We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem ut + [ϕ... more We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem ut + [ϕ(u)] x = 0 in R × (0, T) u = u 0 ≥ 0 in R × {0} , where u 0 a positive Radon measure whose singular part is a finite superposition of Dirac masses, and ϕ ∈ C 2 ([0, ∞)) is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.
Transactions of the American Mathematical Society, Oct 1, 1987
Existence of stationary states is established by means of the method of upper and lower solutions... more Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.
We address the vanishing viscosity limit of the regularized problem studied in Smarrazzo and Tese... more We address the vanishing viscosity limit of the regularized problem studied in Smarrazzo and Tesei [Arch Rat Mech Anal 2012 (in press)]. We show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure-valued solutions of the original problem. In general, these solutions are the sum of a regular term, which is absolutely continuous with respect to the Lebesgue measure, and a singular term, which is a Radon measure singular with respect to the other. By using a family of entropy inequalities, we prove that the singular term is nondecreasing in time. We also characterize the disintegration of the Young measure associated with the regular term, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity ϕ.
Transactions of the American Mathematical Society, Nov 18, 1999
A rather complete study of the existence and qualitative behaviour of the boundaries of the suppo... more A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.
Starting from field investigations and experiments on mimetic butterfly populations a model for t... more Starting from field investigations and experiments on mimetic butterfly populations a model for two mimetic species is developed. The model comprises various features such as the growth rates and carrying capacities of the two species, their unpalatability to predators, the recruitment and the training of the predators and, most important, the similarity of the two mimetic species. The model ranges from pure Batesian to pure Miillerian mimicry over a spectrum of possible cases. The mimetic gain is introduced as the relative increase in equilibrium density in a mimetic situation as compared to a situation where mimicry is not present. The dependence of this quantity on parameters as growth rate, carrying capacity, unpalatability, and similarity is investigated using numerical methods.
We give a necessary and sufficient condition for the validity of the strong maximum principle in ... more We give a necessary and sufficient condition for the validity of the strong maximum principle in one space dimension.
Long-Time Behavior of Solutions to a Class of Forward-Backward Parabolic Equations
Siam Journal on Mathematical Analysis, 2010
We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem f... more We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem for the equation $u_t=[\phi(u)]_{xx}$, where $\phi$ is nonmonotone. These solutions are obtained from the corresponding problem for the regularized equation $u_t=[\phi(u)]_{xx}+\varepsilon u_{xxt}$ ($\varepsilon>0$) by a vanishing viscosity method and satisfy a family of suitable entropy inequalities. Relying on a strong version of these inequalities, we prove exhaustive results concerning the long-time behavior of solutions.
On Some Nonlinear Diffusion Models in Population Dynamics
Lecture notes in biomathematics, 1985
Reaction-diffusion models in population dynamics have been widely investigated in recent years [6... more Reaction-diffusion models in population dynamics have been widely investigated in recent years [6]. In particular, quasilinear models have been recently suggested [11] to deal with situations where diffusivity depends on crowding (see also [7]). In the case of two interacting species such models have the general form $$\begin{array}{*{20}{c}} {{u_t} = {\Delta _y}\left( {u,v} \right) + u\;f\left( {x,u,v} \right)} \\ {{v_t} = {\Delta _j}\left( {u,v} \right) + v\;g\left( {x,u,v} \right)} \\ \end{array}$$ (1) the solution satisfying suitable initial and boundary conditions in a bounded domain (here t is the time variable, x the space variable, Δ the diffusion operator, u=u(t,x), v=v(t,x) are population densities, see [6,7,11]).
Large time behaviour of a diffusion equation with strong convection
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1994
Large time behaviour of a diffusion equation with strong convection S. Claudi R. Natalini A. Tese... more Large time behaviour of a diffusion equation with strong convection S. Claudi R. Natalini A. Tesei 1.-Introduction In this paper we study the large time behaviour of solutions of the equation we always assume q> 1. Equation (1.1) is complemented with initial data and homogeneous Neumann boundary conditions A unique classical solution of problem (1.1)-(1.3) is known to exist, if the initial data are non-negative, bounded and sufficiently smooth (see Section 2). In spite of its simplicity, equation (1.1) allows one to investigate the mutual ...
Sobolev regularization of a class of forward–backward parabolic equations
Journal of Differential Equations, Sep 1, 2014
ABSTRACT We address existence and asymptotic behaviour for large time of Young measure solutions ... more ABSTRACT We address existence and asymptotic behaviour for large time of Young measure solutions of the Dirichlet initial–boundary value problem for the equation ut=∇⋅[φ(∇u)]ut=∇⋅[φ(∇u)], where the function φ need not satisfy monotonicity conditions. Under suitable growth conditions on φ , these solutions are obtained by a “vanishing viscosity” method from solutions of the corresponding problem for the regularized equation ut=∇⋅[φ(∇u)]+ϵΔutut=∇⋅[φ(∇u)]+ϵΔut. The asymptotic behaviour as t→∞t→∞ of Young measure solutions of the original problem is studied by ω -limit set techniques, relying on the tightness of sequences of time translates of the limiting Young measure. When N=1N=1 this measure is characterized as a linear combination of Dirac measures with support on the branches of the graph of φ.
We investigate behavior of the support and decay for large times of nonnegative entropy solutions... more We investigate behavior of the support and decay for large times of nonnegative entropy solutions of the Cauchy problem: a,u + ;dx(u-) =-up in (0,m) X R u = ug in (0) X R. Here m > 1, p > 1, and u0 has compact support. Sharp estimates are given studying the Riemann problem and using comparison results.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Incomplete blowup of solutions of quasilinear hyperbolic balance laws
Archive for Rational Mechanics and Analysis, Nov 1, 1996
We study incomplete blowup of entropy solutions to first-order quasilinear hyperbolic balance law... more We study incomplete blowup of entropy solutions to first-order quasilinear hyperbolic balance laws. A general procedure to continue solutions beyond the blowup time, which makes use of monotonicity methods, is given. The continuations thus obtained are possibly unbounded and satisfy suitable generalized entropy and Rankine-Hugoniot conditions. Then the uniqueness of continuations satisfying such conditions is proved.
On an Evolution Equation Arising in Detonation Theory
ABSTRACT Detonation waves are, for the most part, unstable and it is important to understand the ... more ABSTRACT Detonation waves are, for the most part, unstable and it is important to understand the origins and consequences of the instability. Since activation energy asymptotics is such a valuable tool in small Mach number combustion, it is natural to try it on the detonation problem, a high Mach number phenomenon, and there have been several attempts of this nature. Some of this work will be reviewed here.
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