Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, u _t  = J*u−u := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on . When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behaviors can be presented in a unified way through a suitable global approximation.     

Asymptotic Behaviour near Finite-Time Extinction for the Fast Diffusion Equation

We study the Cauchy problem for the fast diffusion equation when and . For a class of nonnegative radially symmetric finite-mass solutions, which vanish identically at a given time , we show that their asymptotic behaviour as is described by a uniquely determined self-similar solution of the second kind: , where and and . Here is determined from a nonlinear eigenvalue problem involving an ordinary differential equation for the function . Special attention is paid to the case when . Then and the function can be found explicity. The proof is based on a geometric Lyapunov-type argument and comparison arguments based on the intersection properties of the solution graphs. (Accepted March 4, 1996)     

Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

In this paper, we carry out further mathematical studies of nonlocal constrained value problems for a peridynamic Navier equation derived from linear state-based peridynamic models. Given the nonlocal interactions effected in the model, constraints on the solution over a volume of nonzero measure are natural conditions to impose. We generalize previous well-posedness results that were formulated for very special kernels of nonlocal interactions. We also give a more rigorous treatment to the convergence of solutions to nonlocal peridynamic models to the solution of the conventional Navier equation of linear elasticity as the horizon parameter goes to zero. The results are valid for arbitrary Poisson ratio, which is a characteristic of the state-based peridynamic model.     

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.     

Asymptotic States of a Smoluchowski Equation

We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers.Acknowledgement PCs research partially supported by NSF DMS-0202531. IGKs research partially supported by AFOSR (Dynamics and Control) and NSF (ITR). ESTs research partially supported by NSF DMS-0204794, the US CRDF under grant No. RM1-2343-MO-02 and by the S.M. Ulam Visiting Scholar Fellowship at the CNLS Los Alamos National Laboratory under D.O.E. contract W-7405-ENG-36. The comments of the referees are gratefully acknowledged.     

Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].     

Age-Structured Population Dynamics with Nonlocal Diffusion

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

     

Existence,Uniqueness and Asymptotic Stability of Traveling Wavefronts in A Non-Local Delayed Diffusion Equation

In this paper, we study the existence, uniqueness, and global asymptotic stability of traveling wave fronts in a non-local reaction–diffusion model for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. Under realistic assumptions on the birth function, we construct various pairs of super and sub solutions and utilize the comparison and squeezing technique to prove that the equation has exactly one non-decreasing traveling wavefront (up to a translation) which is monotonically increasing and globally asymptotic stable with phase shift.      

Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture

We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width. By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that are consistent with linear elastic fracture mechanics. Submitted to Journal of Elasticity on February 5, 2007. Re-submitted with revisions on May 30, 2007.     

The Asymptotic Speed of Spread and Traveling Waves for a Vector Disease Model

The asymptotic speed of spread is established for a diffusive and time-delayed integro-differential equation modeling vector disease, and its coincidence with the minimal wave speed for monotone traveling waves is proved. An erratum to this article can be found at     

Global Continua of Positive Solutions for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition

This paper is concerned with a quasilinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.     

Global Dynamics and Asymptotic Spreading Speeds for a Partially Degenerate Epidemic Model with Time Delay and Free Boundaries

Journal of Dynamics and Differential Equations - This paper concerns the global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free...     

Large Time Behavior for a Quasilinear Diffusion Equation with Critical Gradient Absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption

$$\begin{aligned} \partial _t u-\Delta _{p}u+|\nabla u|^{q_*}=0 \quad \hbox {in}\, (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

for \(p\in (2,\infty )\) and \(q_*:=p-N/(N+1)\). We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the p-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

     

Exact Asymptotic Behavior of the Solution of a Matrix Difference Equation

An asymptotic expansion of the solution of a nonhomogeneous matrix difference equation of general form is obtained. The case when there is no bound on the differences of the arguments is considered. The effect of the roots of the characteristic equation is taken into account. The asymptotic behavior of the remainder is established depending on the asymptotics of the free term of the equation.

     

Maximal and Minimal Spreading Speeds for Reaction Diffusion Equations in Nonperiodic Slowly Varying Media

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is ${f(x,u) = \mu_0 (\phi (x)) u(1-u)}$ , where??? ₀ is a 1-periodic function and ${\phi}$ is a ${\mathcal{C}^1}$ increasing function that satisfies ${\lim_{x \to+\infty}\phi (x) = +\infty}$ and ${\lim_{x \to +\infty}\phi' (x) =0}$ . Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of ${\phi}$ at infinity. If ${\phi}$ grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If ${\phi}$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation w _??, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.     

Periodic Step-Like Contrast Structures for a Quasilinear Parabolic Equation with Small Diffusion Coefficient

Using the method of boundary functions, for a quasilinear parabolic equation with small diffusion coefficient we construct an asymptotic expansion of a periodic solution with internal transition layer. Sufficient conditions for the existence of this solution are obtained. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 329–350, July–September, 2005.     

Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation as t approaches the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of the initial data.