On the least point of the spectrum of certain cooperative linear systems with a large parameter

We obtain the asymptotic limit of the principal eigenvalue of a cooperative system of linear elliptic equations as a parameter tends to infinity. Our results are much more general than those in the work of Caudevilla and López-Gómez (2008) [2]. We obtain an unusual limit problem.     

The steady states of a non-cooperative model of nuclear reactors

This paper characterizes the existence of coexistence states in a reaction-diffusion model arising in the theory of nuclear reactors. From a mathematical point of view, the importance of this model relies upon the fact that the associated variational systems are of non-cooperative type and, consequently, the comparison techniques available for cooperative systems fail to work out. Although in higher spatial dimensions the dynamics of the model might be rather involved, by the absence of limitations for the number of steady states, we can prove the uniqueness of the steady state in the one-dimensional prototype model. Our results complement and eventually sharpen the findings of Arioli [G. Arioli, Long term dynamics of a reaction-diffusion system, J. Differential Equations 235 (2007) 298-307].     

Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas

In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states (λ,u)=(λ,0) at a nonlinear eigenvalue λ=λ₀ with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz's unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz's context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181).     

Optimal uniqueness theorems and exact blow-up rates of large solutions

In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)u^p, p>1, a>0, must satisfy

     

Blow-up rates of radially symmetric large solutions

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f∈C[0,∞) is positive and non-decreasing, V∈C[0,∞)∩C²(0,∞) satisfies V(0)=0, V^′(u)>0, V^″(u)?0, for every u>0, and V(u)∼Hup−1 as u↑∞, for some H>0 and p>1, then, for each λ?0,

−Δu=λu−f(dist(x,∂Ω))V(u)u     

Properties of the Principal Eigenvalues of a General Class of Non-classical Mixed Boundary Value Problems

In this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations5, 999-1030). In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations138, 86-132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations146, 336-374).     

Steady states of a non-cooperative model arising in nuclear engineering

This paper studies the existence and the uniqueness of coexistence states in a spatially heterogeneous reaction diffusion model originated by the theory of nuclear reactors. The mathematical relevance of our analysis relies on the fact that the underlying variational systems are of non-cooperative type and, consequently, the nice comparison techniques available for cooperative systems fail to be true, making the problem of ascertaining whether or not the model admits a unique coexistence state an extremely challenging task. In this paper we are giving a series of new uniqueness results complementing those available in the literature.     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Variational approach for a class of cooperative systems

The aim of this work is to ascertain the characterization of the existence of coexistence states for a class of cooperative systems supported by the study of an associated non-local equation through classical variational methods. Thanks to those results, we are able to obtain the blow-up behavior of the solutions in the whole domain for certain values of the main continuation parameter.     

Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.     

Numerical experimentation with a human migration model

In this paper, we develop a human migration model with a conjectural variations equilibrium (CVE). In contrast to previous works we extend the model to the case where the conjectural variations coefficients may be not only constants, but also (continuously differentiable) functions of the total population at the destination and of the group’s fraction in it. Moreover, we allow these functions to take distinct values at the abandoned location and at the destination. As an experimental verification of the proposed model, we develop a specific form of the model based upon relevant population data of a three-city agglomeration at the boundary of two Mexican states: Durango (Dgo.) and Coahuila (Coah.). Namely, we consider the 1980–2000 dynamics of population growth in the three cities: Torreón (Coah.), Gómez Palacio (Dgo.) and Lerdo (Dgo.), and propose utility functions of four various kinds for each of the three cities. After having collected necessary information about the average movement and transportation (i.e., migration) costs for each pair of the cities, we apply the above-mentioned human migration model to this example. Numerical experiments have been conducted with interesting results concerning the probable equilibrium states revealed.     

Normal integral bases for cyclic Kummer extensions

Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result.     

Advection-mediated competition in general environments

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large.     

Uniqueness of the positive solution for a non-cooperative model of nuclear reactors

We prove the uniqueness of the positive solution for a non-cooperative reaction–diffusion model of nuclear reactors, by converting the system to an equivalent cooperative one.     

Generalized Convexity and Nonsmooth Problems of Vector Optimization

In this paper it is shown that every generalized Kuhn-Tucker point of a vector optimization problem involving locally Lipschitz functions is a weakly efficient point if and only if this problem is KT- pseudoinvex in a suitable sense. Under a closedness assumption (in particular, under a regularity condition of the constraint functions) it is pointed out that in this result the notion of generalized Kuhn–Tucker point can be replaced by the usual notion of Kuhn–Tucker point. Some earlier results in (Martin (1985), The essence of invexity, J. Optim. Theory Appl., 47, 65–76. Osuna-Gómez et al., (1999), J. Math. Anal. Appl., 233, 205–220. Osuna-GGómez et al., (1998), J. Optim. Theory Appl., 98, 651–661. Phuong et al., (1995) J. Optim. Theory Appl., 87, 579–594) results are included as special cases of ours. The paper also contains characterizations of HC-invexity and KT- invexity properties which are sufficient conditions for KT- pseudoinvexity property of nonsmooth problems.Mathematics Subject Classifications: 90C29, 26B25     

Consensus in the two-state Axelrod model

The Axelrod model is a spatial stochastic model for the dynamics of cultures which, similar to the voter model, includes social influence, but differs from the latter by also accounting for another social factor called homophily, the tendency to interact more frequently with individuals who are more similar. Each individual is characterized by its opinions about a finite number of cultural features, each of which can assume the same finite number of states. Pairs of adjacent individuals interact at a rate equal to the fraction of features they have in common, thus modeling homophily, which results in the interacting pair having one more cultural feature in common, thus modeling social influence. It has been conjectured based on numerical simulations that the one-dimensional Axelrod model clusters when the number of features exceeds the number of states per feature. In this article, we prove this conjecture for the two-state model with an arbitrary number of features.     

A generalized thin-film equation in multidimensional space

This paper deals with the existence and nonnegativity of the weak periodic-domain solution for a degenerate fourth-order parabolic equation modeling the evolution of thin films. This study in the multidimensional domain follows the recent results related to the resolution in one-dimensional space [J.E. Rakotoson, J.M. Rakotoson, C. Verbeke, Generalized lubrification models blow-up and global existence, RACSAM 99 (2) (2005) 235–241; C. Verbeke, Quelques modèles d’équations d’évolution de surfaces: Explosion en temps fini et diverses propriétés qualitatives, Thèse de doctorat de l’Université de Poitiers (15 Décembre 2005); J.E. Rakotoson, J.M. Rakotoson, C. Verbeke, Higher-order equations related to thin film: Blow up and global existence, the influence of the initial data, (submitted for publication)].     

A system of partial differential equations modeling the competition for two complementary resources in flowing habitats

This paper examines a system of reaction-diffusion equations arising from a flowing water habitat. In this habitat, one or two microorganisms grow while consuming two growth-limiting, complementary (essential) resources. For the single population model, the existence and uniqueness of a positive steady-state solution is proved. Furthermore, the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values. Mathematical analysis for the two competing populations is carried out. More precisely, the long-time behavior is determined by using the monotone dynamical system theory when the semi-trivial solutions are both unstable. It is also shown that coexistence solutions exist by using the fixed point index theory when the semi-trivial solutions are both (asymptotically) stable.     

On the existence of ground states for nonlinear Schrödinger-Poisson equation

This paper is concerned with the existence of ground states for the Schrödinger-Poisson equation , where V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities.     

Lorentz estimates for degenerate and singular evolutionary systems

We prove estimates of Calderón–Zygmund type for evolutionary p-Laplacian systems in the setting of Lorentz spaces. We suppose the coefficients of the system to satisfy only a VMO condition with respect to the space variable. Our results hold true, mutatis mutandis, also for stationary p-Laplacian systems.