scholarly journals Linearization of planar homeomorphisms with a compact attractor

2016 ◽  
Vol 48 (1) ◽  
pp. 1 ◽  
Author(s):  
Armengol Gasull ◽  
Jorge Groisman ◽  
Francesc Mañosas
Keyword(s):  
Compact Attractor

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

Fixed Point Principles for Cones of a Linear Normed Space

1980 ◽  
Vol 32 (6) ◽  
pp. 1372-1381 ◽  
Author(s):  
Gilles Fournier
Keyword(s):  
Fixed Point ◽  
Functional Differential Equations ◽  
Compact Attractor ◽  
Completely Continuous ◽  
Continuous Maps ◽  
Compression And Expansion ◽  
Existence Of Periodic Solutions ◽  
Functional Differential ◽  
Generalized Lefschetz Number ◽  
Fixed Point Principles

In [8] and [9], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [11], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to condensing maps and he applied this theorem to prove the existence of periodic solutions of nonlinear autonomous functional differential equations.Nussbaum's proof makes an essential use of the difficult Zabreiko and Krasnosel'skiĭ, and Steinlein (mod p)-theorem for the fixed point index [13 -16]. In [6], Fournier and Peitgen proved two different versions of this theorem for completely continuous maps each one being sufficient for Nussbaum's applications. The proofs of these two theorems are much less involved and, although they are different, they make use of the same easier generalized Lefschetz number calculations (see [12] for (mod p) and [5] for compact attractor).

scholarly journals Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Hamid El Bahja ◽  
Abderrahmane El Hachimi ◽  
Ali Alami Idrissi
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Time Discretization ◽  
Global Compact ◽  
Compact Attractor ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.

scholarly journals An Age-Structured Model of HIV Latent Infection with Two Transmission Routes: Analysis and Optimal Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-22
Author(s):  
Chunyang Qin ◽  
Xia Wang ◽  
Libin Rong
Keyword(s):  
Optimal Control ◽  
Latent Infection ◽  
Dynamical Behavior ◽  
Necessary Condition ◽  
Target Cells ◽  
Uniform Persistence ◽  
Structured Model ◽  
Compact Attractor ◽  
Age Structured ◽  
Age Structured Model

In addition to direct virus infection of target cells, HIV can also be transferred from infected to uninfected cells (cell-to-cell transmission). These two routes might facilitate viral production and the establishment of the latent virus pool, which is considered as a major obstacle to HIV cure. We studied an HIV infection model including the two infection routes and the time since latent infection. The basic reproductive ratio R0 was derived. The existence, positivity, and boundedness of the solution are proved. We investigated the existence of steady states and their stability, which were shown to depend on R0. We established the global asymptotic dynamical behavior by proving the existence of the global compact attractor and uniform persistence of the system and by applying the method of Lyapunov functionals. In the end, we formulated and solved the optimal control problem for the age-structured model. The necessary condition for minimization of the viral level and the cost of drug treatment was obtained, and numerical simulations of various optimal control strategies were performed.

scholarly journals Connections between the convective diffusion equation and the forced Burgers equation

2002 ◽  
Vol 15 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Nejib Smaoui ◽  
Fethi Belgacem
Keyword(s):  
Diffusion Equation ◽  
Parabolic Equations ◽  
Burgers Equation ◽  
Convective Diffusion ◽  
Pseudospectral Method ◽  
Inertial Manifold ◽  
Quasilinear Parabolic Equations ◽  
Compact Attractor ◽  
Convective Diffusion Equation ◽  
Positive Frequency

The convective diffusion equation with drift b(x) and indefinite weight r(x), ∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]+λr(x)ϕ,   (1) is introduced as a model for population dispersal. Strong connections between Equation (1) and the forced Burgers equation with positive frequency (m≥0), ∂u∂t=∂2u∂x2−u∂u∂x+mu+k(x),   (2) are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasilinear parabolic equations given by ∂u∂t=∂2u∂x2+∂(f(v))∂x+g(v)+h(x).  (3) A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed. Equation (2) can be interpreted as the velocity field precursed by Equation (1). Therefore, the dynamics exhibited by the population density in Equation (1) show their effects on the velocity expressed in Equation (2). Numerical results illustrating some aspects of the previous connections are obtained by using a pseudospectral method.

Compact attractor for a nonlinear wave equation with critical exponent

1990 ◽  
Vol 115 (1-2) ◽  
pp. 61-64 ◽  
Author(s):  
Orlando Lopes
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Phase Space ◽  
Critical Exponent ◽  
Growth Condition ◽  
Nonlinear Wave Equation ◽  
Compact Attractor ◽  
Homogeneous Wave ◽  
Coercive Condition ◽  
Homogeneous Wave Equation

SynopsisIn this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.

Attractors for 2D quasi-geostrophic equations with and without colored noise in W2α−,p(ℝ2)

2020 ◽  
pp. 2150017
Author(s):  
Lin Yang ◽  
Yejuan Wang
Keyword(s):  
Asymptotic Behavior ◽  
Colored Noise ◽  
Nonlinear Term ◽  
Global Compact ◽  
Compact Attractor ◽  
Random Attractors

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in [Formula: see text] In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in [Formula: see text] Here, we do not add any modifying factor on the nonlinear term.

scholarly journals Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

2020 ◽  
Vol 10 (1) ◽  
pp. 922-951
Author(s):  
Jinliang Wang ◽  
Renhao Cui
Keyword(s):  
Steady State ◽  
Disease Transmission ◽  
Diffusion Rate ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Compact Attractor ◽  
Positive Steady State ◽  
Host Pathogen ◽  
Large Diffusion

Abstract This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

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