scholarly journals Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid

2020 ◽  
Vol 72 (10) ◽  
pp. 1386-1399
Author(s):  
H. Mahmoudi ◽  
A. Esfahani
Keyword(s):  
Evolution Equation ◽  
Surface Waves ◽  
Asymptotic Properties ◽  
Global Solutions ◽  
Polynomial Decay ◽  
Equation Modeling ◽  
Asymptotic Behavior Of Solutions ◽  
Decay Estimates ◽  
Initial Value ◽  
The Cauchy Problem

UDC 517.9 We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated withthis equation in . We obtain some polynomial decay estimates of the energy.

scholarly journals GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CAUCHY PROBLEM OF A DISSIPATIVE BOUSSINESQ-TYPE EQUATION

2017 ◽  
Vol 22 (4) ◽  
pp. 441-463 ◽  
Author(s):  
Amin Esfahani ◽  
Hamideh B. Mohammadi
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Contraction Mapping ◽  
Type Equation ◽  
Global Solutions ◽  
Contraction Mapping Principle ◽  
Equation Modeling ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle.

scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

scholarly journals Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Equilibrium Solution ◽  
Global Solutions ◽  
Initial Time ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Unique Global Solution ◽  
Initial Value ◽  
The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

scholarly journals On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Hatice Taskesen ◽  
Necat Polat ◽  
Abdulkadir Ertaş
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Type Equation ◽  
Global Solutions ◽  
Initial Energy ◽  
Potential Well ◽  
Global Weak Solutions ◽  
Power Type ◽  
Initial Value ◽  
The Cauchy Problem

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

Very slow grow-up of solutions of a semi-linear parabolic equation

2011 ◽  
Vol 54 (2) ◽  
pp. 381-400 ◽  
Author(s):  
Marek Fila ◽  
John R. King ◽  
Michael Winkler ◽  
Eiji Yanagida
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Large Time ◽  
Global Solutions ◽  
Linear Parabolic Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Supercritical Nonlinearity ◽  
Singular Steady State

AbstractWe consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

scholarly journals Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yinxia Wang ◽  
Hengjun Zhao
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Global Existence ◽  
Decay Estimate ◽  
Boussinesq Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Linear Solution ◽  
Optimal Decay ◽  
The Cauchy Problem

We investigate the Cauchy problem for the generalized damped Boussinesq equation. Under small condition on the initial value, we prove the global existence and optimal decay estimate of solutions for all space dimensionsn≥1. Moreover, whenn≥2, we show that the solution can be approximated by the linear solution as time tends to infinity.

scholarly journals Asymptotic properties for second-order linear evolution problems with fractional laplacian operators

2021 ◽  
Vol 43 ◽  
pp. e14
Author(s):  
Cleverson Roberto da Luz ◽  
Maíra Fernandes Gauer Palma
Keyword(s):  
Initial Data ◽  
Asymptotic Properties ◽  
Low Frequency ◽  
Frequency Region ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Decay Estimates ◽  
High Frequency Region ◽  
Linear Evolution ◽  
Additional Regularity

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on the initial data when compared to previous results in the literature. In certain cases, we observe that the dissipative structure of the equation is of regularity-loss type. Due to that special structure, to get decay estimates in high frequency region in the Fourier space it is necessary to impose additional regularity on the initial data to obtain the same decay estimates as in low frequency region. The results obtained in this work can be applied to several initial value problems associated to second-order equations, as for example, wave equation, plate equation, IBq, among others. 

Existence and Asymptotic Properties of Solutions of a Nonlocal Evolution Equation Modeling Cell-Cell Adhesion

2010 ◽  
Vol 42 (4) ◽  
pp. 1784-1804 ◽  
Author(s):  
Janet Dyson ◽  
Stephen A. Gourley ◽  
Rosanna Villella-Bressan ◽  
Glenn F. Webb
Keyword(s):  
Cell Adhesion ◽  
Evolution Equation ◽  
Asymptotic Properties ◽  
Equation Modeling ◽  
Nonlocal Evolution Equation ◽  
Cell Cell

ASYMPTOTIC PROPERTIES FOR A SEMILINEAR PLATE EQUATION IN UNBOUNDED DOMAINS

2009 ◽  
Vol 06 (02) ◽  
pp. 269-294 ◽  
Author(s):  
CLEVERSON ROBERTO DA LUZ ◽  
RUY COIMBRA CHARÃO
Keyword(s):  
Total Energy ◽  
Decay Rate ◽  
Asymptotic Properties ◽  
Global Solutions ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Rotational Inertia ◽  
Small Data ◽  
Plate Equation ◽  
Inertia Effects

We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated with a semilinear, dissipative, plate equation under rotational inertia effects in ℝn. We obtain polynomial decay rate in time for the total energy. In dimension n ≥ 5 for the linear problem and n = 5 for the semilinear problem with small data, we obtain fast decay of the total energy and a decay rate t-1/2 for L2-norm of the solution and similar decay rates for the L2-norm of higher-order derivatives.

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