemwhere x∈+, y∈n, and n > 1; the precise meaning of the boundary condition at x = 0 is explained in section 1 below. The principal result of [I] can be expressed concisely by saying that singularities do not propagate along the boundary; a detailed statement is given in Theorem 1·9 of the present paper.

Boundedness and stabilization in a multi-dimensional chemotaxis—haptotaxis model

2014 ◽  
Vol 144 (5) ◽  
pp. 1067-1084 ◽  
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Quantitative Result ◽  
Domain Of Attraction ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Flux Boundary Conditions ◽  
The Stability ◽  
Image Position ◽  
Initial Boundary Value ◽  
State 1 ◽  
Uniformly Bounded

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given bywhereχ, ξandμare positive parameters andΩ ⊂ ℝn(n≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for anyμ>χand any sufficiently smooth initial data (u0,w0) satisfyingu0≥ 0 andw0> 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size ofμ. In particular, this will imply that wheneveru0> 0 and 0 <w0< 1 inthere exists a positive constantμ* depending only onχ, ξ, Ω, u0andw0such that for anyμ<μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

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.

Stochastic diffusion equation with fractional Laplacian on the first quadrant

2019 ◽  
Vol 22 (3) ◽  
pp. 795-806
Author(s):  
Jorge Sanchez-Ortiz ◽  
Francisco J. Ariza-Hernandez ◽  
Martin P. Arciga-Alejandre ◽  
Eduard A. Garcia-Murcia
Keyword(s):  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stochastic Evolution ◽  
Stochastic Diffusion Equation ◽  
Representation Of Solutions ◽  
Fokas Method ◽  
Initial Boundary Value ◽  
Main Ideas

Abstract In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

Existence Theorem for the Initial-Boundary Value Problem for a Singular Parabolic Partial Differential Equation

1972 ◽  
Vol 15 (2) ◽  
pp. 229-234
Author(s):  
Julius A. Krantzberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Image Position ◽  
Initial Boundary Value

We consider the initial-boundary value problem for the parabolic partial differential equation1.1in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

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