593 results on '"global attractors"'
Search Results
102. A survey on impulsive dynamical systems
- Author
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Everaldo Bonotto, Matheus Bortolan, Tomas Caraballo, and Rodolfo Collegari
- Subjects
impulsive dynamical systems ,global attractors ,nonautonomous dynamical systems ,cocycle attractors ,navier-stokes equation ,Mathematics ,QA1-939 - Abstract
In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes in their evolution (impulses). But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world.
- Published
- 2016
- Full Text
- View/download PDF
103. Well-posedness and global attractors for a non-isothermal viscous relaxationof nonlocal Cahn-Hilliard equations
- Author
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Joseph L. Shomberg
- Subjects
Nonlocal Cahn-Hilliard equations ,well-posedness ,global attractors ,regularity ,Mathematics ,QA1-939 - Abstract
We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space
- Published
- 2016
- Full Text
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104. Global Attractors for Semilinear Parabolic Problems Involving X-Elliptic Operators
- Author
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Stefanie Sonner
- Subjects
Semilinear degenerate parabolic equations ,global attractors ,sub-elliptic operators ,Analysis ,QA299.6-433 - Abstract
We consider semilinear parabolic equations involving an operator that is X-elliptic with respect to a family of vector fields X with suitable properties. The vector fields determine the natural functional setting associated to the problem and the admissible growth of the non-linearity. We prove the global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.
- Published
- 2015
- Full Text
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105. Global attractors, extremal stability and periodicity for a delayed population model with survival rate on time scales
- Author
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Urszula Ostaszewska, Malgorzata Zdanowicz, Ewa Schmeidel, and Jaqueline G. Mesquita
- Subjects
Physics ,Applied Mathematics ,time scales ,General Medicine ,stability ,periodicity ,Stability (probability) ,Time ,Survival Rate ,Combinatorics ,Computational Mathematics ,delayed population model ,Population model ,Modeling and Simulation ,Attractor ,QA1-939 ,General Agricultural and Biological Sciences ,global attractors ,TP248.13-248.65 ,Mathematics ,Biotechnology - Abstract
In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of the delayed population model with survival rate on isolated time scales given by \begin{document}$ x^{\Delta} (t) = \gamma(t) x(t) + \dfrac{x(d(t))}{\mu(t)}e^{r(t)\mu(t)\left(1 - \frac{x(d(t))}{\mu(t)}\right)}, \ \ t \in \mathbb T. $\end{document} We present many examples to illustrate our results, considering different time scales.
- Published
- 2021
106. Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion
- Author
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Jiaohui Xu, Tomás Caraballo, José Valero, and Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico
- Subjects
Global attractors ,Dafermos transformation ,Applied Mathematics ,Long time memory ,Analysis ,Non-local partial differential equations - Abstract
In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.
- Published
- 2022
107. Vanishing viscosity limit for global attractors for the damped Navier–Stokes system with stress free boundary conditions.
- Author
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Chepyzhov, Vladimir, Ilyin, Alexei, and Zelik, Sergey
- Subjects
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ATTRACTORS (Mathematics) , *NAVIER-Stokes equations , *EULER equations , *DAMPING (Mechanics) , *NONLINEAR partial differential operators , *VISCOUS flow - Abstract
We consider the damped and driven Navier–Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain Ω ⊂ R 2 . We show that the damped Euler system has a (strong) global attractor in H 1 ( Ω ) . We also show that in the vanishing viscosity limit the global attractors of the Navier–Stokes system converge in the non-symmetric Hausdorff distance in H 1 ( Ω ) to the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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108. STRONG ATTRACTORS FOR VANISHING VISCOSITY APPROXIMATIONS OF NON-NEWTONIAN SUSPENSION FLOWS.
- Author
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Kapustyan, Oleksiy V., Kasyanov, Pavlo O., Valero, José, and Zgurovsky, Michael Z.
- Subjects
ATTRACTORS (Mathematics) ,VISCOSITY ,APPROXIMATION theory ,NON-Newtonian flow (Fluid dynamics) ,PHASE space - Abstract
In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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109. Asymptotics of viscoelastic materials with nonlinear density and memory effects.
- Author
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Conti, M., Ma, T.F., Marchini, E.M., and Seminario Huertas, P.N.
- Subjects
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VISCOELASTIC materials , *NONLINEAR difference equations , *ATTRACTORS (Mathematics) , *DAMPING capacity , *ENERGY dissipation - Abstract
This paper is concerned with the nonlinear viscoelastic equation | ∂ t u | ρ ∂ t t u − Δ ∂ t t u − Δ u + ∫ 0 ∞ μ ( s ) Δ u ( t − s ) d s + f ( u ) = h , suitable to modeling extensional vibrations of thin rods with nonlinear material density ϱ ( ∂ t u ) = | ∂ t u | ρ , and presence of memory effects. This class of equations was studied by many authors, but well-posedness in the whole admissible range ρ ∈ [ 0 , 4 ] and for f growing up to the critical exponent were established only recently. The existence of global attractors was proved in presence of an additional viscous or frictional damping. In the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor A ρ for ρ < 4 and critical nonlinearity f . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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110. Porous elastic system with nonlinear damping and sources terms.
- Author
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Freitas, Mirelson M., Santos, M.L., and Langa, José A.
- Subjects
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NONLINEAR theories , *POROUS materials , *ELASTICITY , *DAMPING (Mechanics) , *MONOTONE operators - Abstract
We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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111. Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces.
- Author
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Coti Zelati, Michele
- Subjects
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SOBOLEV spaces , *LAPLACIAN matrices , *GRAPH theory , *NUMERICAL analysis , *BOUNDARY element methods - Abstract
We consider the subcritical SQG equation in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate appears to be new in the literature and allows a sharp use of the subcritical nature of the $$L^\infty $$ bounds for this problem. As a by-product, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper semicontinuity with respect to the strength of the diffusion. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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112. SMOOTHING FOR THE ZAKH AROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS ON EUCLIDEAN SPACES.
- Author
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COMPAAN, E.
- Subjects
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KLEIN-Gordon equation , *SCHRODINGER equation , *ATTRACTORS (Mathematics) , *SMOOTHING (Numerical analysis) , *EUCLIDEAN distance , *NONLINEAR theories - Abstract
This paper studies the existence and regularity properties of solutions to the Zakharov and Klein--Gordon--Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear Xs,b estimate, which is proved using delicate dyadic and angu lar decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-- Gordon--Schrödinger flow in the energy space for dimensions d = 2, 3. Second, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein--Gordon-- Schroödinger evolution on R4 below the energy space for sufficiently small initial data. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
113. Asymptotic behavior of nonlocal partial differential equations with long time memory
- Author
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Universidad de Sevilla. Departamento de Ecuaciones diferenciales y Análisis numérico, Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales, Xu, Jiaohui, Caraballo Garrido, Tomás, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones diferenciales y Análisis numérico, Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales, Xu, Jiaohui, Caraballo Garrido, Tomás, and Valero, José
- Abstract
In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70’s. Thanks to this Dafermos transformation, the original problem with memory is transformed into a nondelay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors to the transformed problem are provided. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem, it is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.
- Published
- 2022
114. Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion
- Author
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Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Xu, Jiaohui, Caraballo Garrido, Tomás, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Xu, Jiaohui, Caraballo Garrido, Tomás, and Valero, José
- Abstract
In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.
- Published
- 2022
115. Existencia de un atractor exponencial para un modelo de p−Kirchhoff con memoria infinita
- Author
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Noel Figueroa, Pablo Fernando and Santaria Leuyacc, Yony Raúl
- Subjects
infinite memory ,atractores globales ,exponential attractors ,memoria infinita ,p−Kirchhoff equation ,atractores exponenciales ,Ecuaci´on de p−Kirchhoff ,global attractors - Abstract
The main objective of this work is to study the long-term dynamics of a p−Kirchhoff model with infinite memory exposed to structural forces on a bounded domain Ω ⊂ ℝn. In particular, the existence of a global attractor with exponential attraction rate and finite fractal dimension is shown, that is, the existence of an exponential attractor is proved., El presente trabajo tiene por objetivo principal estudiar la dinámica a largo plazo de un modelo de p−Kirchhoff con memoria infinita expuesto a fuerzas estructurales sobre un dominio acotado Ω ⊂ ℝn. En particular se muestra la existencia de un atractor global con tasa de atracción exponencial y dimensión fractal finita, es decir, se prueba la existencia de un atractor exponencial.
- Published
- 2022
116. Informational Structures and Informational Fields as a Prototype for the Description of Postulates of the Integrated Information Theory
- Author
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Piotr Kalita, José A. Langa, and Fernando Soler-Toscano
- Subjects
dynamical system ,integrated information theory ,global attractors ,Lotka–Volterra equations ,informational structure ,informational field ,Science ,Astrophysics ,QB460-466 ,Physics ,QC1-999 - Abstract
Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.
- Published
- 2019
- Full Text
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117. Global attractors for a class of semilinear degenerate parabolic equations
- Author
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Kaixuan Zhu and Yongqin Xie
- Subjects
Class (set theory) ,Pure mathematics ,asymptotic higher-order integrability ,35b41 ,35b40 ,General Mathematics ,010102 general mathematics ,Degenerate energy levels ,35k65 ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,Attractor ,degenerate parabolic equations ,QA1-939 ,polynomial growth of arbitrary order ,0101 mathematics ,global attractors ,Mathematics - Abstract
In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f f satisfying the polynomial growth of arbitrary p − 1 p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) \left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega )) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) {L}^{2}\left(\Omega ) with the L p + δ {L}^{p+\delta } -norm for any δ ∈ [ 0 , + ∞ ) \delta \in \left[0,+\infty ) .
- Published
- 2021
118. On the Global Dynamics of an Electroencephalographic Mean Field Model of the Neocortex.
- Author
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Shirani, Farshad, Haddad, Wassim M., and de la Llave, Rafael
- Subjects
- *
FUNCTION spaces , *SEMIGROUPS (Algebra) , *ELECTROENCEPHALOGRAPHY , *BOUNDARY value problems , *PARTIAL differential equations - Abstract
This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley, Cadusch, and Dafilis [Network, 13 (2002), pp. 67-113]. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. Existence, uniqueness, and regularity of weak and strong solutions of the model are established in appropriate function spaces, and the associated initial-boundary value problems are proved to be well- posed. Sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities in the model, as required by their biophysical interpretation. It is shown that the semigroups of weak and strong solution operators possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Challenges involved in establishing a global attractor for the model are discussed and it is shown that there exist parameter values for which the constructed semidynamical systems do not possess a compact global attractor due to the lack of the compactness property. Finally, using the theoretical results of the paper, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
119. UPPER SEMICONTINUITY OF ATTRACTORS AND CONTINUITY OF EQUILIBRIUM SETS FOR PARABOLIC PROBLEMS WITH DEGENERATE p-LAPLACIAN.
- Author
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BRUSCHI, SIMONE M., GENTILE, CLÁUDIA B., and PRIMO, MARCOS R. T.
- Subjects
- *
PARABOLIC differential equations , *LAPLACIAN matrices , *EQUILIBRIUM , *CONTINUITY , *SET theory , *ATTRACTORS (Mathematics) - Abstract
In this work we obtain some continuity properties on the parameter q at p = q for the Takeuchi-Yamada problem which is a degenerate p-laplacian version of the Chafee-Infante problem. We prove the continuity of the ows and the equilibrium sets, and the upper semicontinuity of the global attractors. [ABSTRACT FROM AUTHOR]
- Published
- 2017
120. SMOOTH ATTRACTORS FOR WEAK SOLUTIONS OF THE SQG EQUATION WITH CRITICAL DISSIPATION.
- Author
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ZELATI, MICHELE COTI and KALITA, PIOTR
- Subjects
ENERGY dissipation ,ATTRACTORS (Mathematics) ,VISCOSITY ,DYNAMICAL systems ,SOBOLEV spaces ,COMMUTATORS (Operator theory) ,MATHEMATICAL models - Abstract
We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Holder norms, which complement the existing estimates based on commutator analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
121. Global existence and attractors for the two-dimensional Burgers-Ginzburg-Landau equations.
- Author
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Changhong Guo and Shaomei Fang
- Subjects
EXISTENCE theorems ,BURGERS' equation ,INITIAL value problems - Abstract
This paper investigates the periodic initial value problem for the two-dimensional Burgers-Ginzburg-Landau (2D Burgers-GL) equations, which can be derived from the so-called modulated modulation equations (MME) that govern the dynamics of the modulated amplitudes of some periodic critical modes. The well-posedness of the solutions and the global attractors for the 2D Burgers-GL equations are obtained via delicate a priori estimates, the Galerkin method, and operator semigroup method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
122. The Cahn-Hilliard equation as limit of a conserved phase-field system.
- Author
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Bonfoh, Ahmed and Enyi, Cyril D.
- Subjects
- *
CAHN-Hilliard-Cook equation , *NONLINEAR functions , *PHASE transitions , *ORDINARY differential equations , *HILBERT space - Abstract
Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077-1105], we considered the conserved phasefield system {τφt - Δ(δφt - Δφ + g(φ) - u) = 0, εut + φt - Δu = 0, in a bounded domain of Rd , d = 1, 2, 3, where τ > 0 is a relaxation time, δ > 0 is the viscosity parameter, ε ∈ (0, 1] is the heat capacity, φ is the order parameter, u is the absolute temperature and g : R → R is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at ε = 0. Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at (ε, δ) = (0, 0). To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε, we require ε to be dominated from above by δ. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn-Hilliard equation, improving and extending some previous results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
123. Large time behavior for the fractional Ginzburg-Landau equations near the BCS-BEC crossover regime of Fermi gases.
- Author
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Li, Lang, Jin, Lingyu, and Fang, Shaomei
- Subjects
- *
BCS-BEC crossover , *GALERKIN methods , *SOBOLEV spaces , *ELECTRON gas - Abstract
In this paper, we consider the fractional Ginzburg-Landau equations near the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensate (BCS-BEC) crossover of atomic Fermi gases. This fractional Ginzburg-Landau equations can be viewed as a generalization of the integral differential equations proposed by Machida and Koyama (Phys. Rev. A 74:033603, 2006). By using the Galerkin method and a priori estimates, together with the properties of Sobolev spaces, we first establish the existence and uniqueness of weak solutions to these equations and then we prove the existence of global attractors. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
124. Lipschitz perturbations of the Chafee-Infante equation.
- Author
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Pires, Leonardo
- Published
- 2023
- Full Text
- View/download PDF
125. EXISTENCE, REGULARITY AND APPROXIMATION OF GLOBAL ATTRACTORS FOR WEAKLY DISSIPATIVE P-LAPLACE EQUATIONS.
- Author
-
YANGRONG LI and JINYAN YIN
- Subjects
STOCHASTIC analysis ,LAPLACE'S equation ,EUCLIDEAN algorithm ,DIFFERENTIAL equations ,PARTIAL differential equations - Abstract
A global attractor in L
2 is shown for weakly dissipative p-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than p - 1. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an (L2 ;Lr )-attractor for r belonging to a special interval, which contains the critical exponent p. The obtained attractor is proved to be ap- proximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
126. Asymptotic behavior of the thermoelastic suspension bridge equation with linear memory.
- Author
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Kang, Jum-Ran
- Subjects
- *
ALGEBRAIC equations , *THERMOELASTICITY , *VISCOELASTICITY , *BOUNDARY value problems , *MATHEMATICAL analysis - Abstract
This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
127. N 维空间中一类强阻尼非线性波动方程的 解及其性质.
- Author
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廖扬 and 周晓宇
- Subjects
- *
NONLINEAR wave equations , *THREE-manifolds (Topology) , *GALERKIN methods , *SOBOLEV spaces , *ATTRACTORS (Mathematics) - Abstract
The properties of three dimensional space solution of strongly damped nonlinear wave equation by 3D was expanded to N dimensional space ( N > 3 ) . A standard Galerkin method and the Sobolev embedding theorem were utilized to study the existence of weak solution under the space. The inner product was used to make the solution' s dissipation estimates, and the Gronwall lemma was used to prove the existence of the attractor. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
128. Nonclassical diffusion with memory lacking instantaneous damping
- Author
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Monica Conti, Vittorino Pata, and Filippo Dell'Oro
- Subjects
Physics ,Pure mathematics ,Global attractors ,Diffusion equation ,Semigroup ,Hereditary memory ,Applied Mathematics ,Nonclassical diffusion ,010102 general mathematics ,General Medicine ,Optimal regularity ,01 natural sciences ,Past history ,010101 applied mathematics ,Bounded function ,Domain (ring theory) ,Attractor ,0101 mathematics ,Diffusion (business) ,Analysis - Abstract
We consider the nonclassical diffusion equation with hereditary memory \begin{document}$ u_t-\Delta u_t -\int_0^\infty \kappa(s)\Delta u(t-s)\,{{\rm{d}}} s +f(u) = g $\end{document} on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form \begin{document}$ -\Delta u $\end{document} , contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity.
- Published
- 2020
129. Global attractors for the Benjamin-Bona-Mahony equation with memory
- Author
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Filippo Dell'Oro, Olivier Goubet, Youcef Mammeri, and Vittorino Pata
- Subjects
Pure mathematics ,Benjamin-Bona-Mahony equation, dissipative memory, global attractors ,Semigroup ,General Mathematics ,Benjamin–Bona–Mahony equation ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Space (mathematics) ,01 natural sciences ,dissipative memory ,Nonlinear system ,Mathematics - Analysis of PDEs ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Phase space ,Attractor ,FOS: Mathematics ,Benjamin-Bona-Mahony equation ,0101 mathematics ,Invariant (mathematics) ,global attractors ,Energy (signal processing) ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f $$ where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force $f$, we show that the related solution semigroup possesses the global attractor in the natural weak energy space. The result is obtained by means of a nonstandard approach based on the construction of a suitable family of attractors on certain invariant sets of the phase space.
- Published
- 2020
130. Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls
- Author
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Ciprian G. Gal
- Subjects
Phase separation ,Cahn-Hilliard equations ,dynamic boundary conditions ,exponential attractors ,global attractors ,Laplace-Beltrami differential operators. ,Mathematics ,QA1-939 - Abstract
In a previous article [7], we proposed a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions were derived from a mass conservation law and variational methods. In the present paper, we study the problem further. Using a Faedo-Galerkin method, we obtain the existence and uniqueness of a global solution to our problem, under more general assumptions than those in [7]. We then study its asymptotic behavior and prove the existence of an exponential attractor (and thus of a global attractor) with finite dimension.
- Published
- 2006
131. Semicontinuity of attractors for impulsive dynamical systems.
- Author
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Bonotto, E.M., Bortolan, M.C., Collegari, R., and Czaja, R.
- Subjects
- *
CONTINUITY , *ATTRACTORS (Mathematics) , *DYNAMICAL systems , *IMPULSIVE differential equations , *SET theory - Abstract
In this paper we introduce the concept of collective tube conditions which assures a suitable behaviour for a family of dynamical systems close to impulsive sets. Using the collective tube conditions, we develop the theory of upper and lower semicontinuity of global attractors for a family of impulsive dynamical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
132. Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping.
- Author
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Geredeli, Pelin G. and Webster, Justin T.
- Subjects
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NONLINEAR dynamical systems , *NONLINEAR boundary value problems , *ROTATIONAL motion , *INERTIA (Mechanics) , *ENERGY dissipation - Abstract
The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993). We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
133. On Nonlocal Cahn-Hilliard-Navier-Stokes Systems in Two Dimensions.
- Author
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Frigeri, Sergio, Gal, Ciprian, and Grasselli, Maurizio
- Subjects
- *
NAVIER-Stokes equations , *INTERFACES (Physical sciences) , *INCOMPRESSIBLE flow , *ATTRACTORS (Mathematics) , *VISCOSITY - Abstract
We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak-strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
134. LARGE TIME BEHAVIOR OF A CONSERVED PHASE-FIELD SYSTEM.
- Author
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BONFOH, AHMED and ENYI, CYRIL D.
- Subjects
VISCOSITY ,CAHN-Hilliard-Cook equation ,LOGARITHMIC functions ,EXPONENTIAL functions ,SOBOLEV spaces - Abstract
We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
135. Global attractors for quasilinear parabolic equations on unbounded thin domains.
- Author
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Silva, Ricardo
- Abstract
In this paper we are concerned with the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
136. Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures.
- Author
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Chueshov, Igor, Dowell, Earl, Lasiecka, Irena, and Webster, Justin
- Subjects
- *
NONLINEAR mechanics , *ELASTIC plates & shells , *GAS flow , *SUPERSONIC flow , *DIVERGENCE theorem - Abstract
We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
137. Analyticity for Kuramoto-Sivashinsky-type equations in two spatial dimensions.
- Author
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Ioakim, Xenakis and Smyrlis, Yiorgos‐Sokratis
- Subjects
- *
PARTIAL differential equations , *EQUATIONS , *ATTRACTORS (Mathematics) , *EIGENVALUES , *INTEGERS - Abstract
I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto-Sivashinsky-type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three-dimensional models of a spectral method, which was developed by the authors for the one-dimensional Kuramoto-Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈ C ∞, involving the rate of growth of ∇ n u, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper-Kawahara, Frenkel-Indireshkumar, and Coward-Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
138. A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems.
- Author
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Pimentel, Juliana and Rocha, Carlos
- Subjects
- *
REACTION-diffusion equations , *PARABOLIC differential equations , *ATTRACTORS (Mathematics) , *DECOMPOSITION method , *PERMUTATIONS - Abstract
We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
139. Mathematical aeroelasticity: A survey.
- Author
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Chueshov, Igor, Dowell, Earl H., Lasiecka, Irena, and Webster, Justin T.
- Subjects
- *
FLUID dynamics , *SUPERSONIC aerodynamics , *ULTRASONICS , *VON Karman equations , *BOUNDARY value problems - Abstract
Abstract. A variety of models describing the interaction between flows and oscillating structures are discussed. The main aim is to analyze conditions under which structural instability (flutter) induced by a fluid flow can be suppressed or eliminated. The analysis provided focuses on effects brought about by: (i) different plate and fluid boundary conditions, (ii) various regimes for flow velocities: subsonic, transonic, or supersonic, (iii) different modeling of the structure which may or may not account for in-plane accelerations (full von Karman system), (iv) viscous effects, (v) an assortment of models related to piston-theoretic model reductions, and (iv) considerations of axial flows (in contrast to so called normal flows). The discussion below is based on conclusions reached via a combination of rigorous PDE analysis, numerical computations, and experimental trials. [ABSTRACT FROM AUTHOR]
- Published
- 2016
140. Dynamics of a conserved phase-field system.
- Author
-
Bonfoh, Ahmed
- Subjects
- *
CAHN-Hilliard-Cook equation , *MATHEMATICAL proofs , *EXISTENCE theorems , *ATTRACTORS (Mathematics) , *PERTURBATION theory , *MATHEMATICAL singularities , *LIMITS (Mathematics) - Abstract
Recently, in Bonfoh [Ann. Mat. Pura Appl. 2011;190:105–144], we investigated the dynamics of a nonconserved phase-field system whose singular limit is the viscous Cahn–Hilliard equation. More precisely, we proved the existence of the global attractor, exponential attractors, and inertial manifolds and we showed their continuity with respect to a singular perturbation parameter. In the present paper, we extend most of these results to a conserved phase-field system whose singular limit is the nonviscous Cahn–Hilliard equation. These equations describe phase transition processes. Here, we give a direct proof of the existence of inertial manifolds that differs from our previous method that was based on introducing a change of variables and an auxiliary problem. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
141. Long-time behavior of a class of thermoelastic plates with nonlinear strain.
- Author
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Fatori, L.H., Jorge Silva, M.A., Ma, T.F., and Yang, Zhijian
- Subjects
- *
LAPLACIAN operator , *THERMOELASTICITY , *NONLINEAR analysis , *STRAINS & stresses (Mechanics) , *EXPONENTIAL stability - Abstract
In recent years a class of vibrating plates with nonlinear strain of p -Laplacian type was studied by several authors. The present paper contains a first thermoelastic model of that class of problems including both Fourier and non-Fourier heat laws. Our main result establishes the existence of global and exponential attractors for the strongly damped problem through a stabilizability inequality. In addition, for the weakly damped problem, we establish the exponential stability of its Galerkin semiflows. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
142. About the Structure of Attractors for a Nonlocal Chafee-Infante Problem
- Author
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Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, Valero, José, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Caballero, Rubén, Carvalho, Alexandre N., Marín Rubio, Pedro, and Valero, José
- Abstract
In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.
- Published
- 2021
143. Global attractors for a class of degenerate diffusion equations
- Author
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Shingo Takeuchi and Tomomi Yokota
- Subjects
Global attractors ,p-Laplacian ,degenerate diffusion. ,Mathematics ,QA1-939 - Abstract
In this paper we give two existence results for a class of degenerate diffusion equations with p-Laplacian. One is on a unique global strong solution, and the other is on a global attractor. It is also shown that the global attractor coincides with the unstable set of the set of all stationary solutions. As a by-product, an a-priori estimate for solutions of the corresponding elliptic equations is obtained.
- Published
- 2003
144. Global attractors of non-autonomous quasi-homogeneous dynamical systems
- Author
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David N. Cheban
- Subjects
non-autonomous quasi-homogeneous systems ,global attractors ,Mathematics ,QA1-939 - Abstract
It is shown that non-autonomous quasi-homogeneous dynamical systems admit a compact global attractor. The general results obtained here are applied to differential equations both in finite dimensional spaces and in infinite dimensional spaces, such as ordinary differential equations in Banach space and some types of evolutional partial differential equations.
- Published
- 2002
145. Uniform exponential stability of linear periodic systems in a Banach space
- Author
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David N. Cheban
- Subjects
non-autonomous linear dynamical systems ,global attractors ,periodic systems ,exponential stability ,asymptotically compact systems ,equations on Banach spaces. ,Mathematics ,QA1-939 - Abstract
This article is devoted to the study of linear periodic dynamical systems, possessing the property of uniform exponential stability. It is proved that if the Cauchy operator of these systems possesses a certain compactness property, then the asymptotic stability implies the uniform exponential stability. We also show applications to different classes of linear evolution equations, such as ordinary linear differential equations in the space of Banach, retarded and neutral functional differential equations, some classes of evolution partial differential equations.
- Published
- 2001
146. Uniform exponential stability of linear almost periodic systems in Banach spaces
- Author
-
David N. Cheban
- Subjects
non-autonomous linear dynamical systems ,global attractors ,almost periodic system ,exponential stability ,asymptotically compact systems. ,Mathematics ,QA1-939 - Abstract
This article is devoted to the study linear non-autonomous dynamical systems possessing the property of uniform exponential stability. We prove that if the Cauchy operator of these systems possesses a certain compactness property, then the uniform asymptotic stability implies the uniform exponential stability. For recurrent (almost periodic) systems this result is precised. We also show application for different classes of linear evolution equations: ordinary linear differential equations in a Banach space, retarded and neutral functional differential equations, and some classes of evolution partial differential equations.
- Published
- 2000
147. Existence of smooth global attractors for nonlinear viscoelastic equations with memory.
- Author
-
Conti, Monica and Geredeli, Pelin
- Abstract
We consider the memory relaxation of an Euler-Bernoulli plate equation with nonlinear source term and internal frictional damping of arbitrary polynomial growth. The main focus is the existence of a smooth global attractor for the associated dynamical system. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
148. Cahn-Hilliard equation with nonlocal singular free energies.
- Author
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Abels, Helmut, Bosia, Stefano, and Grasselli, Maurizio
- Abstract
We consider a Cahn-Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is characterized by a Helmholtz free energy density, which can be of logarithmic type. Moreover, the spatial interactions between the different phases are modeled by a singular kernel. As a consequence, the chemical potential $$\mu $$ contains an integral operator acting on the concentration difference $$c$$ , instead of the usual Laplace operator. We analyze the equation on a bounded domain subject to no-flux boundary condition for $$\mu $$ and by assuming constant mobility. We first establish the existence and uniqueness of a weak solution and some regularity properties. These results allow us to define a dissipative dynamical system on a suitable phase-space, and we prove that such a system has a (connected) global attractor. Finally, we show that a Neumann-like boundary condition can be recovered for $$c$$ , provided that it is supposed to be regular enough. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
149. CONVECTIVE NONLOCAL CAHN-HILLIARD EQUATIONS WITH REACTION TERMS.
- Author
-
PORTA, FRANCESCO DELLA and GRASSELLI, MAURIZIO
- Subjects
CAHN-Hilliard-Cook equation ,NUMERICAL analysis ,MATHEMATICAL equivalence ,VON Neumann algebras - Abstract
We introduce and analyze the nonlocal variants of two Cahn- Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
150. On the Existence of Regular Global Attractor for $$p$$ -Laplacian Evolution Equation.
- Author
-
Geredeli, Pelin
- Subjects
- *
EXISTENCE theorems , *ATTRACTORS (Mathematics) , *LAPLACIAN operator , *NONLINEAR evolution equations , *PARABOLIC differential equations , *PROOF theory - Abstract
In this study, we consider the nonlinear evolution equation of parabolic type We analyze the long time dynamics (in the sense of global attractors) under very general conditions on the nonlinearity $$f$$ . Since we do not assume any polynomial growth condition on it, the main difficulty arises at first in the proof of well-posedness. In fact, the very first contribution to this problem is a pioneering paper (Efendiev and Ôtano, Differ Int Equ 20:1201-1209, ) where the well-posedness result has been shown by exploiting the technique from the theory of maximal monotone operators. However, from some physical aspects, to obtain the solution in variational sense might be demanding which requires limiting procedure on the approximate solutions. In this work, we are interested in variational (weak) solution. The critical issue in the proof of well-posedness is to deal with the limiting procedure on $$f$$ which is overcome utilizing the weak convergence tecniques in Orlicz spaces (Geredeli and Khanmamedov, Commun Pure Appl Anal 12:735-754, ; Krasnosel'skiĭ and Rutickiĭ, Convex functions and Orlicz spaces, ). Then, proving the existence of the global attractors in $$L^2(\Omega )$$ and in more regular space $$W_0^{1,p}(\Omega )$$ , we show that they coincide. In addition, if $$f$$ is monotone and $$g=0$$ , we give an explicit estimate of the decay rate to zero of the solution. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
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