Large Time Behavior Research Articles

Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model

Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model

Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers

For certain materials science scenarios arising in rubber technology, one-dimensional moving boundary problems with kinetic boundary conditions are capable of unveiling the large-time behavior of the diffusants penetration front, giving a direct estimate on the service life of the material. Driven by our interest in estimating how a finite number of diffusant molecules penetrate through a dense rubber, we propose a random walk algorithm to approximate numerically both the concentration profile and the location of the sharp penetration front. The proposed scheme decouples the target evolution system in two steps: (i) the ordinary differential equation corresponding to the evaluation of the speed of the moving boundary is solved via an explicit Euler method, and (ii) the associated diffusion problem is solved by a random walk method. To verify the correctness of our random walk algorithm we compare the resulting approximations to computational results based on a suitable finite element approach with a controlled convergence rate. Our numerical results recover well penetration depth measurements of a controlled experiment designed specifically for this setting.

Global Strong Solutions and Large-time Behavior of 2D Tropical Climate Model

Global Strong Solutions and Large-time Behavior of 2D Tropical Climate Model

Generalised solution to a 2D parabolic-parabolic chemotaxis system for urban crime: Global existence and large-time behaviour

AbstractWe consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter $\chi =2$ . The available analytical findings for this urban crime model including $\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any $\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces RN−k×(0,∞)k, k≥0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L2(RN−k×(0,∞)k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k=0, to any k≥0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

Asymptotic profile for the interaction of a rigid ball and an incompressible viscous fluid

In this paper, we study the large time behavior of the solutions of the Stokes fluid-solid system. We compute the asymptotic expansion of the solution for Lq integrable initial data U0. We show that the asymptotic profile of the solution is a linear combination of the Stokes fundamental solution for the data with extra condition |x|U0∈L1. We also show that L1 integrability of the solution is strongly related to the net force exerted by the fluid on the boundary of the solid.

The existence and nonlinear stability of stationary solutions to inflow problem for a reduced gravity two and a half layer model

The main concern of this paper is to study large-time behavior of solutions to the reduced gravity two and a half layer model for the oceanic fluid dynamic in a half-space. We proved the existence and the nonlinear stability of stationary solutions for an inflow problem under some assumptions on the boundary data and the initial perturbation. The proof is based on the detailed analysis of the corresponding stationary solution and elaborate energy estimates to the nonlinear system.

Convergence of genealogies through spinal decomposition with an application to population genetics

Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when N rightarrow infty . Informally, the moment of order k of the population is obtained by observing the genealogy of k individuals chosen uniformly at random after size-biasing the population at time N by its kth factorial moment. We show that the sampled genealogy can be expressed in terms of a k-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying k-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright–Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

Global well-posedness of the Navier–Stokes equations in homogeneous Besov spaces on the half-space

Consider the Stokes equations in the half-space R+n, n≧2. It is shown that the negative of the Stokes operator defined on the homogeneous Besov space B˙p,q,σs(R+n) generates a bounded strongly continuous semigroup in B˙p,qs(R+n) provided that 1<p<∞, 1≦q<∞, and −1+1/p<s<1/p. As a by-product, the maximal Lq-regularity of the Stokes operator is obtained, admitting the limiting case q=1. This will be applied to develop the global well-posedness result for the incompressible Navier–Stokes equations in the maximal L1-regularity class provided that the initial data are small in B˙p,1−1+n/p(R+n) with n−1<p<∞. The dependence of the solution on initial data and the large time behavior of the solution are investigated.

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.

Large-time behavior of solutions to three-dimensional bipolar Euler–Poisson equations with time-dependent damping

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects for . Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions.

A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System

Abstract A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.

Asymptotic expansion of solutions to the wave equation with space-dependent damping

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy methods with suitable supersolutions of the corresponding parabolic problem.

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation ut=Δu+f(u) with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that f(u)=0 has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution (u(t,|x|),h(t)) for x∈RN with free boundary |x|=h(t) such that, as t→∞, |x|≤h(t) expands to the whole space RN and u(t,⋅) converges to a positive stable equilibrium for f(u) uniformly in any compact set of RN. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem.Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when u(t,⋅) converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate.

Motion of Rigid Bodies of Arbitrary Shape in a Viscous Incompressible Fluid: Wellposedness and Large Time Behaviour

Motion of Rigid Bodies of Arbitrary Shape in a Viscous Incompressible Fluid: Wellposedness and Large Time Behaviour

Current fluctuations in an interacting active lattice gas

We study the fluctuations of the integrated density current across the origin up to time T in a lattice model of active particles with hard-core interactions. This model is amenable to an exact description within a fluctuating hydrodynamics framework. We focus on quenched initial conditions for both the density and the magnetization fields and derive expressions for the cumulants of the density current, which can be matched with direct numerical simulations of the microscopic lattice model. For the case of uniform initial profiles, we show that the variance of the integrated current displays three regimes: an initial rise with a coefficient given by the symmetric simple exclusion process, a cross-over regime where the effects of activity increase the fluctuations, and a large-time behavior with a prefactor that depends on the initial conditions, the Péclet number, and the mean density of particles. Additionally, we study the limit of zero diffusion, where the fluctuations intriguingly exhibit a T 2 behavior at short times. However, at large times, the fluctuations still grow as , with a coefficient that can be calculated explicitly. For low densities, we show that this coefficient can be expressed in terms of the effective diffusion constant D eff for non-interacting active particles.

Well-posedness of the discrete collision-induced breakage equation with unbounded fragment distribution

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel ai,j satisfying ai,j≤Aij for some A>0. More precisely, it is proved that, given suitable initial conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large time behaviour of solutions.

The global classical solution to compressible system with fractional viscous term

We consider the initial value problem to the fractional system of motions for compressible viscous fluids in this paper. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the large-time behavior of the solution, where solutions converge to a constant steady state exponentially in time.

Analysis of a radial free boundary tumor model with time-dependent absorption efficiency

We investigate an n-dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β(t) is considered. The system consists of a reaction-diffusion equation for the nutrient u(r,t) with nonlinear consumption and an integro-differential equation for the tumor radius R(t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β(t). We prove that for any sufficiently small c>0, the tumor radius R(t) will remain bounded or shrink to zero when β(t) is uniformly bounded or tends to zero, respectively; and if β(t)→β⁎, the solution (u,R) will converge to the unique steady state (u⁎,R⁎) associating with β⁎. We also give examples of R(t) blowing up as t→∞ even if β(t)→0 when c is not small.