arXiv subjects

Monica Conti

Publications and source records attributed to Monica Conti.

At least 19 recordsLinked to original sources

arXiv2026

Global weak solutions to the Cahn-Hilliard equation with degenerate mobility and singular diffusion

We study the initial-boundary value problem for the Cahn-Hilliard equation with degenerate mobility and singular diffusion at pure phases. This model describes the dynamics of phase separation in polymer blends with associated Flory-Huggins-de Gennes free energy. We prove the existence of suitable global weak solutions in three-dimensional bounded, smooth, and convex domains, assuming that the initial datum has finite energy. A key novelty of our analysis is the derivation of $L^4(0,T; H^2(\Omega))$ estimates for both the solution $u$ and the function $\phi(u)=\arcsin(u)$, obtained without relying on the classical entropy-based approach.

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arXiv2025

On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this work, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions. In two spatial dimensions, we prove uniqueness of weak solutions, their smoothing effect for positive times, and convergence to equilibrium as time tends to infinity. In three dimensions, we show local well-posedness of strong solutions for arbitrary initial data and global existence for data close to energy minimizers, yielding a Lyapunov stability principle. A key ingredient of our analysis is a Lojasiewicz-Simon inequality tailored to the nonlinear diffusion case, which enables us to characterize the longtime dynamics.

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arXiv2024

New results for the Cahn-Hilliard equation with non-degenerate mobility: well-posedness and longtime behavior

We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.

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arXiv2021

A note on the energy transfer in coupled differential systems

We study the energy transfer in the linear system $$ \begin{cases} \ddot u+u+\dot u=b\dot v\\ \ddot v+v-\epsilon \dot v=-b\dot u \end{cases} $$ made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter $b$.

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arXiv2021

On the Moore-Gibson-Thompson equation with memory with nonconvex kernels

We consider the MGT equation with memory $$\partial_{ttt} u + \alpha \partial_{tt} u - \beta \Delta \partial_{t} u - \gamma\Delta u + \int_{0}^{t}g(s) \Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $\alpha\beta>\gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g'$, namely, $$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that $g$ vanishes exponentially fast.

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arXiv2021

The MGT-Fourier model in the supercritical case

We address the energy transfer in the differential system $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u = -\eta \Delta \theta \\ \theta_t - \kappa \Delta \theta =\eta \Delta u_{tt}+ \alpha\eta \Delta u_t \end{cases} $$ made by a Moore-Gibson-Thompson equation in the supercritical regime, hence antidissipative, coupled with the classical heat equation. The asymptotic properties of the related solution semigroup depend on the strength of the coupling, ruling the competition between the Fourier damping and the MGT antidamping. Exponential stability will be shown always to occur, provided that the coupling constant is sufficiently large with respect to the other structural parameters. A fact of general interest will be also discussed, namely, the impossibility of attaining the optimal exponential decay rate of a given dissipative system via energy estimates.

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arXiv2016

Averaging of equations of viscoelasticity with singularly oscillating external forces

Given $ρ\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-ρ}g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,τ)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $ρ<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established.

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arXiv2016

Viscoelasticity with time-dependent memory kernels. Part II: asymptotic behavior of solutions

We continue the analysis on the model equation arising in the theory of viscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]Δu(t) -\int_0^\infty k'_t(s)Δu(t-s) d s + f(u(t)) = g $$ in the presence of a (convex, nonnegative and summable) memory kernel $k_t(\cdot)$ explicitly depending on time. Such a model is apt to describe, for instance, the dynamics of aging viscoelastic materials. The earlier paper [4] was concerned with the correct mathematical setting of the problem, and provided a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Di Plinio {\it et al.}\ [14] In this second work, we focus on the asymptotic properties of the solutions, proving the existence and the regularity of the time-dependent global attractor for the dynamical process generated by the equation. In addition, when $k_t$ approaches a multiple $mδ_0$ of the Dirac mass at zero as $t\to\infty$, we show that the asymptotic dynamics of our problem is close to the one of its formal limit $$\partial_{tt} u(t)-Δu(t) -mΔ\partial_t u(t)+ f(u(t)) = g$$ describing viscoelastic solids of Kelvin-Voigt type.

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arXiv2016

A model of viscoelasticity with time-dependent memory kernels

We consider the model equation arising in the theory of viscoelasticity $$\partial_{tt} u-h_t(0)Δu -\int_{0}^\infty h_t'(s)Δu(t-s)d s+ f(u) = g.$$ Here, the main feature is that the memory kernel $h_t(\cdot)$ depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.

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arXiv2014

On the Cahn-Hilliard-Brinkman system

We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field $ϕ$, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface surface force proportional to $ϕ\nabla μ$, where $μ$ is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman (CHB) system for $(ϕ,\mathbf{u})$. Then we establish the existence of a global attractor and the convergence of a given (weak) solution to a single equilibrium via Łojasiewicz-Simon inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS) system. We first prove the existence of a weak solution to the CHHS system as limit of CHB solutions. Then, in dimension two, we estimate the difference of the solutions to CHB and CHHS systems in terms of the viscosity constant appearing in CHB.

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arXiv2013

Global attractors for nonlinear viscoelastic equations with memory

We study the asymptotic properties of the semigroup S(t) arising from a nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain written in the past history framework of Dafermos. We establish the existence of the global attractor of optimal regularity for S(t) for a wide class of nonlinearities as well as within the most general condition on the memory kernel.

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arXiv2013

Timoshenko systems with fading memory

The decay properties of the semigroup generated by a linear Timoshenko system with fading memory are discussed. Uniform stability is shown to occur within a necessary and sufficient condition on the memory kernel.

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arXiv2013

Totally dissipative dynamical processes and their uniform global attractors

We discuss the existence of the global attractor for a family of processes $U_σ(t,τ)$ acting on a metric space $X$ and depending on a symbol $σ$ belonging to some other metric space $Σ$. Such an attractor is uniform with respect to $σ\inΣ$, as well as with respect to the choice of the initial time $τ\in\R$. 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.

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arXiv2012

Attractors for processes on time-dependent spaces. Applications to wave equations

For a process U(t,s) acting on a one-parameter family of normed spaces, we present a notion of time-dependent attractor based only on the minimality with respect to the pullback attraction property. Such an attractor is shown to be invariant whenever the process is T-closed for some T>0, a much weaker property than continuity (defined in the text). As a byproduct, we generalize the recent theory of attractors in time-dependent spaces developed in [10]. Finally, we exploit the new framework to study the longterm behavior of wave equations with time-dependent speed of propagation.

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arXiv2010

Global minimizers of coexistence for competing species

A class of variational models describing ecological systems of k species competing for the same resources is investigated. The occurrence of coexistence in minimal energy solutions is discussed and positive results are proven for suitably differentiated internal dynamics.

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arXiv2009

On the regularity of global attractors

This note is focused on a novel technique in order 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 of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-Δu_t-Δu+ϕ(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.

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arXiv2008

Minimal coexistence configurations for multispecies systems

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.

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arXiv2007

Coexistence and Segregation for Strongly Competing Species in Special Domains

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species), as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems.

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