scholarly journals One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

2021 ◽  
Vol 175 (1) ◽  
Author(s):  
Eugenia Franco ◽  
Mats Gyllenberg ◽  
Odo Diekmann
Keyword(s):  
Mathematical Biology ◽  
Population Dynamics ◽  
Asymptotic Behaviour ◽  
Dimensional Reduction ◽  
Large Time ◽  
Renewal Equation ◽  
Time Behaviour ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Renewal Equations

AbstractDespite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas

1996 ◽  
Vol 126 (6) ◽  
pp. 1277-1296 ◽  
Author(s):  
L. Hsiao ◽  
T. Luo
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Time Behaviour ◽  
Real Gas ◽  
Large Time Behaviour ◽  
One Dimensional ◽  
Viscous Heat

We investigate the large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas. A conclusive answer to the problem of asymptotic behaviour is given in Theorem 1.2.

Subexponential solutions of linear integro-differential equations and transient renewal equations

2002 ◽  
Vol 132 (3) ◽  
pp. 521-543 ◽  
Author(s):  
John A. D. Appleby ◽  
David W. Reynolds
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Renewal Equation ◽  
Asymptotically Stable ◽  
Renewal Equations ◽  
Integro Differential Equation ◽  
All Solutions ◽  
Image Position

This paper studies the asymptotic behaviour of the solutions of the scalar integro-differential equation The kernel k is assumed to be positive, continuous and integrable.If it is known that all solutions x are integrable and x(t) → 0 as t → ∞, but also that x = 0 cannot be exponentially asymptotically stable unless there is some γ > 0 such that Here, we restrict the kernel to be in a class of subexponential functions in which k(t) → 0 as t → ∞ so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) → 0 as t → ∞ is given by The result is proved by determining the asymptotic behaviour of the solution of the transient renewal equation If the kernel h is subexponential, then

Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping

2015 ◽  
Vol 145 (5) ◽  
pp. 959-978 ◽  
Author(s):  
Shifeng Geng ◽  
Lina Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Nonlinear Damping ◽  
Boundary Effects ◽  
Time Behaviour ◽  
Quasilinear Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Quarter Plane

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

scholarly journals SUR LA STABILITÉ POUR L’ÉQUATION MONODIMENSIONNELLE D’UN GAZ VISQUEUX ET CALORIFÈRE AVEC LA FRONTIÈRE VARIABLE

2001 ◽  
Vol 44 (2) ◽  
pp. 295-315
Author(s):  
Rachid Benabidallah
Keyword(s):  
Boundary Conditions ◽  
Positive Constant ◽  
Large Time ◽  
Subject Classification ◽  
Heat Conducting ◽  
Time Behaviour ◽  
Initial Amplitude ◽  
Variable Boundary ◽  
One Dimensional ◽  
Viscous Heat

AbstractWe consider the equation of a one-dimensional viscous heat-conducting compressible gas in the variable domain with the appropriate boundary conditions. We study the large-time behaviour of the solution in the particular case where the displacement of the variable boundary is given by $L(t)=L_0(1+at)^\alpha$ with $0lt\alphalt1$, where $a$ is a positive constant and $L_0$ is the initial amplitude of our domain.AMS 2000 Mathematics subject classification: Primary 35B40; 76N15

Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Miguel Yangari
Keyword(s):  
Anomalous Diffusion ◽  
Large Time ◽  
Mild Solutions ◽  
Reaction Diffusion ◽  
Cooperative Systems ◽  
Initial Condition ◽  
Time Behaviour ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
The One

AbstractThe aim of this paper is to study the large-time behaviour of mild solutions to the one-dimensional cooperative systems with anomalous diffusion when at least one entry of the initial condition decays slower than a power. We prove that the solution moves at least exponentially fast as time goes to infinity. Moreover, the exponent of propagation depends on the decay of the initial condition and of the reaction term.

Large-time behaviour of solutions of Burgers' equation

2002 ◽  
Vol 132 (4) ◽  
pp. 843-878 ◽  
Author(s):  
Daniel B. Dix
Keyword(s):  
Initial Data ◽  
Asymptotic Behaviour ◽  
Initial Value Problem ◽  
Large Time ◽  
Burgers Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
Large Time Behaviour ◽  
Image Position ◽  
Sharp Error

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

scholarly journals About the Use of Entropy Production for the Landau–Fermi–Dirac Equation

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
R. Alonso ◽  
V. Bagland ◽  
L. Desvillettes ◽  
B. Lods
Keyword(s):  
Dirac Equation ◽  
Classical Limit ◽  
Large Time ◽  
A Priori Estimates ◽  
Landau Equation ◽  
A Priori ◽  
Time Behaviour ◽  
Entropy Dissipation ◽  
Priori Estimates ◽  
Potential Case

AbstractIn this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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