Search

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single.

In recent years, it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences. In social sciences, the appearance of lognormal distribution has been noticed, among others, when looking at body weight, and at women’s age at first marriage. Likewise, in economics, lognormal distribution appears when looking at consumption in a western society, at call-center service times, and others. The common feature of these situations, which describe the distribution of a certain people’s hallmark, is the presence of a desired target to be reached by repeated choices. In this paper, we discuss a possible explanation of lognormal distribution forming in human activities by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic variation of the hallmark around its target value, leading to a linear Fokker–Planck-type equation with lognormal equilibrium density, is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

The main goal in this paper is to study asymptotic behavior in L p ( R N ) for the solutions of the fractional version of the discrete in time N-dimensional diffusion equation, which involves the Caputo fractional h-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.

We study the decay/growth rates in all $L^p$ norms of solutions to an inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power $\beta$ of the Laplacian when the dimension is large, $N> 4\beta$. Rates depend strongly on the space-time scale and on the time behavior of the spatial $L^1$ norm of the forcing term., Comment: 16 pages, 1 figure

We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle approximations with piecewise constant densities introduced in~\cite{Di Francesco-Rosini} starting from general bounded initial densities converge strongly in $L^1$ to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded domains and for arbitrary nonnegative bounded initial densities, thus extending the results in \cite{Gosse-Toscani, Matthes-Osberger, Mathes-Soellner} (in which a no-vacuum condition is required) and giving an alternative approach to \cite{Carrillo-Craig-Patacchini} in the one-dimensional case, including also subquadratic and superquadratic diffusions., 31 pages

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker--Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincar\'e and logarithmic Sobolev type inequalities for this class of densities.

International audience; This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.

We treat 2D and 3D tumor invasion models with quasi-variational structures, which are composed of two PDEs, one ODE and certain constraint conditions. Although the original model was proposed by M. R. A. Chaplain and A. R. A. Anderson in 2003, the difference between their original model and ours is that the constraint conditions for the distributions of tumor cells and the extracellular matrix are imposed in our model, which give a quasi-variational structure. For 2D and 3D tumor invasion models with quasi-variational structures, we show the existence of global-in-time solutions and consider their large-time behaviors. Especially, for the large-time behaviors, we show that there exists at least one global-in-time solution such that it converges to a constant steady state in an appropriate function space as time goes to $\infty$.

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 to the system and a time-asymptotic profile, where the profile is the superposition of diffusion waves and exponentially decaying waves., Refined version

We prove the large time behavior of solutions to a coupled thermo-diffusion arising in the modelling of the motion of hot colloidal particles in porous media. Additionally, we also ensure the uniqueness of solutions of the target problem. The main mathematical difficulty is due to the presence in the right-hand side of the equations of products between temperature and concentration gradients. Such terms mimic the so-called thermodynamic Soret and Dufour effects. These are cross-coupling terms emphasizing in this context a strong interplay between heat conduction and molecular diffusion., Comment: 19 pages

We study large-time asymptotics for a class of noncoercive Hamilton–Jacobi equations with Dirichlet boundary condition in one space dimension. We prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset. We show that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. This work is an extension with more rigorous analysis of a recent paper by E. Yokoyama, Y. Giga and P. Rybka, in which a growing crystal model is established and the asymptotic behavior described above is first discovered.

The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray–Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again.

We present a collision potential for the Vlasov–Poisson–Boltzmann system near vacuum in plasma physics case. This potential measures the future possible collisions between charged particles with different velocities and satisfies a time-decay estimate. We use this time-decay property of the functional to show that the dynamics of the Vlasov–Poisson–Boltzmann is time-asymptotically equivalent to that of the corresponding linear Vlasov equation, when initial datum is small and decays fast enough in phase space as in [Y. Guo, The Vlasov–Poisson–Boltzmann system near vacuum, Comm. Math. Phys. 218 (2001) 293–313].

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented., Comment: Accepted for publication in ESAIM: Mathematical Modelling and Numerical Analysis

We are concerned with the large-time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws. We present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions. These approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come. The relation between the large-time behavior of entropy solutions and the uniqueness of Riemann solutions leads to an extensive study of the uniqueness problem. We use a direct method to show the large-time behavior of large L ∞ solutions for a class of m × m systems including a model in multicomponent chromatography; we employ the uniqueness of Riemann solutions and the convergence of self-similar scaling sequence of solutions to show the asymptotic behavior of large BV solutions for the 3 × 3 system of Euler equations in thermoelasticity. These results indicate that the Riemann solution is the unique attractor of large discontinuous entropy solutions, whose initial data are L ∞ ∩ L 1 or BV ∩ L 1 perturbation of the Riemann data, for these systems. These approaches also work for proving the large-time behavior of approximate solutions to hyperbolic conservation laws.

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton–Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (Commun. Partial. Differ. Equ. 24(5–6):883–893, 1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given.

International audience; In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the ''weak KAM approach'' which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.

International audience; The aim of the paper is to study qualitative behavior of solutions to a hyperbolic balance law. The main new feature with respect to previous works is that the flux function may have finitely many inflection points, intervals in which it is affine, and corner points. The reaction function is supposed to be zero at 0 and 1, and positive in between. We prove existence of heteroclinic travelling waves connecting the two constant states for opportune choice of speeds. Finally we analyze the large-time behavior of the Riemann problem with values 0 and 1, showing convergence to one of the travelling waves. The speed of the limiting profile is explicitly characterized.

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in Lp and the velocity u converges to the steady state velocity in W1,p for any 1⩽p

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic magnetohydrodynamic flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L 2 , and that the initial density and the gradients of initial velocity/magnetic field are bounded in L ∞ and L 2 , respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.

In this paper, assuming suitable hypotheses on the transport coefficients, we prove the large-time behavior, as time tends to infinity, of solutions in H i = H i × H 0 i × H i × H i + 1 ( i = 1 , 2 ) for the one-dimensional infrarelativistic model of a compressible viscous gas with radiation.

International audience; We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.