Existence and uniqueness of global solutions of onR 2

We consider the pullbackD-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullbackD-absorbing set is obtained. By proving the properties of compactness about the nonlinear operatorg1(·),g2(·), and then proving the pullbackD-condition (C), the existence of the pullbackD-attractor of the equations previously mentioned is given.

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Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation

The Scientific World JOURNAL ◽

10.1155/2014/716740 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Daewook Kim ◽

Dojin Kim ◽

Keum-Shik Hong ◽

Il Hyo Jung

Keyword(s):

Wave Equation ◽

Existence And Uniqueness ◽

Global Solutions ◽

Growth Conditions ◽

Decay Rates ◽

Decay Estimates ◽

Nonlinear Dissipation ◽

Kirchhoff Type ◽

Type Wave ◽

Energy Decay Rates

The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form under suitable assumptions on . Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipationg. Lastly, numerical simulations in order to verify the analytical results are given.

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Global existence for the generalised 2D Ginzburg-Landau equation

The ANZIAM Journal ◽

10.1017/s1446181100008099 ◽

2003 ◽

Vol 44 (3) ◽

pp. 381-392 ◽

Cited By ~ 3

Author(s):

Hongjun Gao ◽

Keng-Huat Kwek

Keyword(s):

Existence And Uniqueness ◽

Landau Equation ◽

Sufficient Conditions ◽

Global Solutions ◽

Ginzburg Landau ◽

Spatial Variables ◽

Type Complex ◽

Ginzburg Landau Equation ◽

Fifth Order ◽

Formation Systems

AbstractGinzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.

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Boundary linear stabilization of the modified generalized Korteweg–de Vries–Burgers equation

Advances in Difference Equations ◽

10.1186/s13662-019-2387-7 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 3

Author(s):

Nejib Smaoui ◽

Boumediène Chentouf ◽

Ala’ Alalabi

Keyword(s):

Exponential Stability ◽

Numerical Simulations ◽

Boundary Control ◽

Existence And Uniqueness ◽

Burgers Equation ◽

Global Solutions ◽

Spatial Variable ◽

Stabilization Problem ◽

De Vries ◽

Adaptive Boundary Control

Abstract The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in $[0,1]$ [ 0 , 1 ] . First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in $L^{2} (0,1)$ L 2 ( 0 , 1 ) . Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.

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The asymptotic behaviour of fractional lattice systems with variable delay

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0038 ◽

2019 ◽

Vol 22 (3) ◽

pp. 681-698

Author(s):

Linfang Liu ◽

Tomás Caraballo ◽

Peter E. Kloeden

Keyword(s):

Asymptotic Behaviour ◽

Existence And Uniqueness ◽

Time Delays ◽

Lipschitz Constant ◽

Time Derivative ◽

Global Solutions ◽

Variable Delay ◽

Lattice Systems ◽

Functional Differential ◽

Fractional Functional

Abstract The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a certain condition on the Lipschitz constant.

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