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Marta Strani

Publications and source records attributed to Marta Strani.

At least 19 recordsLinked to original sources
arXiv2025

Stability properties of solutions to convection-reaction equations with nonlinear diffusion

In this paper we study a convection-reaction-diffusion equation of the form \begin{equation*} u_t=\varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \quad t>0, \end{equation*} with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $\varepsilon>0$ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $\varepsilon$) time in a transition non-stable phase, before converging to a stable configuration.

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arXiv2023

Layered patterns in reaction-diffusion models with Perona-Malik diffusions

In this paper we deal with a reaction-diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona-Malik's type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria $\pm1$, described by a parameter $\theta>1$. If $\theta\in(1,2)$, we prove existence of steady states oscillating (and touching) $\pm1$, called $compactons$, while in the case $\theta=2$ we prove the presence of $metastable$ $solutions$, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for $\theta>2$, solutions with an unstable transition layer structure persist only for an algebraically long time.

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arXiv2022

On a generalized Cahn-Hilliard model with p-Laplacian

A generalized Cahn-Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label "generalized" refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p>1$ and a double well potential of the form $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$, with $\theta>1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F"(\pm1)>0$, which are typical of the standard Cahn-Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $\theta\geq p>1$. In the $critical$ $\theta=p>1$, we prove $exponentially$ $slow$ $motion$ of profiles with a transition layer structure, thus extending the well know results of the standard model, where $\theta=p=2$; conversely, in the $supercritical$ case $\theta>p>1$, we prove $algebraic$ $slow$ $motion$ of layered profiles.

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arXiv2021

Dispersive regularization for phase transitions

We introduce a dispersive regularization of the compressible Euler equations in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive regularization is of Schroedinger type. In particular, the regularized system is complex-valued. It has a conservation law, which, for real unknowns, is identical to the energy of the unregularized physical system. The regularized system supports high-frequency solutions, with an existence time or an amplitude which depend strongly on the pressure law.

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arXiv2020

Stability properties of the steady state for the isentropic compressible Navier-Stokes equations with density dependent viscosity in bounded intervals

We prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequent investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.

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arXiv2020

Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms

This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$ and $p>1$ are constants, driven by the action of a family of double-well potentials of the form \[ F(u)=\frac{1}{2n} |1-u^2|^{n}, \] indexed by $n>1$, $n\in\mathbb{R}$ with minima at two pure phases $u = \pm 1$. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$, persist for an exponentially long time in the critical case with $n=p$, and for an algebraically long time in the supercritical (or degenerate) case with $n > p$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are established. In contrast, in the subcritical case with $n<p$, the transition layer solutions are stationary.

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arXiv2019

Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an exponentially long time, i.e. a time proportional to $\exp(C/\e)$ where $C,\e$ are strictly positive constants and $\e^2$ is the diffusion coefficient. In this paper, we extend such results to the case when the linear diffusion flux is substituted by the mean curvature operator both in Euclidean and Lorentz--Minkowski spaces. More precisely, for both models, we prove existence of metastable states which maintain a transition layer structure for an exponentially long time and we show that the speed of the layers is exponentially small. Numerical simulations, which confirm the analytical results, are also provided.

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arXiv2019

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz-Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as $t\to+\infty$. Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.

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arXiv2018

Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

In this paper we study the long time dynamics of the solutions to the initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of 'metastability', whereby the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.

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arXiv2018

A gradient flow approach to relaxation rates for the multi-dimensional Cahn-Hilliard equation

The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy, the dissipation, and the squared $H^{--1}$ distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first fast phase of the order $t^{ -- 1/2}$ where one sees convergence to some kink, followed by a slow relaxation phase with rate $t^{-- 1/ 4}$ where convergence to the centered kink is observed.

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arXiv2017

Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations

We study the existence of monotone heteroclinic traveling waves for the $1$-dimensional reaction-diffusion equation $$ u_t = (| u_x |^{p-2} u_x + | u_x |^{q-2} u_x)_x + f(u), $$ where the non-homogeneous operator appearing on the right-hand side is known as $(p, q)$-Laplacian. Here we assume that $2 \leq q < p$ and $f$ is a nonlinearity of Fisher type, namely it is always positive out of its zeros. We give an estimate of the critical speed and we comment on the roles of $p$ and $q$ in the dynamics, providing some numerical simulations.

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arXiv2017

Heteroclinic traveling fronts for a generalized Fisher-Burgers equation with saturating diffusion

We study the existence of monotone heteroclinic traveling waves for a general Fisher-Burgers equation with nonlinear and possibly density-dependent diffusion. Such a model arises, for instance, in physical phenomena where a saturation effect appears for large values of the gradient. We give an estimate for the critical speed (namely, the first speed for which a monotone heteroclinic traveling wave exists) for some different shapes of the reaction term, and we analyze its dependence on a small real parameter when this brakes the diffusion, complementing our study with some numerical simulations.

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arXiv2016

Nonlinear metastability for a parabolic system of reaction-diffusion equations

We consider a system of reaction-diffusion equations in a bounded interval of the real line, with emphasis on the metastable dynamics, whereby the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval as the viscosity coefficient $\varepsilon>0$ goes to zero. To rigorous describe such behavior, we analyze the dynamics of solutions in a neighborhood of a one-parameter family of approximate steady states, and we derive an ODE for the position of the internal interfaces.

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arXiv2016

Slow dynamics in reaction-diffusion systems

We consider a system of reaction-diffusion equations in a bounded interval of the real line, with emphasis on the metastable dynamics, whereby the time-dependent solution approaches the steady state in an asymptotically exponentially long time interval as the viscosity coefficient $\varepsilon>0$ goes to zero. To rigorous describe such behavior, we analyze the dynamics of layered solutions localized far from the stable configurations of the system, and we derive an ODE for the position of the internal interfaces.

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arXiv2015

Asymptotic behavior of interface solutions to quasilinear parabolic equations with nonlinear forcing terms

We investigate the asymptotic behavior of solutions for quasilinear parabolic equations in bounded intervals. In particular, we are concerned with a special class of solutions, called interface solutions, which exhibit e metastable behavior, meaning that their convergence towards the asymptotic configuration of the system is exponentially slow. The key of our analysis is a linearization around an approximation of the steady state of the problem, and the reduction of the dynamics to a one-dimensional motion, describing the slow convergence of the interfaces towards the equilibrium.

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arXiv2015

Time-delayed instabilities in complex Burgers equations

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, Amer.~J.~Math.~2010] proved that only analytical data generate local $C^2$ solutions. The corresponding instabilities are however not observed numerically; rather, numerical simulations show an exponential growth only after a delay in time. We argue that numerical diffusion is responsible for this time delay, as we prove that for Burgers equations in the torus with small viscosity and a complex forcing, oscillating data generate solutions which grow linearly in time before growing exponentially. Numerical simulations illustrate the results.

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arXiv2015

Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation

We study a one dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation \begin{equation*} \partial_t u = \varepsilon \partial_x^2 u -\partial_x f(u)+ f'(u). \end{equation*} A metastable behavior appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequent approaches its steady state in a very long time interval. A rigorous analysis is used to study such behavior, by means of the construction of a one-parameter family $\{ U^\varepsilon(x;ξ)\}_ξ$ of approximate stationary solutions and of a linearization of the original system around an element of this family. We obtain a system consisting in an ODE for the parameter $ξ$, describing the position of the interface, coupled with a PDE for the perturbation $v$, defined as the difference $v:=u-U^\varepsilon$. The key of our analysis are the spectral properties of the linearized operator around an element of the family $\{ U^\varepsilon \}$: the presence of a first eigenvalue, small with respect to $\varepsilon$, leads to a metastable behavior when $\varepsilon \ll 1$.

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