scholarly journals Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation

2003 ◽  
Vol 2003 (9) ◽  
pp. 521-538
Author(s):  
Nikos Karachalios ◽  
Nikos Stavrakakis ◽  
Pavlos Xanthopoulos
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Evolution Equation ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Evolution Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Nonlinear Parabolic

We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

scholarly journals Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains

2020 ◽  
pp. 2050020
Author(s):  
Renhai Wang ◽  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Random Coefficients ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Additive White Noise ◽  
Approach Zero ◽  
Drift Terms ◽  
Laplacian Equations

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic [Formula: see text]-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain [Formula: see text]. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in [Formula: see text]. This attractor is further proved to be a bi-spatial [Formula: see text]-attractor for any [Formula: see text], which is compact, measurable in [Formula: see text] and attracts all random subsets of [Formula: see text] with respect to the norm of [Formula: see text]. Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in [Formula: see text] for [Formula: see text] in order to overcome the non-compactness of Sobolev embeddings on [Formula: see text] and the nonlinearity of the fractional [Formula: see text]-Laplace operator.

scholarly journals Asymptotic Behavior of the Solutions of the Generalized Globally Modified Navier–Stokes Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Yonghong Duan ◽  
Xiaojuan Chai
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Global Attractors ◽  
Navier Stokes ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equations

The paper is concerned with the existence and the asymptotic behavior of solutions to a class of generalized Navier–Stokes equations, which generalises the so-called globally modified Navier–Stokes equations. The existence and uniqueness of solutions are proved under different assumptions on the dissipation and modification factors. For the asymptotic behavior of solutions, we prove the existence of global attractors in proper spaces. The results generalize some results derived in our previous work Ann. Polon. Math. 122(2):101–128(2019).

scholarly journals Long Term Behavior for a Class of Stochastic Delay Lattice Systems in Xρ Space

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yijin Zhang ◽  
Zongbing Lin
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Asymptotic Behavior Of Solutions ◽  
Random Dynamical System ◽  
Lattice Systems ◽  
Stochastic Delay ◽  
Long Term Behavior

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor in Xρ space. Finally, ergodicity of the systems is achieved.

scholarly journals Global existence and uniqueness of the solution to a nonlinear parabolic equation

2018 ◽  
Vol 14 (2) ◽  
pp. 7860-7863
Author(s):  
Alexander G. Ramm
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Unique Global Solution ◽  
Nonlinear Parabolic ◽  
Uniqueness Of The Solution ◽  
Global Existence And Uniqueness

Consider the equation  u’ (t)  -  u + | u |p u = 0, u(0) = u0(x), (1), where u’ := du/dt , p = const > 0, x E R3, t > 0.  Assume that u0 is a smooth and decaying function,           ||u0|| =            sup             |u(x, t)|.                                         x E R3 ,t E R+      It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate                              ||u(x, t)|| < c, where c > 0 does not depend on x, t.

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