Corrado M. Mascia- PhD
- Professor (Full) at Sapienza University of Rome
Corrado M. Mascia
- PhD
- Professor (Full) at Sapienza University of Rome
About
93
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1,390
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Introduction
Evolutive partial differential equations.
Propagating fronts.
Stability and instability issues.
Personal web page: http://www1.mat.uniroma1.it/people/mascia
Current institution
Additional affiliations
November 2019 - present
November 2019 - present
November 1993 - October 1997
Publications
Publications (93)
We consider systems of conservation laws derived from coupled fluid-kinetic equations intended to describe particle-laden flows. By means of Chapman–Enskog type expansion, we determine second order corrections and we discuss the existence and stability of shock profiles. Entropy plays a central role in this analysis.
This approach is implemented o...
Two (consecutive) reductions of the complete Gatenby–Gawlinski model for cancer invasion are proposed in order to investigate the mathematical framework, mainly from a computational perspective. After a brief overview of the full model, we proceed by examining the case of a two-equations-based and one-equation-based reduction, both obtained by mean...
Global geodynamics is the result of the long-lasting action of large-scale stress sources with different origin. Plate boundary forces and mantle circulation are usually considered to be dominant in driving plate tectonics. However, these contributions are not sufficient to explain absolute plate dynamics. Plates follow a westward mainstream and th...
Tidal forces are generally neglected in the discussion about the mechanisms driving plate tectonics despite a worldwide geodynamic asymmetry also observed at subduction and rift zones. The tidal drag could theoretically explain the westerly shift of the lithosphere relative to the underlying mantle. Notwithstanding, viscosity in the asthenosphere i...
Starting from coupled fluid-kinetic equations for the modeling of laden flows, we derive relevant viscous corrections to be added to asymptotic hydrodynamic systems, by means of Chapman-Enskog expansions and analyse the shock profile structure for such limiting systems. Our main findings can be summarized as follows. Firstly, we consider simplified...
Tidal forces are generally neglected in the discussion about the mechanisms driving plate tectonics despite a worldwide geodynamic asymmetry also observed at subduction and rift zones. The tidal drag could theoretically explain the westerly shift of the lithosphere relative to the underlying mantle. Notwithstanding, viscosity in the asthenosphere i...
This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form ∂tfi+∑j=1dλij∂xjfi=Qi(f). where fi=fi(x,t) (i=1,⋯,n) and x=(x1,⋯,xd)∈IRd (n≥2,d≥1). Under assumption of the existence of a conserved quantity ∑iαifi for some α1,⋯,αn>0, of (strong) quasimonotonicity and an additional assumption on the spe...
In this paper, we analyse propagating fronts in the context of hyperbolic theories of dissipative processes. These can be considered as a natural alternative to the more classical parabolic models. Emphasis is given toward the numerical computation of the invasion velocity.
Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model ∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and by (U±,V±) its asymptotic states at ±∞, we invest...
Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby--Gawlinski model \[ \partial_t U = U\{f(U)-dV\}, \qquad \partial_t V = \partial_x \{f(U)\,\partial_x V\} + r V f(V), \] where $f(u) = 1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal...
The Gatenby-Gawlinski model for cancer invasion is object of analysis in order to investigate the mathematical framework behind the model working by means of suitable reductions. We perform numerical simulations to study the sharpness/smoothness of the traveling fronts starting from a brief overview about the full model and proceed by examining the...
![The graph of h1(t)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/336968957/figure/fig1/AS:961746038689793@1606309532170/The-graph-of-h1tdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn–Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an approximately invariant manifold\(\mathcal {M}_0\) for such boundary value probl...
In this article, we propose a mathematical model -- based on a cellular automaton -- for the redistribution of patients within a network of hospitals with limited available resources, in order to reduce the risks of a local/global collapse of the healthcare system. We attempt at developing a conceptual tool to support making rational decisions rele...
Motivated by radiation hydrodynamics, we analyse a 2x2 system consisting of a one-dimensional viscous conservation law with strictly convex flux -- the viscous Burgers' equation being a paradigmatic example -- coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the p...
Motivated by radiation hydrodynamics, we analyse a \begin{document}$ 2\times2 $\end{document} system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and...
This paper deals with the numerical approximation of a stick–slip system, known in the literature as Burridge–Knopoff model, proposed as a simplified description of the mechanisms generating earthquakes. Modelling of friction is crucial and we consider here the so-called velocity-weakening form. The aim of the article is twofold. Firstly, we establ...
We introduce a variant of the Gatenby-Gawlinski model for acid-mediated tumor invasion, accounting for anisotropic and heterogeneous diffusion of the lactic acid across the surrounding healthy tissues. Numerical simulations are performed for two-dimensional data by employing finite volume schemes on staggered Cartesian grids, and parallel implement...
The basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-par...
This paper concerns with the motion of the interface for a damped hyperbolic Allen--Cahn equation, in a bounded domain of R^n, for n=2 or n=3. In particular, we focus the attention on radially symmetric solutions and extend to the hyperbolic framework some well-known results of the classic parabolic case: it is shown that, under appropriate assumpt...
Looking at the outbreak of SARS-CoV-2 and the global state of emergency imposed due to its pandemic spread, the necessity for antiviral drugs to be immediately available is a priority for the scientific community. Considering that research and implementation of new antiviral therapies or vaccines usually take a long time, the World Health Organizat...
Interleukin IL-6 is a cytokine produced in response to various types of damage (infections, tissue injuries, autoimmune diseases, ...) whose action contributes to host defense through stimulation of acute phase responses and immune reactions. However, experimental data show that high concentration levels of interleukin IL-6, and the subsequent infl...
In the present paper, a Vortex Particle Method is combined with a Boundary Element Method for the study of viscous incompressible planar flow around solid bodies. The method is based on Chorins operator splitting approach for the Navier–Stokes equations written in vorticity–velocity formulation, and consists of an advection step followed by a diffu...
The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn-Hilliard equation in one space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an "energy approach", already proposed for various evolution PDEs, including the Allen-...
The capability of cells to alter their phenotype in response to signals is crucial to the understanding of different morphogenetic pathways. We focus presently on the case of Epithelial-to-Mesenchynaml Transition (EMT) and its reverse Mesenchymal-to-Epithelial Transition (MET), which are considered as a plausible mechanism at the base of tumours on...
The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn-Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an "approximately invariant manifold" $\mathcal{M}_0$ for such boundary value probl...
Given \(A,\, B\in \mathbb R^{n\times n}\), we consider the Cauchy problem for partially dissipative hyperbolic systems having the form $$\begin{aligned} \partial _{t}u+A\partial _{x}u+Bu=0, \end{aligned}$$with the aim of providing a detailed description of the large-time behavior. Sharp \(L^p\)-\(L^q\) estimates are established for the distance bet...
This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen--Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump proc...
This paper addresses the existence and spectral stability of traveling fronts for nonlinear hyperbolic equations with a positive "damping" term and a reaction function of bistable type. Particular cases of the former include the relaxed Allen-Cahn equation and the nonlinear version of the telegrapher's equation with bistable reaction term. The exis...
This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen--Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump proc...
Consider the Allen-Cahn equation $u_t=\varepsilon^2\Delta u-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$...
In this report, we aim at presenting a viable strategy for the study of Epithelial-Mesenchymal Transition (EMT) and its opposite Mesenchymal-Epithelial Transition (MET) by means of a Systems Biology approach combined with a suitable Mathematical Modeling analysis. Precisely, it is shown how the presence of a metastable state, that is identified at...
We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity \(\varepsilon \). In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:308...
The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn-Hilliard equation in one space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an "energy approach", already proposed for various evolution PDEs, including the Allen-...
Given $A,B\in\mathbb{R}^{n\times n}$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} \partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description of the large-time behavior. We establish sharp $L^p$-$L^q$ estimates for the distance of the solution...
Given $A,B\in M_n(\mathbb R)$, we consider the Cauchy problem for partially dissipative hyperbolic systems having the form \begin{equation*} \partial_{t}u+A\partial_{x}u+Bu=0, \end{equation*} with the aim of providing a detailed description of the large-time behavior. Sharp $L^p$-$L^q$ estimates are established for the distance between the solution...
A Predictor-Corrector strategy is employed for the numerical simulation of the one-dimensional Burridge-Knopoff model of earthquakes. This approach is totally explicit and allows to reproduce the main features of the model. The results achieved are compared with those of several previous works available in the literature, in order to state the effe...
Metastable dynamics of a hyperbolic variation of the Allen-Cahn equation with homogeneous Neumann boundary conditions are considered. Using the "dynamical approach" proposed by Carr-Pego [10] and Fusco-Hale [19] to study slow-evolution of solutions in the classic parabolic case, we prove existence and persistence of metastable patterns for an expon...
Metastable dynamics of a hyperbolic variation of the Allen-Cahn equation with homogeneous Neumann boundary conditions are considered. Using the "dynamical approach" proposed by Carr-Pego [10] and Fusco-Hale [19] to study slow-evolution of solutions in the classic parabolic case, we prove existence and persistence of metastable patterns for an expon...
Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in an half-space w...
Motivated by a vulcanological problem, we establish a sound mathematical approach for surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in an half-space w...
This paper examines a class of linear hyperbolic systems which generalizes the Goldstein-Kac model to an arbitrary finite number of speeds vi with transition rates μij. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the differences vi-vj generate all the space, the system exhibits a large-time behavior desc...
In this paper, we address a simplified version of a problem arising from volcanology.
Specifically, as reduced form of the boundary value problem for the Lamé system,
we consider a Neumann problem for harmonic functions in the half-space with a cavity C.
Zero normal derivative is assumed at the boundary of the half-space; differently, on the bound...
This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu, with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to...
This paper analyzes heat equation with memory in the case of kernels that are
linear combinations of Gamma distributions. In this case, it is possible to
rewrite the non-local equation as a local system of partial differential
equations of hyperbolic type. Stability is studied in details by analyzing the
corresponding dispersion relation, providing...
A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution...
Una chiacchierata (semplice) sul confronto tra modelli diversi per descrivere due tipologie diverse di modelli diffusivi: parabolico ed iperbolico.
The paper examines a class of first order linear hyperbolic systems, proposed
as a generalization of the Goldstein-Kac model for velocity-jump processes and
determined by a finite number of speeds and corresponding transition rates. It
is shown that the large-time behavior is described by a corresponding scalar
diffusive equation of parabolic type,...
The aim article is to contribute to the definition of a versatile
language for metastability in the context of partial differential
equations of evolutive type. A general framework suited for parabolic
equations in one dimensional bounded domains is proposed, based on
choosing a family of approximate steady states, and on the spectral
properties of...
We consider the compressible Navier-Stokes equations for isentropic
dynamics with real viscosity on a bounded interval. In the case of
boundary data defining an admissible shock wave for the corresponding
unviscous hyperbolic system, we determine a scalar differential equation
describing the motion of the internal transition layer. In particular,
f...
![Fig. 2. Regime δ → 2/γ and [u] → −∞. Upper line: the phase plane (x, y)...](profile/Corrado-Mascia/publication/257006653/figure/fig2/AS:764569588883456@1559299003495/Regime-d-2-g-and-u-Upper-line-the-phase-plane-x-y-with-the-graph-of-the_Q320.jpg)
We examine the existence of shock profiles for a hyperbolic–elliptic system arising in radiation hydrodynamics. The algebraic–differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in detail showing the existence of a heteroclinic connection between the two singular points of the system for any dista...
The initial-boundary-value problem for a viscous scalar conservation law in a
bounded interval is considered, with emphasis on metastable dynamics, whereby
the time-dependent solution approaches its steady state in an asymptotically
exponentially long time interval as the viscosity coefficient goes to zero. A
rigorous analysis is used to analyze su...
![Figure 2. Regime δ → 2/γ and [u] → −∞. Upper line: the phase plane (x,...](profile/Corrado-Mascia/publication/221663901/figure/fig2/AS:669291510784013@1536582939309/Regime-d-2-g-and-u-Upper-line-the-phase-plane-x-y-with-the-graph-of-the_Q320.jpg)
We examine the existence of shock profiles for a hyperbolic-elliptic system
arising in radiation hydrodynamics. The algebraic-differential system for the
wave profile is reduced to a standard two-dimensional form that is analyzed in
details showing the existence of heteroclinic connection between the two
singular points of the system for any distan...
We analyze numerically a forward–backward diffusion equation with a cubic-like diffusion function — emerging in the framework of phase transitions modeling — and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second- and the third-or...
In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima ([SK]) coupling
condition. This investigation is motivated by the fact that this condition is not satisfied by various physical systems, and
almost all the time in several space dimensions. First, we explore the role of entropy functionals a...
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4. Vita in silicio e multidisciplinarità
This article deals with the Cauchy problem for a forward–backward parabolic equation, which is of interest in physical and
biological models. Considering such an equation as the singular limit of an appropriate pseudoparabolic third-order regularization,
we consider the framework of entropy solutions, namely weak solutions satisfying an additional...
We consider a general model of chemotaxis with finite speed of propagation in one space dimension. For this model we establish a general result of global stability of some constant states both for the Cauchy problem on the whole real line and for the Neumann problem on a bounded interval. These results are obtained using the linearized operators an...
This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green functi...
Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun [88.
Plaza , R. ,
Zumbrun , K. ( 2004 ). An Evans function approach to spectral stability of small-amplitude...
A hyperbolic system with relaxation is a system of partial differential equations of hyperbolic type with a zero-order term,
describing the relaxation mechanism toward a given equilibrium. After hyperbolic rescaling, the system can be thought as a
dynamical system with two time scales: the fast one is governed mainly by the kinetic part of the syst...
In this chapter we deal with a hyperbolic system of conservation laws with a nonlinear coupling with a scalar elliptic equation, modeling radiation dynamics. In particular, we study existence and regularity of radiative shock waves for that system, that is (possibly discontinuous) traveling wave-type solutions, both in the strictly convex and in th...
The Saint-Venant model, introduced in [A.-J.-C. B. de Saint-Venant, C. R. Acad. Sci. Paris 73, 147–154, 237–240 (1871; JFM 03.0482.04)], gives rise to a simple and, nevertheless, rich hyperbolic system of conservation laws with a zero-order reaction term. Apart from its interest as a model itself, we consider the analysis of the Saint-Venant system...
Travelling fronts for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity " 0 exist for all velocities greater than or equal to an "-dependent minimal value, both in the parabolic case when " > 0 and in the hyperbolic case when " = 0. We prove that as " ! 0, the minimal velocity c " converges to c , the minimal val...
We consider strongly degenerate convection-diffusion equations which mix possible parabolic and hyperbolic behaviour. We prove some qualitative properties of the solutions, in the one-dimensional case. In particular we study the evolution in time of the number of connected components of parabolic and hyperbolic regions and the continuity of the int...
The present paper deals with the following hyperbolic--elliptic coupled system, modelling dynamics of a gas in presence of radiation, $u_{t}+ f(u)_{x} +Lq_{x}=0, -q_{xx} + Rq +G\cdot u_{x}=0,$ where $u\in\R^{n}$, $q\in\R$ and $R>0$, $G$, $L\in\R^{n}$. The flux function $f : \R^n\to\R^n$ is smooth and such that $\nabla f$ has $n$ distinct real eigen...
Building on previous analyses carried out in (24, 27), we establish L1 \ H2 ! Lp nonlinear orbital stability, 1 p 1, with sharp rates of decay, of large-amplitude Lax-type shock proles for a general class of relaxation systems that includes most models in common use, under the necessary conditions of strong spectral stability, i.e., stable point sp...
Combining pointwise Green's function bounds obtained in a companion paper [36] with earlier, spectral stability results obtained in [16], we establish nonlinear orbital stability of small-amplitude Lax-type viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [20], notably including comp...
We establish nonlinear L1H3Lp orbital stability, 2p, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the lineariz...
Following the pointwise semigroup approach of [ZH,MZ.1], we establish sharp pointwise Green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic-parabolic systems of conservation laws of dissipative type, under the necessary assumptions ([Z.1,Z.3,Z.4]) of spectral stability, i.e., stable point spectru...
Under the weak assumption of spectral stability, or stable point spectrum of the linearized operator about the wave, we establish sharp pointwise Green’s function bounds and consequent hnear and nonhnear stability for shock profiles of relaxation and real viscosity systems satisfying the dissipativity condition of Zeng/Kawashima. These include in p...
This paper deals with the singular limit for Lɛu:=ut−Fx(u,ɛux)−ɛ−1g(u)=0, where the function F is assumed to be smooth and uniformly elliptic, and g is a “bistable” nonlinearity. Denoting with um the unstable zero of g, for any initial datum u0 for which u0−um has a finite number of zeroes, and u0−um changes sign crossing each of them, we show the...
The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type$$$$
in the case of nonhomogeneous boundary data a
0. Here u=u(x,t)∈ℝ, with (x,t)∈Q=Ω× (0,T), where Ω is a bounded domain in ℝN
with smooth boundary and T>0. The function b is assumed to be nondecreasing (allowing degeneration zones wher...
Combining pointwise Green's function bounds obtained in a companion paper [MZ.2] with earlier, spectral stability results obtained in [HuZ], we establish nonlinear orbital stability of small amplitude viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [Kaw], notably including compressi...
We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator a...
The purpose of the paper is to analyze large time behavior of solutions to the Dirichlet problem for a scalar hyperbolic balance law under the key assumption that zero characteristic speed never corresponds to an equilibrium state of the underlying ordinary differential equation.
In questa tesi si intende studiare il comportamento qualitativo di soluzioni entropiche per equazioni di reazione-convezione, cioe' per equazioni quasilineari iperboliche della forma
\partial_t u+\partial_x f(u)=g(u),
dove $u=u(x,t)\in\R$ e $(x,t)\in\R\times (0,+\infty)$.
In particolare l'attenzione, per la maggior parte del lavoro, e' rivolta allo...
The aim of the paper is to study qualitative properties of entropy solutions to the Cauchy problem for the following reaction-convection equation with nonconvex flux f In particular we prove that under suitable assumptions the solution is continuous for large t.New resultsconcerning the asymptotic behavior of solutions are also given, proving conve...
. The aim of the paper is to study qualitative behavior of solutions to equation @u @t + @f(u) @x = g(u); where (x; t) 2 R Theta R+ , u = u(x; t) 2 R. The main new feature with respect to previous works is that the flux function f can have finitely many inflections, intervals in which it is affine, and corner points. The function g is supposed to b...
. We study the asymptotic behaviour of the bounded solutions of a hyperbolic conservation law with source term @ t u(x; t) + @xf(u(x; t)) = g(u(x; t)); x 2 IR; t 0: where the flux f is convex and the source term g has simple zeros. We assume that the initial value coincides outside a compact set with an initial value of Riemann type. We prove that...
. We investigate the existence and the asymptotic stability of travelling wave solutionsfor a hyperbolic 2 \Theta 2 system with a relaxation source term. Using the subcharacteristic stabilitycondition, which implies special monotonicity properties of the solutions, we are able to establishthe L1asymptotic attractivity of these solutions.Key words a...
. We study entropy travelling wave solutions for first order hyperbolic balance laws. Results concerning existence, regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros. 1. Introduction It is the purpose of this paper to study existence, regularity and stability of travelling...
The aim of the present manuscript is to present a result on global existence and asymptotic stability for traveling wave solutions to a special 2x2 hyperbolic system with relaxation. The paper is written in italian. Complete results, presented in english, are contained in Mascia C., Natalini R., L^1 nonlinear stability of traveling waves for a hype...
In this tutorial, we attempt to furnish a basic introduction on shallow water modelling with specific attention to Saint-Venant equations. We propose a selection of results, including derivation of the model, well-posedness of the Cauchy problem, existence and stability of roll-waves, kinetic formulation and the corresponding hydrodynamical limit,...
We review some recent work concerning an ill-posed forward-backward parabolic equation, which arises, e.g., in the theory of phase transitions. Some new results are also presented, concerning local existence and uniqueness of solutions within a certain class of physical interest, and a hint of their proofs is given.
Questions
Questions (2)
I am interested in general approach/specific examples where Floquet multipliers for linear system of ode with periodic coefficients are determined, at least approximately, by perturbation arguments. Specifically, I have in mind a linear system, parametrized by epsilon, such that: 1. for epsilon=0 the system is defined by a constant-coefficient diagonalizable matrix; 2. for epsilon>0, the system has periodic coefficients. How may I "follow" the motion of the multiplier as epsilon increases (at least for small epsilon). More than that: what may happen in case of singular perturbations?
How would you call the set generated by the linear combination with integer coefficients of a finite set of vectors in R^n? Apparently, a "lattice" requires that the vectors are linearly independent. I am looking for the "correct" name when the latter condition is not required. As an example, how would you call the set in R given by {a*sqrt{2}+b*\pi: a,b integers}?
