Asymptotic behavior of a parabolic-hyperbolic system
M. Grasselli V. Pata
Communications on Pure & Applied Analysis 2004, 3(4): 849-881 doi: 10.3934/cpaa.2004.3.849
We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.
keywords: damped semilinear wave equation. absorbing sets Phase-field models exponential attractors upper semi- continuity universal attractors
A reaction-diffusion equation with memory
M. Grasselli V. Pata
Discrete & Continuous Dynamical Systems - A 2006, 15(4): 1079-1088 doi: 10.3934/dcds.2006.15.1079
We consider a one-dimensional reaction-diffusion type equation with memory, originally proposed by W.E. Olmstead et al. to model the velocity $u$ of certain viscoelastic fluids. More precisely, the usual diffusion term $u_{x x}$ is replaced by a convolution integral of the form $\int_0^\infty k(s) u_{x x}(t-s)ds$, whereas the reaction term is the derivative of a double-well potential. We first reformulate the equation, endowed with homogeneous Dirichlet boundary conditions, by introducing the integrated past history of $u$. Then we replace $k$ with a time-rescaled kernel $k_\varepsilon$, where $\varepsilon>0$ is the relaxation time. The obtained initial and boundary value problem generates a strongly continuous semigroup $S_\varepsilon(t)$ on a suitable phase-space. The main result of this work is the existence of the global attractor for $S_\varepsilon(t)$, provided that $\varepsilon$ is small enough.
keywords: Reaction-diffusion equations global attractors. memory effects
Weakly dissipative semilinear equations of viscoelasticity
Monica Conti V. Pata
Communications on Pure & Applied Analysis 2005, 4(4): 705-720 doi: 10.3934/cpaa.2005.4.705
We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity, in which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. Setting the equation in the history space framework, we prove the existence of a regular global attractor.
keywords: dynamical systems gradient systems Lyapunov functionals global attractors Hyperbolic equations with memory
A result on the existence of global attractors for semigroups of closed operators
V. Pata Sergey Zelik
Communications on Pure & Applied Analysis 2007, 6(2): 481-486 doi: 10.3934/cpaa.2007.6.481
In this note, we establish a general result on the existence of global attractors for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that $S(t)$ be a closed map.
keywords: global attractors Semigroups of operators connected attractors. closed operators abstract Cauchy problems
Robust exponential attractors for a family of nonconserved phase-field systems with memory
S. Gatti M. Grasselli V. Pata M. Squassina
Discrete & Continuous Dynamical Systems - A 2005, 12(5): 1019-1029 doi: 10.3934/dcds.2005.12.1019
We consider a family of phase-field systems with memory effects in the temperature $\vartheta$, depending on a parameter $\omega\geq 0$. Setting the problems in a suitable phase-space accounting for the past history of $\vartheta$, we prove the existence of a family of exponential attractors $\mathcal E_\omega$ which is robust as $\omega\to 0$.
keywords: absorbing sets memory effects robust exponential attractors. strongly continuous semigroups Phase-field models
A remark on the damped wave equation
V. Pata Sergey Zelik
Communications on Pure & Applied Analysis 2006, 5(3): 611-616 doi: 10.3934/cpaa.2006.5.611
In this short note we present a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation with a nonlinearity of critical growth.
keywords: critical nonlinearity global attractor. Damped wave equation
Global attractors for nonlinear viscoelastic equations with memory
Monica Conti Elsa M. Marchini V. Pata
Communications on Pure & Applied Analysis 2016, 15(5): 1893-1913 doi: 10.3934/cpaa.2016021
We study the asymptotic properties of the semigroup $S(t)$ arising from the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain \begin{eqnarray} |\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\ -\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u +\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h \end{eqnarray} written in the past history framework of Dafermos [10]. We establish the existence of the global attractor of optimal regularity for $S(t)$ when $\rho\in [0,4)$ and $f$ has polynomial growth of (at most) critical order 5.
keywords: solution semigroup global attractor. memory kernel Nonlinear viscoelastic equations

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