Asymptotic Profile Research Articles

Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism

<p style='text-indent:20px;'>In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.</p>

A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{R}_{0} $\end{document}</tex-math></inline-formula> which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.</p>

Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

Abstract This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

A reaction–diffusion SIS epidemic model with saturated incidence rate and logistic source

A reaction–diffusion SIS epidemic model with saturated incidence rate and logistic source for the susceptible individuals is considered. We establish the uniform bounds of parabolic system and investigate the extinction and persistence of infectious diseases in terms of the basic reproduction number. We further analyze the asymptotic profiles of the endemic equilibrium for small and large movement rates and large saturate rate. In particular, it is shown that large saturation may cause the elimination of disease and the logistic source can enhance persistence of infectious disease.

Uniqueness and non-uniqueness of steady states for a diffusive predator-prey-mutualist model with a protection zone

This paper is concerned with the stationary problem for a diffusive Lotka-Volterra predator-prey-mutualist model with a protection zone under homogeneous Neumann boundary conditions. Compared with the case where the mutualist is absent in [12], this paper aims to reveal the effects of mutualism coefficients α and β on the existence, number and stability of steady states. It turns out that when α is large, the model has at most one steady state and it is stable (if it exists); however existence of multiple steady states is examined for large β, moreover asymptotic profiles of steady states are established as β tends to infinity.

Higher-order asymptotic profiles of the solutions to the viscous Fornberg–Whitham equation

We consider the initial value problem for the viscous Fornberg–Whitham equation which is one of the nonlinear and nonlocal dispersive–dissipative equations. In this paper, we establish the global existence of the solutions and study its asymptotic behavior. We show that the solution to this problem converges to the self-similar solution to the Burgers equation called the nonlinear diffusion wave, due to the dissipation effect by the viscosity term. Moreover, we analyze the optimal asymptotic rate to the nonlinear diffusion wave and the detailed structure of the solution by constructing higher-order asymptotic profiles. Also, we investigate how the nonlocal dispersion term affects the asymptotic behavior of the solutions and compare the results with the ones of the KdV–Burgers equation.

Blow-up analysis and spatial asymptotic profiles of solutions to a modified two-component hyperelastic rod system

In this paper, we consider a modified two-component hyperelastic rod system, which is a generalization of the Camassa–Holm equation modeling shallow water waves moving over a linear shear flow. We show blow-up analysis and exact spatial asymptotic profiles of solutions to the system. Especially, for the special case $$\gamma =1$$ , we present the exact blow-up rate of the breaking-wave solution.

Structural analysis of an L-infinity variational problem and relations to distance functions

In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the $L^2$-subdifferential of ${\cal J}$ and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of ${\cal J}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of ${\cal J}$. Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.

Optical solitons in a saturable nonlinear medium in the presence of an asymmetric complex potential

We report on the existence of families of stable spatial solitons in a saturable nonlinear medium characterized by a refractive index with asymmetric distribution of gain and loss. The properties of the nonlinear modes bifurcating from the eigenvalue of the underlying linear problem are thoroughly investigated. The eigenvalue ranges in the power-eigenvalue diagrams for different gain/loss profiles are inspected. We find that the saturable nonlinearity severely restricts these ranges, as the eigenvalues tend to move quite fast to an asymptotic profile, as power increases. Numerical simulations of the wave equations are carried out and examples of the dynamics of the asymmetric solitons obtained exhibit a remarkable agreement with the analytic stability results.

Qualitative analysis on a diffusive SIRS epidemic model with standard incidence infection mechanism

In this paper, we are concerned with an SIRS epidemic reaction–diffusion system with standard incidence infection mechanism in a spatially heterogeneous environment. We first establish the uniform bounds of solutions and then derive the threshold dynamics in terms of the basic reproduction number $$\mathcal {R}_0$$ . Our main focus is on the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible, infected or recovered population is small, and our results show that the disease always exists in the entire habitat in each case of small diffusion rate. This suggests that restricting the diffusion (migration) rate of population is not an effective strategy of disease eradication.

Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effective damping case

We consider the Cauchy problem for the linear beam equation with two variable coefficients:utt+b(t)ut+uxxxx−a(t)uxx=0,(t,x)∈R+×R. The purpose of this study is to classify the behavior of solution with respect to (α,β) when the coefficients a and b behave as a(t)∼(1+t)α and b(t)∼(1+t)β. In this article we shall give the asymptotic profile formula in effective damping cases, by using the method of scaling variables developed by Gallay and Raugel [9].

Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media

Utilizing the similarity between the spinor representation of the Dirac and the Maxwell equations that has been recognized since the early days of relativistic quantum mechanics, a quantum lattice algorithm (QLA) representation of unitary collision-stream operators of Maxwell's equations is derived for both homogeneous and inhomogeneous media. A second-order accurate 4-spinor scheme is developed and tested successfully for two-dimensional (2-D) propagation of a Gaussian pulse in a uniform medium whereas for normal (1-D) incidence of an electromagnetic Gaussian wave packet onto a dielectric interface requires 8-component spinors because of the coupling between the two electromagnetic polarizations. In particular, the well-known phase change, field amplitudes and profile widths are recovered by the QLA asymptotic profiles without the imposition of electromagnetic boundary conditions at the interface. The QLA simulations yield the time-dependent electromagnetic fields as the wave packet enters and straddles the dielectric boundary. QLA involves unitary interleaved non-commuting collision and streaming operators that can be coded onto a quantum computer: the non-commutation being the very reason why one perturbatively recovers the Maxwell equations.

A Quantitative Description of Skyrmions in Ultrathin Ferromagnetic Films and Rigidity of Degree $$\pm \,1$$ Harmonic Maps from $${\mathbb {R}}^2$$ to $${\mathbb {S}}^2$$

We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree $\pm 1$ harmonic maps from the two-dimensional sphere to itself.

Asymptotic profiles of solutions to sixth order Boussinesq-type equations with damping

We investigate the initial value problem for sixth-order Boussinesq-type equations with damping. The decay structure of this equation is of the regularity-loss type, which causes difficulty in proving global solutions in the high frequency region. We establish global existence and time-decay rates of solutions for spaces with dimension n≥2 by introducing the time-weighted norm and a version of the Banach fixed-point theorem. Moreover, the asymptotic profile of global solutions is obtained for n≥3.

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.

Asymptotic profiles for damped plate equations with rotational inertia terms

We consider the Cauchy problem for plate equations with rotational inertia and frictional damping terms. We derive asymptotic profiles of the solution in [Formula: see text]-sense as [Formula: see text] in the case when the initial data have high and low regularity, respectively. Especially, in the low regularity case of the initial data one encounters the regularity-loss structure of the solutions, and the analysis is more delicate. We employ the so-called Fourier splitting method combined with the explicit formula of the solution (high-frequency estimates) and the method due to [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations 257 (2014) 2159–2177.] (low-frequency estimates). In this paper, we will introduce a new threshold [Formula: see text] on the regularity of the initial data that divides the property of the corresponding solution to our problem into two parts: one is wave-like, and the other is parabolic-like.

Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping

We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in \(\mathbf{R}^{n}\) and study the asymptotic profile and optimal decay rates of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense. The damping terms considered in this paper is not studied so far, and in the low-frequency parameters, the damping is rather weakly effective than that of well-studied power type one such as \((-\Delta )^{\theta }u_{t}\) with \(\theta \in (0,1)\). When getting the optimal rate of decay, we meet the so-called hypergeometric functions with special parameters, so the analysis seems to be more difficult and attractive.

Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis

This paper is concerned with the stationary problem of an aero-taxis system with physical boundary conditions proposed by Tuval et al (2005 Proc. Natl Acad. Sci. 102 2277–82) to describe the boundary layer formation in the air–fluid interface in any dimensions. By considering a special case where fluid is free, the stationary problem is essentially reduced to a singularly perturbed nonlocal semi-linear elliptic problem. Denoting the diffusion rate of oxygen by ɛ > 0, we show that the stationary problem admits a unique classical solution of boundary-layer profile as ɛ → 0, where the boundary-layer thickness is of order ɛ. When the domain is a ball, we find a refined asymptotic boundary layer profile up to the first-order approximation of ɛ by which we find that the slope of the layer profile in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases w.r.t. the curvature.

The strong Malthusian behavior of growth-fragmentation processes

Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.