Nonconstant Solutions Research Articles

Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity

. We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in ℝ. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels K which are convex and kernels for which K(τ 1∕2) is a completely monotonic function of τ. This last new class arose in our previous work on nonlocal Delaunay surfaces in ℝ n . Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle 𝕊 1 established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in C β and when 2 s + β < 1 (2s being the order of the operator), the solution is not always C 2 s + β − ϵ for all ϵ > 0.

ABOUT THE CORE STRUCTURE OF THE SCHWARTZ PROBLEM FOR FIRST-ORDER ELLIPTIC SYSTEMS ON A PLANE

The Schwarz problem for 𝐽-analytic functions in an arbitrary ellipse is considered. The matrix 𝐽 is assumed to be two-dimensional, with different eigenvalues lying above the real axis. An example of a nonconstant solution of the homogeneous Schwarz problem in the form of a vector-polynomial of degree three is given. The numerical parameter 𝑙 of the matrix 𝐽, which is expressed through its eigenvectors, is introduced. After that, one relation derived earlier by the author is analyzed. Based on this analysis, a method for computing the dimension and structure of the kernel of the Schwarz problem in an arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed through the ellipse parameters, the eigenvalues of the matrix 𝐽 and the parameter 𝑙 are obtained. Examples of one-dimensional and trivial kernels are given.

Barotropic equations of state in 4D Einstein-Maxwell-Gauss-Bonnet stellar distributions

We investigate the role of a linear barotropic equation of state (p=γρ) on the structure of charged stars under higher curvature effects induced by the Gauss-Bonnet invariants in 4 dimensions. Assuming a constant spatially directed potential which gives isothermal behavior in the standard theory, the master equation is solved in terms of hypergeometric functions but a viable model could not be constructed. Setting the temporal potential to a constant, comparable to the defective Einstein static universe, interestingly admits nontrivial nonconstant exact solutions due to the higher curvature terms unlike in general relativity. Next the existence of a one-parameter group of conformal motions in the spacetime geometry was investigated. The master differential equation is solvable exactly in implicit form and explicit solutions are found for special cases. For the case of a stiff fluid p=ρ a stellar model with pleasing physical attributes is found. When the potential is assumed to vary linearly with the radius, an exact incoherent radiation model p=13ρ emerges. The physical properties of both these solutions are analyzed comprehensively with the aid of graphical plots in conjunction with suitably defined parameter spaces. It was found that both exact models passed elementary astrophysical tests for physical plausibility.

Arithmetic Ramsey theory over the primes

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$ , where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

Dynamics of a single population model with memory and nonlinear boundary condition

In this paper, a heterogeneous single population model with memory effect and nonlinear boundary condition is investigated. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. Through the distribution of the eigenvalues and bifurcation analysis, sufficient conditions for the occurrence of the Hopf bifurcation associated with one spatially nonconstant positive steady state are obtained. Results about the stability associated with the other one are also yielded. It is found that with the interaction of memory delay, spatial heterogeneity and nonlinear boundary condition, the Hopf bifurcation will happen in such diffusive model. To be specific, the memory delay could induce a single stability switch from stability to instability. As contrast to the diffusive model only with memory effect, the stability switch is independent of memory delay. The obtained results are applied to some specific models, from which we can find that the theoretical analysis is verified and the results enrich the existing ones.

The study of non-constant steady states and pattern formation for an interacting population model in a spatial environment

This manuscript accounts for an investigation of the complex dynamics of a spatial model for interacting populations. We discuss the existence and boundedness of solutions for the proposed spatio-temporal system. The global stability of the co-existing steady state of the proposed system is analyzed with the help of a suitable Lyapunov function. We provide results on the existence and non-existence of positive non-constant solutions of the model. The priori estimate for the positive steady state is obtained for the nonexistence of the non-constant positive steady state by using the maximum principle. The existence of a non-constant positive steady state is studied with the help of Leray–Schauder degree theory. The stability and Hopf bifurcation are briefly revisited for the co-existing steady state in the corresponding temporal model, where a bubble-like structure is observed. The onset of Hopf bifurcation has been analyzed, and different conditions for the formation of the Turing pattern have been established through diffusion-driven instability analysis. Numerical simulations are performed in detail to figure out the effects of saturated harvesting on Turing patterns. The Turing as well as non-Turing patterns in their respective domains are also examined. Finally, the criteria of Turing–Hopf bifurcation is briefly demonstrated with relevant numerical examples and corresponding plots that give a better illustration of this work.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Effects of extra resource and harvesting on the pattern formation for a predation system

We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.

Hopf bifurcation and stability analysis for a delayed equation with φ-Laplacian

A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, implies the non-existence of nonconstant periodic solutions. Our conditions improve previous results known in the literature for the standard case $\varphi(x)=x$.

Complex Dynamics of a Simple Tumor-Immune Model with Tumor Malignancy

One main feature of a malignant tumor is its uncontrolled growth. In this paper, we propose a simple tumor-immune model to study the progressive characteristics of malignant and benign tumors, where the anti-tumor immunity can be described by the Michaelis–Menten function or the mass action law. The model includes only two state variables for the tumor cells and the effector cells representing the immune system. Three quantities with clear biological meanings are given to determine the asymptotic states of the tumor progression. Moreover, differences in asymptotic states between the two anti-tumor immunity descriptions are drawn. Differently from existing simple models, on the one hand, the model exhibits rich dynamical behaviors including super-critical and sub-critical Bogdanov–Takens bifurcations (consisting of Hopf bifurcation, saddle–node bifurcation, and homoclinic bifurcation) and saddle–node bifurcation of nonconstant periodic solutions (leading to the appearance of two periodic orbits) as the parameters vary; on the other hand, the malignant feature, dormancy, and immune escape of the tumor are revealed with numerical simulations. Furthermore, from the perspective of qualitative analysis and numerical simulations, how the obtained results can be applied to the treatment and control of tumors is illustrated.

Minimum Principles for Sturm–Liouville Inequalities and Applications

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

Non-constant periodic solutions of the Ricker model with periodic parameters

In this paper, we study the Ricker model x n + 1 = x n exp ⁡ [ r n ( 1 − x n ) ] , n = 0 , 1 , 2 … , ( ∗ ) where x 0 ≥ 0 , and { r n } n = 0 ∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that ( ∗ ) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results.

All meromorphic solutions of the autonomous Schwarzian differential equations

AbstractThis paper considers a specific autonomous Schwarzian differential equation given by and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case.

On a diffusive epidemic model with the tendency to move away from the infectious diseases

This paper is purported to investigate an epidemic system with the tendency of the susceptible to move away from the infectious diseases in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without cross-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are considered for this epidemic system. The results show that there exists spatiotemporal pattern formation when the tendency of susceptible individuals is strong to move away from the infectious diseases with 1<R0<1+min⁡{dSd,βσ1Aσ2}, while there does not exist spatiotemporal pattern formation when the self pressure is big enough with a weak tendency of the susceptible individuals to keep away from the infectious individual. Moreover, there exists positive solution bifurcation from the semi-trivial solution when the basic reproduction number is equal to one, which implies that endemic disease exists locally.

Steady States of a Diffusive Population-Toxicant Model with Negative Toxicant-Taxis

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4).

Dynamics of neural fields with exponential temporal kernel.

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

Monotonic convergence of positive radial solutions for general quasilinear elliptic systems

We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form Δpu=c1|x|m1⋅g1v⋅|∇u|αin Rn,Δpv=c2|x|m2⋅g2v⋅g3|∇u|in Rn, where Δp denotes the p-Laplace operator, p > 1, n⩾2 , c1,c2>0 and m1,m2,α⩾0 . For a general class of functions gj which grow polynomially, we show that every non-constant positive radial solution (u, v) asymptotically approaches (u0,v0)=(Cλ|x|λ,Cμ|x|μ) for some parameters λ,μ,Cλ,Cμ>0 . In fact, the convergence is monotonic in the sense that both u/u0 and v/v0 are decreasing. We also obtain similar results for more general systems.

Some qualitative analyses on a vegetation-water model with cross-diffusion and internal competition

This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions. First, we found that the equilibrium with small vegetation density is always unstable, and if the cross-diffusion coefficient is suitably large, the equilibrium with relatively large vegetation density loses its stability, and Turing instability occurs. A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations. Moreover, some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail. Space decomposition and the implicit function theorem are used for double eigenvalues. In particular, the global continuation is obtained, and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large. Finally, numerical simulations are provided to prove and supplement theoretic research results, and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed, where the transition appears: gap [Formula: see text] stripe [Formula: see text] spot.

Steady-state bifurcations and patterns formation in a diffusive toxic-phytoplankton–zooplankton model

In this paper, we study a diffusive toxic-phytoplankton–zooplankton model with prey-taxis under Neumann boundary condition. By analyzing the characteristic equation, we discuss the local stability of the positive constant solutions and show the repulsive prey-taxis is the key factor that destabilizes the solutions. By means of the abstract bifurcation theorem, we investigate the existence of non-constant positive steady-state solutions bifurcating from the constant coexistence equilibrium. Furthermore, we obtain the criterion for the stability of the branching solutions near the bifurcation point. Numerical simulations support our theoretical results, together with some interesting phenomena, stable heterogeneous periodic solutions emerge when prey-tactic sensitivity coefficient is well below the critical value, and zooplankton populations present extinction and continued transitions as habitat size increases.