A multidimensional nonlinear sixth-order quantum diffusion equation

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.     

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity

We present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity. The upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. It is also shown that the minimal time blow-up occurs when initial data attains its maximum at space infinity.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.     

A weighted quasilinear equation related to the mean curvature operator

This paper studies a weighted quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady states of the model, though their derivatives must be somewhere unbounded. The dynamics of the positive solutions of the model is fully described.     

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.     

Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders

This paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders. Assume that the equation admits three equilibria: two stable equilibria 0 and 1, and an unstable equilibrium θ. It is well known that there are different wave fronts connecting any two of those three equilibria. By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions, we establish three different types of entire solutions. Finally, we analyze a model for shear flows in cylinders to illustrate our main results.     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. (Received 11 October 2000; in revised form 13 March 2001)     

Roles of weight functions in a nonlinear nonlocal parabolic system

This paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. We establish the conditions for global and non-global solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non-global, but also whether (for the non-global solutions) the blowing up occurs for any positive initial data or just for large ones.     

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.     

Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems

In this paper, we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x=0. We carry out a series of weighted estimates to the boundary layer equations—Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions.     

The interface between regions where u<0 andu > 0in the porous medium equation

We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines     

The effect of predator competition on positive solutions for a predator–prey model with diffusion

A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large m$m$. Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when k$k$ goes to ∞$\infty$, we find that there is at most a positive solution for any c∈R$c\in R$ when k$k$ is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension.     

Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.Supported by the National Natural Science Foundation of China.Received: December 12, 2001; revised: May 6, December 3, 2002     

On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat

In this paper we study a system of reaction-diffusion equations arising from competition of two microbial populations for a single-limited nutrient with internal storage in an unstirred chemostat. The conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient. The reduced system (limiting system) generates a strongly monotone dynamical system in its feasible domain under a partial order. We construct suitable upper, lower solutions to establish the existence of positive steady-state solutions. Given the parameters of the reduced system, we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors. The primary conclusion is that the survival of species depends on species's intrinsic biological characteristics, the external environment forces and the principal eigenvalues of some scalar partial differential equations. We also lift the dynamics of the limiting system to the full system.     

Permanence under strong aggressions is possible

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7].     

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria.The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.