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In this paper, we investigate a two-species competition model with seasonal succession and different harvesting strategies. We aim to comprehend the impact of the proportion of good season on species competition outcomes and harvesting practices. For the case of weak harvest, we establish proportion thresholds for the good season for the species. Based on these threshold settings, we determine conditions for the existence and uniqueness of two semi-trivial periodic solutions and the existence, uniqueness and stability of positive periodic solutions under certain conditions. Numerical simulations and brief discussions are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study a perturbation of the Cahn–Hilliard equation with nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed in [5]. We prove the existence, regularity and uniqueness of solutions, as well as (strong) separation properties of the solutions from the pure states, also in three space dimensions. We finally prove the convergence to the Cahn–Hilliard equation, on finite time intervals. [ABSTRACT FROM AUTHOR]

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C ∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm. [ABSTRACT FROM AUTHOR]

In this paper, I consider a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the center-stable and center unstable manifolds, one gets the existence of a positive traveling wave connecting the two constant steady states (0 , 0) and (b , α b β) with a small wave speed ϵc. In addition, the traveling wave is monotone for b ≥ 1 and is not monotone for 0 < b < 1. Moreover, by the spectral analysis it shows that the above traveling wave is spectrally unstable in some exponentially weighted spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

In this paper, a nonlocal reaction-diffusion equation modeling the growth of phytoplankton species with changing-sign advection in a vertical water column is investigated, where the species depends solely on light for its metabolism. We mainly study the concentration phenomenon of the phytoplankton with large advection amplitude and small diffusion rate. Firstly, we study the threshold-type dynamics of the population by critical death rate d ⁎. Secondly, we examine the concentration phenomenon with large advection amplitude and small diffusion in two cases: (i) the advection function h (x) changes sign only once from positive to negative in water column [ 0 , 1 ]. We find that the phytoplankton will concentrate at certain critical point with large advection amplitude and small diffusion; (ii) the advection function h (x) < 0 in [ 0 , κ) with ∫ 0 x h (s) a (s) d s < 0 for all x ∈ (0 , 1 ] , the phytoplankton will concentrate at the surface of water column with large advection amplitude and small diffusion. We also investigate the limiting distribution of phytoplankton as diffusion rate D → + ∞ : the phytoplankton tends to even distribution in water column. [ABSTRACT FROM AUTHOR]

This paper focuses on the effects of environmental improvement and worsening on the spread and invasion of populations with birth pulse. We propose an impulsive reaction-diffusion model with a shifting environment to describe the dynamics of species with distinct reproduction stage and dispersal stage. First, the impulsive reaction-diffusion model is reduced to a discrete-time recursive system defined by a discrete map. Next, with the aid of the appropriate test function and comparison principle, we obtain some sufficient conditions on the nonexistence and uniqueness of nontrivial fixed points of the discrete map. This, combined with the abstract theory of spatially non-translation dynamical systems, enables us to establish the existence of traveling wave solutions and the asymptotic propagation properties of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study the propagation dynamics for a class of integro-difference equations with a shifting habitat. We first use the moving coordinates to transform such an equation to an integro-difference equation with a new kernel function containing the shifting speed c. In two directions of the spatial variable, the resulting equation has two limiting equations with spatial translation invariance. Under the hypothesis that each of these two limiting equations has both leftward and rightward spreading speeds, we establish the spreading properties of solutions and the existence of nontrivial forced waves for the original equation by appealing to the abstract theory of nonmonotone semiflows with asymptotic translation invariance. Further, we prove the stability and uniqueness of forced waves under appropriate conditions. [ABSTRACT FROM AUTHOR]

This paper investigates a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity (⋆) { u t = D 1 u x x − χ 1 (u k 1 v x) x + μ 1 u (λ 1 − u + a 1 v) , v t = D 2 v x x + χ 2 (v k 2 u x) x + μ 2 v (λ 2 − v − a 2 u) , under homogeneous boundary conditions of Neumann type for u and v , in an open bounded interval Ω ⊂ R , where D i , χ i , λ i , a i > 0 and μ i ≥ 0 for i ∈ { 1 , 2 }. In Tao and Winkler (2021) [1] ; (2022) [2] , Tao and Winkler studied the system (⋆) with k 1 = k 2 = 1 and obtained the global existence and asymptotic behavior. We study the system (⋆) with k 1 , k 2 ≠ 1 and prove that: • If k 1 ∈ (1 , 40 31) and k 2 ∈ [ 1 4 , 12 − 9 k 1 8 − 5 k 1 ) , then the model (⋆) possesses global weak solutions for arbitrarily large positive initial data in H 1 (Ω). • If k 1 ∈ (1 , 11 10) and k 2 ∈ [ 2 3 , 12 − 9 k 1 8 − 5 k 1 ) , then the global weak solution (u , v) of (⋆) with μ 1 = μ 2 = 0 stabilizes toward homogeneous equilibria (1 | Ω | ∫ Ω u 0 , 1 | Ω | ∫ Ω v 0). • If k 1 ∈ (1 , 9 8) and k 2 ∈ [ 1 2 , 12 − 9 k 1 8 − 5 k 1 ) , then the global weak solution (u , v) of (⋆) with μ 1 , μ 2 > 0 converges to constant stable steady state (λ 1 , 0) provided that λ 2 ≤ a 2 λ 1 and the tactic sensitivities χ 1 and χ 2 are suitably small. [ABSTRACT FROM AUTHOR]

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality | div (A (x , t) ∇ u) − u t | ≤ M | x | 2 | u | , where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator. [ABSTRACT FROM AUTHOR]

The paper studies the large-time behavior of solutions to the Robin problem for PDEs with critical nonlinearities. For the considered problems, nonexistence results are obtained, which complements the interesting recent results by Ikeda et al. (2020) [7] , where critical cases were left open. Moreover, our results provide partially answers to some other open questions previously posed by Zhang (2001) [21] and Jleli and Samet (2019) [13]. [ABSTRACT FROM AUTHOR]

To connect topological entropy with rotation theory, Botelho [2] introduced the notion of topological rotational entropy for annulus maps which are homotopic to the identity and later Geller and Misiurewicz [5] showed that it indeed vanished. In this paper, we generalize the notion of rotational entropy to any torus map, give its calculation formula and hence show that it is a homotopy invariant. We also introduce a notion of measure-theoretic rotational entropy and show that it is always less than or equal to the rotational entropy. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the transonic shock solutions in divergent nozzles as the expanding angle goes to zero. It is well-known that there exist infinite shock solutions for steady 1-D flows in a flat nozzle with the position of the shock front being arbitrary, while there exists a unique shock solution in an divergent nozzle as the pressure at the exit is given within an appropriate interval. By analyzing the asymptotic behavior of the shock solutions as the expanding goes to zero, we are also trying to figure out a criterion which may be used to select the physical one among all shock solutions in the flat nozzle. It finally turns out that the limit shock solution as the expanding angle goes to zero strongly depends on the asymptotic behavior of the receiver pressure, which is assumed to be a function of the expanding angle, imposed at the exit. In particular, as the Mach number of the flow at the entrance is larger than γ + 3 2 , the function of the receiver pressure can be set identical to the value of the pressure behind the shock front for the flat nozzle. Then the limit shock solution may be considered as the physical one for the flat nozzle, and the limit position of the shock front being the admissible position of the shock front. To show these results, one of the key steps is to establish quantitative estimates for the position of the shock front for the given pressure at the exit. To this end, a free boundary problem for the linearized Euler system will be proposed which gives an approximation of the shock solution, then a nonlinear iteration scheme could be carried out to approach the shock solution. Moreover, the error estimates between the exact shock solution and the approximation are also established, which give quantitative information on the position of the shock front. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis system with power-type singular sensitivities: u t = Δ u − χ ∇ ⋅ (u v α ∇ v) and v t = Δ v − v + u in a bounded domain Ω ⊂ R 2 , with χ > 0 and α > 0. It is notable that, whenever α = 0 , this system reduces to the minimal Keller-Segel model; the persistent Dirac-type singularities are known to occur at least in radial parabolic-elliptic setting. In this work, our first result shows that for any χ > 0 , if either α ≥ 1 4 , or α ∈ (0 , 1 4) and ‖ u 0 ‖ L 1 is small appropriately, then the corresponding initial-boundary value problem possesses a global generalized solution (u , v) with the property that u belongs to L 1 (Ω × (0 , T)) for any T > 0 , which further implies that the general sensitivities 1 v α , with α ≥ 1 4 , can rule out the emergence of Dirac-type singularities. Our second result indicates that, if α ∈ (0 , 1 2) and ‖ u 0 ‖ L 1 is small properly, then such global generalized solution becomes bounded and smooth at least eventually, and approaches the spatial equilibria in the large time limit. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary solutions to the one-dimensional full (non-isentropic) compressible Navier-Stokes-Korteweg equations with far field states and boundary data in the half line. When the fluids enter into and blow out the region through the boundary, respectively, the unique existence of the stationary solutions to the one-dimensional full compressible Navier-Stokes-Korteweg equations in the half line is shown provided that the boundary strength is small enough. Moreover, we also give the spatial decay rates of the stationary solutions. The main ingredient of the proof is the manifold theory and the center manifold theorem that take the accurate analysis of the cubic characteristic equations. [ABSTRACT FROM AUTHOR]

In this paper we consider determining a minimal surface embedded in a Riemannian manifold Σ × R. We show that if Σ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine Σ up to an isometry. [ABSTRACT FROM AUTHOR]

In this paper, we obtain families of uniformly rotating vorticity in the unit disk with a sufficiently large angular velocity. The solution is either a small nearly-ellipse vortex patch highly concentrated near the origin or a 2 + 1 vorticity configuration where another two vortical domains are sufficiently close to the boundary of fluid domain. The vorticity components are not necessarily symmetric nor small in the same order. These solutions are constructed via a perturbational approach and exhibit an interesting multi-scale phenomenon in the planar vortex flows. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of propagation dynamics for a large class of nonautonomous cooperative reaction-diffusion systems in a time-periodic shifting environment. We first establish the spreading properties of solutions and the existence of forced time-periodic waves for such a system by appealing to the abstract theory developed for monotone semiflows with asymptotic translation invariance. Then we prove the uniqueness of the forced wave and its attractivity under appropriate conditions. Finally, we apply our analytical results to a reaction-diffusion-advection model of malaria transmission. [ABSTRACT FROM AUTHOR]

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

This paper introduces a new notion of dispersion for kinetic equations solely based on the conservation laws and independent of the specific type of interactions. We present new a-priori estimates for kinetic PDEs and improve the Bony-type functional approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights ϕ α (x) = | x | − α for some α > 0. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from the BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure p (ρ) = ρ γ , γ = 1 + 2 d from that kinetic equation. [ABSTRACT FROM AUTHOR]

The Hamiltonian systems considered in this paper are obtained by weakly coupling two systems having completely different behaviors. The first one satisfies the twist assumptions usually considered for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered lower and upper solutions. In the higher dimensional case, we also treat a coupling situation where the classical Hartman condition is assumed. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data. [ABSTRACT FROM AUTHOR]

This paper develops stability and stabilization for systems of fully coupled jump diffusions. Such systems frequently arise in numerous applications where each subsystem (component) is operated under the influence of other subsystems (components). It derives sufficient conditions under which the underlying system of coupled jump diffusions is stable. The results are then applied to investigate the stability of linearizable jump diffusions, fast-slow coupled jump diffusions. Moreover, weak stabilization of interacting systems and consensus of leader-following systems are examined. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fully nonlinear wave equations on the product space R 3 × T with quadratic nonlinearities and on R 2 × T with cubic nonlinearities, respectively. It is shown that for the small initial data satisfying some space-decay rates at infinity, these nonlinear equations admit global smooth solutions when the corresponding partial null conditions hold and while have almost global smooth solutions when the partial null conditions are violated. Our proof relies on the Fourier mode decomposition of the solutions with respect to the periodic direction, the efficient combinations of time-decay estimates for the solutions to the linear wave equations and the linear Klein-Gordon equations, and the global weighted energy estimates. In addition, an interesting auxiliary energy is introduced. As a byproduct, our results can be applied to the 4D irrotational compressible Euler equations of polytropic gases or Chaplygin gases on R 3 × T , the 3D relativistic membrane equation and the 3D nonlinear membrane equation on R 2 × T. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals T λ f (x) = p.v. ∫ R n Ω (y) | y | n − λ f (x − y) d y for any f ∈ L q (R n) , 1 < q < ∞ and 0 < λ < n , which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation (0.1) ∂ t θ + u ⋅ ∇ θ + κ Λ 2 β θ = 0 , (x , t) ∈ R 2 × R + , κ > 0 , where u = − ∇ ⊥ Λ − 2 + 2 α θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator T λ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [ b , T λ ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases. [ABSTRACT FROM AUTHOR]

This paper is concentrated upon the qualitative analysis of two specific cases of the compressible Euler equations with linear damping: self-balanced non-isentropic system and externally driven isentropic system. We consider initial-boundary value problems of the models on bounded domains in R d. Both systems are supplemented with the no-normal-flow boundary condition. Under smallness assumptions on the initial perturbation and/or external force, it is shown that non-trivial steady states associated with the initial-boundary value problems are asymptotically stable in the long run. [ABSTRACT FROM AUTHOR]

Releasing Wolbachia-infected mosquitoes to invade the wild mosquito population is a method of mosquito control. In this paper, a stochastic mosquito population model with Wolbachia invasion perturbed by environmental fluctuation is studied. Firstly, the well-posedness, positivity, and Markov-Feller property of the solution for this model are proved. Then a group of sharp threshold-type conditions is provided to characterize the long-term behavior of the model, which pinpoints the almost necessary and sufficient conditions for the persistence and extinction of Wolbachia-infected and uninfected mosquito populations. Our results indicate that even for a low initial Wolbachia infection frequency, a successful Wolbachia invasion into the wild mosquito population can be driven by stochastic environmental fluctuations. Finally, some numerical experiments are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of spreading properties and traveling wave solutions for a vector-host disease system, which models the invasion of vectors and hosts to a new habitat. Combining the uniform persistence idea from dynamical systems with the properties of the corresponding entire solutions, we investigate the propagation phenomena in two different cases: (1) fast susceptible vector; (2) slow susceptible vector when the disease spreads. It turns out that in the former case, the susceptible vector may spread faster than the infected vector and host under appropriate conditions, which leads to multi-front spreading with different speeds; while in the latter case, the infected vector and host always catch up with the susceptible vector, and they spread at the same speed. We further obtain the existence and nonexistence of traveling wave solutions connecting zero to the endemic equilibrium. We also conduct numerical simulations to illustrate our analytic results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spatial behavior of the strongly competing systems involving the square root of the Laplacian. The coexistence of minimal energy solutions is discussed and a mechanism to ensure coexistence is given. Moreover, in the case of two densities the global minimizer converges, as the interspecies interaction tends to infinity, to a spatially segregated distribution where the two densities coexist and solve a non-coupled variational problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n c ∇ c) + n c , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + (u ⋅ ∇) u = Δ u + ∇ P + n ∇ Φ , x ∈ Ω , t > 0 , subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain Ω ⊂ R 2 , where + n c in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach (1 | Ω | ∫ Ω (n 0 + c 0) , 0 , 0) as t → ∞. As compared to a precedent result obtained without cell proliferation, the complexity caused by + n c in the first equation requires some conditions on the initial data c 0 rather only on n 0 in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by + n c. [ABSTRACT FROM AUTHOR]

In this paper, we study the statistical property of Anosov systems on surface driven by an external force. By utilizing the Birkhoff cone method, we show that if the systems on surface satisfying the Anosov and topological mixing on fibers property, then the quenched random correlation for Hölder observables with respect to the unique random SRB measures decays exponentially. [ABSTRACT FROM AUTHOR]

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established. [ABSTRACT FROM AUTHOR]