CPAA
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
DCDS
On the regularity of global attractors
Monica Conti Vittorino Pata
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1209-1217 doi: 10.3934/dcds.2009.25.1209
This note is focused on a novel technique to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application, we consider the semigroup generated by the strongly damped wave equation with critical nonlinearity, whose attractor is shown to possess the optimal regularity.
keywords: regularity global attractor absorbing set semigroup strongly damped wave equation. Solution operator
DCDS
A minimal approach to the theory of global attractors
Vladimir V. Chepyzhov Monica Conti Vittorino Pata
Discrete & Continuous Dynamical Systems - A 2012, 32(6): 2079-2088 doi: 10.3934/dcds.2012.32.2079
For a semigroup $S(t):X\to X$ acting on a metric space $(X,d)$, we give a notion of global attractor based only on the minimality with respect to the attraction property. Such an attractor is shown to be invariant whenever $S(t)$ is asymptotically closed. As a byproduct, we generalize earlier results on the existence of global attractors in the classical sense.
keywords: absorbing and attracting sets Semigroups global attractors invariant sets.
DCDS-S
Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions
Monica Conti Stefania Gatti Alain Miranville
Discrete & Continuous Dynamical Systems - S 2012, 5(3): 485-505 doi: 10.3934/dcdss.2012.5.485
This paper deals with the longtime behavior of the Caginalp phase-field system with coupled dynamic boundary conditions on both state variables. We prove that the system generates a dissipative semigroup in a suitable phase-space and possesses the finite-dimensional smooth global attractor and an exponential attractor.
keywords: Caginalp system exponential attractor. phase transition global attractor dynamic boundary conditions
PROC
Singular limit of dissipative hyperbolic equations with memory
Monica Conti Vittorino Pata M. Squassina
Conference Publications 2005, 2005(Special): 200-208 doi: 10.3934/proc.2005.2005.200
We consider a class of weakly damped semilinear hyperbolic equations with memory, expressed by a convolution integral. We study the passage to the singular limit when the memory kernel collapses into the Dirac mass at zero, and we establish a convergence result for a proper family of exponential attractors.
keywords: exponential attractors. Hyperbolic equations with memory singular limit
DCDS
A singular cahn-hilliard-oono phase-field system with hereditary memory
Monica Conti Stefania Gatti Alain Miranville
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 3033-3054 doi: 10.3934/dcds.2018132

We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.

keywords: Phase-field models Cahn-Hilliard and Oono equations memory effects singular potential strict separation property
DCDS-B
Exponential stability for a class of linear hyperbolic equations with hereditary memory
Monica Conti Elsa M. Marchini Vittorino Pata
Discrete & Continuous Dynamical Systems - B 2013, 18(6): 1555-1565 doi: 10.3934/dcdsb.2013.18.1555
We establish a necessary and sufficient condition of exponential stability for the contraction semigroup generated by an abstract version of the linear differential equation $$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0 $$ modeling hereditary heat conduction of Gurtin-Pipkin type.
keywords: semigroups of linear contractions exponential stability. memory kernels Hereditary heat conduction
DCDS
Semilinear wave equations of viscoelasticity in the minimal state framework
Monica Conti Elsa M. Marchini Vittorino Pata
Discrete & Continuous Dynamical Systems - A 2010, 27(4): 1535-1552 doi: 10.3934/dcds.2010.27.1535
A semilinear integrodifferential equation of hyperbolic type is studied, where the dissipation is entirely contributed by the convolution term accounting for the past history of the variable. Within a novel abstract framework, based on the notion of minimal state, the existence of a regular global attractor is proved.
keywords: Hyperbolic equation with memory minimal state global attractor. viscoelasticity
CPAA
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
CPAA
Totally dissipative dynamical processes and their uniform global attractors
Vladimir V. Chepyzhov Monica Conti Vittorino Pata
Communications on Pure & Applied Analysis 2014, 13(5): 1989-2004 doi: 10.3934/cpaa.2014.13.1989
We discuss the existence of the global attractor for a family of processes $U_\sigma(t,\tau)$ acting on a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$. Such an attractor is uniform with respect to $\sigma\in\Sigma$, as well as with respect to the choice of the initial time $\tau\in R$. The existence of the attractor is established for totally dissipative processes without any continuity assumption. When the process satisfies some additional (but rather mild) continuity-like hypotheses, a characterization of the attractor is given.
keywords: absorbing and attracting sets Dynamical processes uniform global attractors.

Year of publication

Related Authors

Related Keywords

[Back to Top]