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In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko''s result on Hamiltonian differential equations. The main tools are Picard–Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A''Campo''s theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss–Manin connection on the so-called Brieskorn lattice/Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and (*) condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given. [Copyright &y& Elsevier]

This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice Z1. The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically. [Copyright &y& Elsevier]

This paper is part of the program launched in (J. Differential Equations 110(1) (1994) 86) to prove the finiteness part of Hilbert''s 16th problem for quadratic system, which consists in proving that 121 graphics have finite cyclicity among quadratic systems. We show that any pp-graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity. Such graphics may have additional saddles and/or saddle-nodes. Altogether we show the finite cyclicity of 15 graphics of (J. Differential Equations 110(1) (1994) 86). In particular we prove the finite cyclicity of a pp-graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity ≠1 which appears in generic 3-parameter families of vector fields and hence belongs to the zoo of Kotova and Stanzo (Concerning the Hilbert 16th problem, American Mathematical Society Translation Series 2, Vol. 165, American Mathematical Society, Providence, RI, 1995, pp. 155–201). [Copyright &y& Elsevier]

We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on Ω¯. The proofs are based on various techniques related to the maximum principle for elliptic equations. [Copyright &y& Elsevier]

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models ut=[D(u)ux]x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c* and we show the appearance of a sharp-type profile when c=c*. These results solve recent conjectures formulated by Sa´nchez-Gardun˜o and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). [Copyright &y& Elsevier]

This paper deals with the following inter-connected subjects: (i) Separation of positive radial solutions is studied for the elliptic equation Δu+K(|x|)up=0 in Rn. In case r−ℓK(r) is decreasing, with suitable decay rate, to a positive constant as r→∞ for some ℓ>−2, the asymptotic behavior near ∞ of radial solutions is described in detail when n and p are large enough. When the equation has separation structure (any two positive radial solutions do not intersect), the obtained asymptotic behaviors decide natural topology for stability of positive radial solutions regarded as steady states of the corresponding semilinear heat equation. (ii) By using local separation of regular solutions, we establish the existence of a singular solution when every solution with positive initial data at 0 exists globally. (iii) Through Kelvin''s transform, separation structure of regular solutions reflects that of singular solutions. We present a new exponent, which is critical in the study of separation and intersection of singular solutions. [Copyright &y& Elsevier]

In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczyn´ski (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto–Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions. [Copyright &y& Elsevier]

In this paper, we prove the existence of solutions with multiple interior layers for an elliptic Neumann problem. [Copyright &y& Elsevier]

The aim of this paper is to study necessary conditions for existence of weak solutions of the inequalityL(x,y,Dx,Dy)u⩾|u|q/|x|θ1|y|θ2, x∈Rd, y∈Rk,where L is a quasi-homogeneous differential operator, q>1, θ1.θ2∈R and k,d⩾1. As special cases, our results can be applied when L is Tricomi or Grushin-type operators. [Copyright &y& Elsevier]

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations:ut=Δu+F(u,Dxu,Dx2u), u∈Rn,u(t,x)|t=0=u0(x), x∈RN, N⩾1.Under the optimal growth conditions on the smooth nonlinear function F(u,Dxu,Dx2u), we obtain the global existence results to the above Cauchy problem. The influence of the nonlinear function F(u,Dxu,Dx2u) on the large time behaviour of the global smooth solution u(t,x) is also studied. [Copyright &y& Elsevier]

In this paper, we study the L1 stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169–179. Under the suitable assumptions on the initial data, we construct a nonlinear functional H(t) which measures L1 distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of H(t), we show that mild solutions are L1-stable:||f(· ,· ,t)−f¯(· ,· ,t)||L1(R2)⩽G||f0(· ,·)−f¯0(· ,·)||L1(R2),where G is a positive constant independent of time t, f and f¯ are mild solutions corresponding to initial data f0 and f¯0 in L1(R2)∩L+∞(R2), respectively. [Copyright &y& Elsevier]

The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka–Volterra model, both in the conventional sense (as t→∞) and in the “pullback” sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually die out, ruling out any conventional type of permanence. In contrast, we introduce the notion of “pullback permanence” and show that this property is enjoyed by our model. This is not just a mathematical artifice, but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented (and their numbers are bounded below), even if the final fate of one of them is less happy. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations. [Copyright &y& Elsevier]

In this paper we develop Kaplan–Yorke''s method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times. [Copyright &y& Elsevier]

This paper deals with a two-species Lotka–Volterra competition model with discrete delays but without instantaneous negative feedbacks. Motivated by Wright''s 3/2 global attractivity result for the delayed scalar logistic equation, we establish some new 3/2 global attractivity result for the delayed scalar logistic equation, we establish some new 3/2-type criteria for global attractivity of the positive equilibrium of the system. These criteria provide convenient and better (than some existing) estimates for the diagonal delays. [Copyright &y& Elsevier]

The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy''s law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations. [Copyright &y& Elsevier]

The Falkner–Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336–394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur. [Copyright &y& Elsevier]

In this paper we are concerned with a differential equation of the form x¨+cx˙+q(t)g(x)=0, t∈(a,b), where −∞⩽a

Sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions. The conditions assume the existence of a strong upper solution–lower solution pair, a concept that is defined in the paper. The differential equation has nonlinear dependence on all lower order derivatives of the unknown; in particular, appropriate Nagumo conditions are obtained and employed. [Copyright &y& Elsevier]

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained. [Copyright &y& Elsevier]