Global existence and long-time behavior of the initial–boundary value problem for the dissipative Boussinesq equation

2016 ◽  
Vol 31 ◽  
pp. 552-568 ◽  
Author(s):  
Shubin Wang ◽  
Xiao Su
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Boussinesq Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Initial Boundary Value

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.

scholarly journals Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation

1997 ◽  
Vol 2 (3-4) ◽  
pp. 281-299 ◽  
Author(s):  
Vladimir V. Varlamov
Keyword(s):  
Boundary Value Problem ◽  
Boussinesq Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Small Initial Data ◽  
Time Asymptotics ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Initial Boundary Value

For the damped Boussinesq equationutt−2butxx=−αuxxxx+uxx+β(u2)xx,x∈(0,π),t>0;α,b=const>0,β=const∈R1, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limitb→+0in the constructed solution is investigated.

scholarly journals Global attractor for a class of nonlinear generalized Kirchhoff models

2016 ◽  
Vol 12 (8) ◽  
pp. 6452-6462 ◽  
Author(s):  
Penghui Lv ◽  
Jingxin Lu ◽  
Guoguang Lin
Keyword(s):  
Boundary Value Problem ◽  
Global Attractor ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Natural Energy ◽  
Long Time ◽  
Initial Boundary Value

The paper studies the long time behavior of solutions to the initial boundary value problem(IBVP) for a class of Kirchhoff models flow  .We establish the well-posedness, theexistence of the global attractor in natural energy space

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

2018 ◽  
Author(s):  
Shkelqim Hajrulla ◽  
Leonard Bezati ◽  
Fatmir Hoxha
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Sobolev Inequality ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Initial Boundary Value ◽  
The Galerkin Method

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

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