Browsing by Browse by FOR 2008 "010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems"

We estimate the blow-up rate and then improve some existing uniqueness results for boundary blow-up solutions to certain quasilinear elliptic equations with a weight function. The weight function is allowed to vanish on the part of the boundary where the solution blows up. Our approach is based on the construction of certain upper and lower solutions on small annuli with partial boundary blow-up, and on a modified version of an iteration technique due to Safonov.

In this paper, we discuss the asymptotic behaviour of the positive solutions to the problem -Δu=au-bu^p, u|δΩ=0 as p→1+0 and as p→∞. We show that, for each case, the behavior is determined by a limiting problem. Moreover, the limiting problem is of free boundary nature when p→∞.

To understand the heterogeneous spatial effect on predator–prey models, we study the behaviour of the positive steady states of a predator–prey model as certain parameters are small or large. We compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy. Our results show that the effect of the degeneracy can be clearly observed in one limiting case, but not in the others.

Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain over the entire R^N.

We prove that for large λ>0, the boundary blow-up problem -Δu = -u(u - a)(1 - u) in Ω, u∣∂Ω=∞ has a solution with a spike located near the "most centered part" of Ω. Our proof is based on the reduction method and uses a "cut-off" technique to avoid the difficulties for the estimates near the boundary of Ω. Crown Copyright © 2004 Published by Elsevier Inc. All rights reserved.

We show that for large λ>0, the 'generalized' boundary value problem −Δu = λu(u − a(x))(1 − u) in Ω, u|∂Ω = ø, where 1≼ø(x)≼∞, behaves like the special case ø ≡ 1. In particular, we show the existence of solutions with sharp boundary layers and interior spikes, and determine the location of the spikes.

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded Engel CR manifold.

In this paper, we consider the following elliptic problem... where a(x) is a continuous function satisfying 0 < a(x)<1 for x∈...,Ω is a bounded domain in R^N with smooth boundary, ε > 0 is a small number. A solution of (1.1) can be interpreted as a steady state solution of the corresponding problem: ut=ε²Δu+f(x,u), where f(x,t)=t(t-a(x))(1-t), which arises in a number of places such as population genetics [24,33]. The readers can refer to [22] for more background material for problem (1.1).

We study O(d)-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flowfrom B⁴(0, 1) into S⁴. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible O(4)-equivariant biharmonic maps from R⁴ into S⁴, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from B⁴(0, 1) into S⁴ that wind around S⁴ as many times as we wish. We believe that the ideas developed herein could be useful in the study of other higher-order parabolic equations.

We study the degenerate logistic model described by the equation ut-Δu=au-b(x)u^p with standard boundary conditions, where p>1, b vanishes on a nontrivial subset Ω₀ of the underlying bounded domain Ω⊂ℝ^N and b is positive on Ω₊... We consider the difficult case where ∂Ω₀⋂∂Ω≠... and ∂Ω₊⋂∂Ω≠... and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model.

This paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor. Our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor, the weak species almost always survives, but it cannot survive when the protection zone is below the critical size and its competitor is strong enough. While this is the main feature of the model, the actual dynamical behavior of the reaction-diffusion system is more complicated. The key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady-state solutions. Our mathematical analysis shows that, compared with the predator-prey model discussed in [Yihong Du, Junping Shi, A diffusive predator-prey model with a protect zone, J. Differential Equations 226 (2006) 63-91], the protection zone has some essentially different effects on the fine dynamics of the competition model.

In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case.

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator–prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue λ₁D(Ω₀)of the Laplacian operator over Ω₀ with with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if λ₁D(Ω₀) is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator–prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.

We study an elliptic problem involving critical Sobolev exponent in domains with small holes. We prove the existence of solutions which blow up like a volcano near the centre of each hole.

We show that the critical problem {-∆υ + λυ = υ²*⁻¹ + αυq⁻¹, υ > 0 in Ω, ∂υ∕∂ν = 0 on ∂Ω, 2 < q < 2* = 2N/(N − 2), has no positive solutions concentrating, as λ → ∞, at interior points of Ω if a = 0, but for a class of symmetric domains Ω, the problem with α > 0 has solutions concentrating at interior points of Ω.

Using Caputo fractional derivative of order α we build the fractional α tangent fiber bundle and its main geometrical structures. We consider Poisson realizations for Maxwell-Bloch and Rabinovich differential equations that allow us to construct a Leibniz algebroid structure on R³ Fractional differential equations on algebroids are defined, some conclusions and numeric simulations are provided.

We consider a minimization problem associated with the elliptic systems of FitzHugh–Nagumo type and prove that the minimizer of this minimization problem has not only a boundary layer, but also may oscillate in a set of positive measure.

We consider the elliptic problem -Δu - λu = a(x)g(u), with a(x) sign-changing and g(u) behaving like u^p, p > 1. Under suitable conditions on g(u) and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space ℝ^N. More precisely, we show that there exists Λ > 0 such that this equation on ℝ^N has no positive solution for λ > Λ, at least two positive solutions for λ ∈ (o,Λ), and at least one positive solution for λ ∈ (-∞,0]U{A}. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.

In this paper, we prove the existence of solutions with multiple interior layers for an elliptic Neumann problem.

In this paper, we shall construct multibump solutions for an elliptic problem on this expanding domain, such that all the local maximum points of the solution are close to the set...

This parabolic system in one space dimension is a simplification of the original Hodgkin-Huxley nerve conduction equations. Here u denotes an activator and v acts as its inhibitor. This system has also been studied in many applied areas. The readers can find more references in for the background of the systems of FitzHugh--Nagumo type. Here we mention some early results on the systems of FitzHugh--Nagumo type, obtained by Klaasen and Troy, Klaasen and Mitidieri and DeFigueiredo and Mitidieri.

We consider the elliptic problem -Δu − λu = a(x)u^p, with p >1 and (x) sign-changing. Under suitable conditions on p and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space R^N. More precisely, we show that there exists ∧>0 such that this equation on R^N has no positive solution for λ>∧, at least two positive solutions for λ ∈ (0,Λ), and at least one positive solution for λ ∊ (−∞, 0] ∪ {Λ}.

We consider hypersurfaces of finite type in a direct product space R²×R², which are analogues to real hypersurfaces of finite type in C². We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in C². For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted ℓ-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.

We establish the existence of solutions of the problem...

We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.

This is volume one of a two volume series. The intention is to provide a reference book for researchers in nonlinear partial differential equations and nonlinear functional analysis, especially for postgraduate students who want to be led to some of the current research topics. It could be used as a textbook for postgraduate students, either in formal classes or in working seminars. In these two volumes, we attempt to use order structure as a thread to introduce the various versions of the maximum principles, the fixed point index theory, and the relevant part of critical point theory and conley index theory. The emphasize is on their applications, and we try to demonstrate the usefulness of these tools by choosing applications to problems in partial differential equations that are of considerable concern of current research.

The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

We consider a radially symmetric free boundary problem with logistic nonlinear term. The spatial environment is assumed to be asymptotically periodic at infinity in the radial direction. For such a free boundary problem, it is known from [7] that a spreading-vanishing dichotomy holds. However, when spreading occurs, only upper and lower bounds are obtained in [7] for the asymptotic spreading speed. In this paper, we investigate one-dimensional pulsating semi-waves in spatially periodic media. We prove existence, uniqueness of such pulsating semi-waves, and show that the asymptotic spreading speed of the free boundary problem coincides with the speed of the corresponding pulsating semi-wave.

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,...,Dm of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D:=D₁∪...∪Dm and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model u₁(x,t)-d₁(x)Δu(x,t)=λa₁(x)u-[ε⁻¹b(x)+1]u²-c(x)uv... under natural boundary conditions on δΩ, possesses an asymptotically stable positive steady-state solution... that has pattern D, that is, roughly speaking, as ε→0, u... converges to a positive function over D, while it converges to 0 over the rest of Ω; on the other hand, v... converges to 0 over D but converges to some positive function in the rest of Ω. In other words, the two competing species u... and v... become spatially segregated as ε→0, with u... concentrating on D and v... concentrating on ΩD.

We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

We present several recent results obtained in our attempts to understand the influence of spatial heterogeneity in the predator-prey models. Two different approaches are taken. The first approach is based on the observation that the behavior of many diffusive population models is very sensitive to certain coefficient functions becoming small in part of the underlying spatial region. We apply this observation to three predator-prey models to reveal fundamental differences from the classical homogeneous case in each model, and demonstrate the essential differences of these models from each other. In the second approach, we examine the influence of a protection zone in a Holling type II diffusive predator-pre model, which introduces different mathematical problems from those in the first approach, and reveals important impacts of the protection zone.

For a wide class of nonlinearities f(u) satisfying f(0)=f(a)=0, f(u)>0 in (0,a) and f(u)<0 in (a,∞), we show that any nonnegative solution of the quasilinear equation -Δpu=f(u) over the entire ℝ^N must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.

For several classes of functions including the special case f(u)=u-u³, we obtain boundedness and symmetry results for solutions of the problem -∆u=f(u) defined on Rⁿ. Our results complement a number of recent results related to a conjecture of De Giorgi.

In this paper, we review some recent results on the effects of heterogeneous spatial environment on various population models. By making use of the observation that the population models make deep changes of behavior when certain coefficient functions vanish in the underlying domain, it is shown that sharp stable patterns can be obtained if the heterogeneous environment is designed suitably. The one species logistic model, two species competition and predator-prey models are discussed.

In this paper, we construct three types of symmetric peaked solutions for a Neumann problem involving critical Sobolev exponent: the interior peaked solution, the boundary peaked solution and the interior-boundary peaked solution.

For a wide class of nonlinearities f (u) satisfying f (u) > 0 in (0, a) and f(u) < 0 in (a,∞), but not necessarily Lipschitz continuous, we study the quasi-linear equation -Δ pu = f(u) in T, u│∂T = 0, where T = {x = (x₁,x₂,...., xN )ϵ ℝ^2 RN :x₁ > 0}≽ with N > 2. By using a new approach based on the weak maximum principle, we show that any positive solution on T must be a function of x1 only. Under our assumptions, the strong maximum principle does not hold in general and the solution may develop a flat core; our symmetry result allows an easy and precise determination of the flat core.

We consider the logistic equation -∆u=a(x)u-b(x)u^p on all of R^N with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of R^N have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has unique positive solution under suitable growth restrictions for its coefficients.