scholarly journals A GRADIENT-LIKE NONAUTONOMOUS EVOLUTION PROCESS

2010 ◽  
Vol 20 (09) ◽  
pp. 2751-2760 ◽  
Author(s):  
TOMÁS CARABALLO ◽  
JOSÉ A. LANGA ◽  
FELIPE RIVERO ◽  
ALEXANDRE N. CARVALHO
Keyword(s):  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Evolution Process ◽  
Pullback Attractor ◽  
Damped Wave Equation ◽  
Bounded Solutions ◽  
Pullback Attractors ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form [Formula: see text] in a bounded smooth domain Ω ⊂ ℝn with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping β : ℝ → (0, ∞) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small nonautonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small nonautonomous perturbation of any autonomous gradient-like evolution processes. Moreover, we are also able to prove that the pullback attractor for (1) is also a forwards attractor and that the rate of attraction is exponential.

scholarly journals On a System of Equations of a Non-Newtonian Micropolar Fluid

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
G. M. de Araújo ◽  
M. A. F. de Araújo ◽  
E. F. L. Lucena
Keyword(s):  
Weak Solutions ◽  
Micropolar Fluid ◽  
Dirichlet Boundary ◽  
Coupled System ◽  
Dirichlet Boundary Conditions ◽  
System Of Equations ◽  
Existence Of Weak Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Compactness Arguments

We investigate a problem for a model of a non-Newtonian micropolar fluid coupled system. The problem has been considered in a bounded, smooth domain ofR3with Dirichlet boundary conditions. The operator stress tensor is given byτ(e(u))=[(ν+ν0M(|e(u)|2))e(u)]. To prove the existence of weak solutions we use the method of Faedo-Galerkin and compactness arguments. Uniqueness and periodicity of solutions are also considered.

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Alessandro Trombetta
Keyword(s):  
Elliptic Equation ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Dirichlet Boundary Conditions ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Moving Plane ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

AbstractWe prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equationin bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

The Pullback Asymptotic Behavior of the Solutions for 2D NonautonomousG-Navier-Stokes Equations

2012 ◽  
Vol 4 (2) ◽  
pp. 223-237 ◽  
Author(s):  
Jinping Jiang ◽  
Yanren Hou ◽  
Xiaoxia Wang
Keyword(s):  
Asymptotic Behavior ◽  
Measure Of Noncompactness ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Fractal Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Pullback Attractors ◽  
Navier Stokes Equations

AbstractThe pullback asymptotic behavior of the solutions for 2D Nonau-tonomousG-Navier-Stokes equations is studied, and the existence of itsL2-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2DG-Navier-Stokes equations is given.

scholarly journals Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yonghai Wang ◽  
Minhui Hu ◽  
Yuming Qin
Keyword(s):  
Wave Equation ◽  
Upper Semicontinuity ◽  
Pullback Attractor ◽  
Damped Wave Equation ◽  
Pullback Attractors ◽  
Strongly Damped Wave Equation ◽  
Strongly Damped

AbstractIn this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ { A ε ( t ) } t ∈ R of Eq. (1.1) with $\varepsilon \in [0,1]$ ε ∈ [ 0 , 1 ] satisfies $\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L^{2}}(A_{\varepsilon }(t),A_{ \varepsilon _{0}}(t))=0$ lim ε → ε 0 sup t ∈ [ a , b ] dist H 0 1 × L 2 ( A ε ( t ) , A ε 0 ( t ) ) = 0 for any $[a,b]\subset \mathbb{R}$ [ a , b ] ⊂ R and $\varepsilon _{0}\in [0,1]$ ε 0 ∈ [ 0 , 1 ] .

Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

2003 ◽  
Vol 133 (5) ◽  
pp. 1137-1153 ◽  
Author(s):  
M. A. Jendoubi ◽  
P. Poláčik
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Cylindrical Domain ◽  
Damped Wave Equation ◽  
Bounded Solutions ◽  
Single Function

We consider two types of equations on a cylindrical domain Ω × (0, ∞), where Ω is a bounded domain in RN, N ≥ 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Ω × (0, ∞) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Ω × (0, ∞) and satisfy a Dirichlet boundary condition on ∂Ω × (0, ∞). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function φ: Ω → R, as t → ∞; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t → ∞. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.

scholarly journals Blow up of solutions of semilinear heat equations in general domains

2015 ◽  
Vol 17 (02) ◽  
pp. 1350042 ◽  
Author(s):  
Valeria Marino ◽  
Filomena Pacella ◽  
Berardino Sciunzi
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Initial Value ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent [Formula: see text]. Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

scholarly journals The second eigenvalue of the fractional p-Laplacian

2016 ◽  
Vol 9 (4) ◽  
pp. 323-355 ◽  
Author(s):  
Lorenzo Brasco ◽  
Enea Parini
Keyword(s):  
Dirichlet Boundary ◽  
Optimal Shape ◽  
Dirichlet Boundary Conditions ◽  
Mutual Distance ◽  
Minimizing Sequence ◽  
Local Case ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Second Eigenvalue ◽  
Variational Formulations

AbstractWe consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ${\Omega\subset\mathbb{R}^{n}}$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue ${\lambda_{2}(\Omega)}$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem$\inf\{\lambda_{2}(\Omega):|\Omega|=c\}.$We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume ${c/2}$ whose mutual distance tends to infinity.

scholarly journals Strong Exponential Attractors for Weakly Damped Semilinear Wave Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Cuncai Liu ◽  
Fengjuan Meng ◽  
Chang Zhang
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Strichartz Estimate ◽  
Exponential Attractor ◽  
Damped Wave Equation ◽  
Exponential Attractors ◽  
Bounded Smooth Domain ◽  
Strong Energy ◽  
Bounded Smooth ◽  
Recent Extension

In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain of ℝ3. The exponential attractor is investigated in a strong energy space for the case of subquintic nonlinearity, which is based on the recent extension of the Strichartz estimate for the case of a bounded domain. The results obtained complete some previous works.

scholarly journals Long-term behaviour in a chemotaxis-fluid system with logistic source

2016 ◽  
Vol 26 (11) ◽  
pp. 2071-2109 ◽  
Author(s):  
Johannes Lankeit
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Logistic Source ◽  
State Formula ◽  
Bounded Smooth ◽  
Homogeneous Dirichlet Boundary Conditions

We consider the coupled chemotaxis Navier–Stokes model with logistic source terms: [Formula: see text] [Formula: see text] [Formula: see text] in a bounded, smooth domain [Formula: see text] under homogeneous Neumann boundary conditions for [Formula: see text] and [Formula: see text] and homogeneous Dirichlet boundary conditions for [Formula: see text] and with given functions [Formula: see text] satisfying certain decay conditions and [Formula: see text] for some [Formula: see text]. We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state [Formula: see text].

On the system of p-Laplacian equations with critical growth

2018 ◽  
Vol 29 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Xiangqing Liu ◽  
Junfang Zhao ◽  
Jiaquan Liu
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Growth ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Laplacian Equations

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

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