Discrete and Continuous Dynamical Systems
June 2020 , Volume 40 , Issue 6
DCDS-A special issue to honor Wei-Ming Ni's 70th birthday
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The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [
We first study the so-called Heat equation with two families of elliptic operators which are fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equation with operators including the "large" eigenvalues has strong similarities with a Heat equation in lower dimension whereas, surprisingly, for operators including "small" eigenvalues it shares some properties with some transport equations. In particular, for these operators, the Heat equation (which is nonlinear) not only does not have the property that "disturbances propagate with infinite speed" but may lead to quenching in finite time. Last, based on our analysis of the Heat equations (for which we provide a large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.
In this paper, we study the global dynamics of a general
We are concerned with the blow-up analysis of mean field equations. It has been proven in [
We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.
We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.
The strong version of the Krein-Rutman theorem requires that the positive cone of the Banach space has nonempty interior and the compact map is strongly positive, mapping nonzero points in the cone into its interior. In this paper, we first generalize this version of the Krein-Rutman theorem to the case of "semi-strongly positive" operators; and we prove it in a totally elementary fashion. We then prove the equivalence of semi-strong positivity and irreducibility in a Banach lattice, linking the afore-mentioned result with the Krein-Rutman theorem for irreducible operators. One of the things we emphasize is to use "upper and lower spectral radii" to characterize, in the fashion of Collatz-Wielandt formula for nonnegative irreducible matrices, the principal eigenvalue of these operators. For reducible operators, we prove that the lower spectral radius always serves as the least upper bound of the set of eigenvalues pertaining to positive eigenvectors, and the upper spectral radius the greatest lower bound of the set. Finally, we demonstrate the full power of these Krein-Rutman theorems on some PDE examples such as elliptic eigenvalue problems on non-smooth domains, and cooperative systems which may or may not be fully coupled, by using as few PDE tools as possible.
The purpose of this paper is to study the solutions of
Under the assumption
We prove that all harmonic maps from
In this paper, we study qualitative properties of the fractional
In this paper, we consider the singular limit of an energy minimizing problem which is a semi-limit of a singular elliptic equation modeling steady states of thin film equation with both Van der Waals force and Born repulsion force. We show that the singular limit of energy minimizers is a Dirac mass located on the boundary point with the maximum curvature.
In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [
In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [
We consider the Cauchy problem for the energy critical heat equation
We find that for given points
This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.
This paper deals with a free boundary problem for a reaction-diffusion equation in a one-dimensional interval whose boundary consists of a fixed end-point and a moving one. We put homogeneous Dirichlet condition at the fixed boundary, while we assume that the dynamics of the moving boundary is governed by the Stefan condition. Such free boundary problems have been studied by a lot of researchers. We will take a nonlinear reaction term of positive bistable type which exhibits interesting properties of solutions such as multiple spreading phenomena. In fact, it will be proved that large-time behaviors of solutions can be classified into three types; vanishing, small spreading and big spreading. Some sufficient conditions for these behaviors are also shown. Moreover, for two types of spreading, we will give sharp estimates of spreading speed of each free boundary and asymptotic profiles of each solution.
Consider the Cauchy problem for a nonlinear diffusion equation
This paper considers a two-patch system with asymmetric diffusion rates, in which exploitable resources are included. By using dynamical system theory, we exclude periodic solution in the one-patch subsystem and demonstrate its global dynamics. Then we exhibit uniform persistence of the two-patch system and demonstrate uniqueness of the positive equilibrium, which is shown to be asymptotically stable when the diffusion rates are sufficiently large. By a thorough analysis on the asymptotic population abundance, we demonstrate necessary and sufficient conditions under which the asymmetric diffusion rates can lead to the result that total equilibrium population abundance in heterogeneous environments is larger than that in heterogeneous/homogeneous environments with no diffusion, which is not intuitive. Our result extends previous work to the situation of asymmetric diffusion and provides new insights. Numerical simulations confirm and extend our results.
We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be
We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.
Tremendous efforts have been devoted to the development and analysis of mathematical models to assess the efficacy of the endosymbiotic bacterium Wolbachia in the control of infectious diseases such as dengue and Zika, and their transmission vector Aedes mosquitoes. However, the larval stage has not been included in most models, which causes an inconvenience in testing directly the density restriction on population growth. In this work, we introduce a system of delay differential equations, including both the adult and larval stages of wild mosquitoes, interfered by Wolbachia infected males that can cause complete female sterility. We clarify its global dynamics rather completely by using delicate analyses, including a construction of Liapunov-type functions, and determine the threshold level
We consider the structure and the stability of positive radial solutions of a semilinear inhomogeneous elliptic equation with multiple growth terms
which is a generalization of Matukuma's equation describing the dynamics of a globular cluster of stars. Equations similar to this kind have come up both in geometry and in physics, and have been a subject of extensive studies. Our result shows that any positive radial solution is stable or weakly asymptotically stable with respect to certain norm.
We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system
in a bounded domain
We consider the eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a spherical domain. Especially, we investigate the case when the domain is a large zonal one and letting the zone larger so that the zone covers the whole sphere as a limit. We discuss the behavior of eigenvalues according to the rate of expansion of the zone.
In 1979, Shigesada, Kawasaki and Teramoto [
Biological organisms leave their habitat when the environment becomes harsh. The essence of a biological dispersal is not in the rate, but in the capability to adjust to the environmental changes. In nature, conditional asymmetric dispersal strategies appear due to the spatial and temporal heterogeneity in the environment. Authors show that such a dispersal strategy is evolutionary selected in the context two-patch problem of Lotka-Volterra competition model. They conclude that, if a conditional asymmetric dispersal strategy is taken, the dispersal is not necessarily disadvantageous even for the case that there is no temporal fluctuation of environment at all.
The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.
We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem
in the unit ball
This paper is concerned with the existence and stability of nontrivial positive steady states of Shigesada-Kawasaki-Teramoto competition model with cross diffusion under zero Neumann boundary condition. By applying the special perturbation argument based on the Lyapunov-Schmidt reduction method, we obtain the existence and the detailed asymptotic behavior of two branches of nontrivial large positive steady states for the specific shadow system when the random diffusion rate of one species is near some critical value. Further by applying the detailed spectral analysis with the special perturbation argument, we prove the spectral instability of the two local branches of nontrivial positive steady states for the limiting system. Finally, we prove the existence and instability of the two branches of nontrivial positive steady states for the original SKT cross-diffusion system when both the cross diffusion rate and random diffusion rate of one species are large enough, while the random diffusion rate of another species is near some critical value.
We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.
We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator
We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.
In this paper we survey some results concerning the construction of spike-layers, namely solutions to singularly perturbed equations that exhibit a concentration behaviour. Their study is motivated by the analysis of pattern formation in biological systems such as the Keller-Segel or the Gierer-Meinhardt's. We describe some general perturbative variational strategy useful to study concentration at points, and also at spheres in radially symmetric situations.
We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches
Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution
Some reaction-diffusion models describing the cell polarity are proposed, where the system has two independent variables standing for the concentration of proteins in the membrane and the cytosol respectively. In this article we deal with such a polarity model consisting of one equation on a unit sphere and the other one in the ball inside the sphere. The two equations are coupled through a nonlinear boundary condition and the total mass is conserved. We investigate the linearized stability of a constant steady state and provide conditions under which a Turing type instability takes place, namely, the constant state is stable against spatially uniform perturbations on the sphere for all choices of diffusion rates, while unstable against nonuniform perturbations on the sphere as the diffusion coefficient of the equation on the sphere becomes small relative to the one in the ball.
We study the following Neumann problem in one dimension,
We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as
We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.
We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which
In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.
When the Ohta-Kawasaki theory for diblock copolymers is applied to a bounded domain with the Neumann boundary condition, one faces the possibility of micro-domain interfaces intersecting the system boundary. In a particular parameter range, there exist stationary assemblies, stable in some sense, that consist of both perturbed discs in the interior of the system and perturbed half discs attached to the boundary of the system. The circular arcs of the half discs meet the system boundary perpendicularly. The number of the interior discs and the number of the boundary half discs are arbitrarily prescribed and their radii are asymptotically the same. The locations of these discs and half discs are determined by the minimization of a function related to the Green's function of the Laplace operator with the Neumann boundary condition. Numerical calculations based on the theoretical findings show that boundary half discs help lower the energy of stationary assemblies.
For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [
We look for solutions
The number of clines (i.e., nonconstant equilibria) maintained by viability selection, migration, and partial global panmixia in a step-environment with a geographical barrier is investigated. Our results extend the results of T. Nagylaki (2016, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol. 109) from the no dominance case to arbitrary dominance and to various other selection functions. Unexpectedly, besides the usual monotone clines, we discover nonmonotone clines with both equal and unequal limits at
We consider the degenerate haptotaxis system
endowed with no-flux boundary conditions in a bounded open interval
We now prove that under the additional restriction
Understanding the effects of hurricanes and other large storms on ecological communities and the post-event recovery in these communities can guide management and ecosystem restoration. This is particularly important for communities impacted by invasive species, as the hurricane may affect control efforts. Here we consider the effect of a hurricane on tree communities in southern Florida that has been invaded by Melaleuca quinquevervia (melaleuca), an invasive Australian tree. Biological control agents were introduced starting in the 1990s and are reducing melaleuca in habitats where they are established. We used size-structured matrix modeling as a tool to project the continued possible additional effects of a hurricane on a pure stand of melaleuca that already had some level of biological control. The model results indicate that biological control could suppress or eliminate melaleuca within decades. A hurricane that does severe damage to the stand may accelerate the trend toward elimination of melaleuca with both strong and moderate biological control. However, if the biological control is weak, the stand is resilient to all but extremely severe hurricane damage. Although only a pure melaleuca stand was simulated in this study, other plants, such as natives, are likely to accelerate the decline of melaleuca due to competition. Our model provides a new tool to simulate post-hurricanes effect on invasive species and highlights the essential role that biological control has played on invasive species management.
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