In this paper singularly perturbed quadratic polynomial differential systems εẋ=Pε(x,y)=P(x,y,ε),ẏ=Qε(x,y)=Q(x,y,ε)with x,y∈R,ε⩾0 and (Pε,Qε)=1 for ε>0, are considered. We prove that there are 10 classes of equivalence for these systems. We describe the dynamics of these 10 classes on the Poincaré disc when ε=0. For ε>0, we present the possible local behavior of the solutions near of a finite and

This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka–Volterra. The optimality of these uniqueness theorems reveals the

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces Bp,rs(Rd), d≥2, with 11+d∕p and r∈[1,+∞], or s=1+d∕p and r=1. A key ingredient is the reformulation of the system via the so-called Elsässer variables. On the other hand, we give a rigorous justification of quasi-homogeneous

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.

In this paper, we investigate the controllability for a semilinear heat equation with memory and internal control in a bounded domain of Rn. The semilinearity has gradient terms and is globally Lipschitzian. Since memory term does not allow applying Fabre’s unique continuation theorem directly, we made certain adjustments to the terms of the equation. Additionally, the proof of the controllability

In this paper, the optimal decay rates of solutions for three-dimensional incompressible Navier–Stokes equations with damping term uβ−1u are established. Bounds on the H.−s negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Based on the zero curvature equation as well as recursive operators, a new spectral problem and the associated multi-component Gerdjikov–Ivanov (GI) integrable hierarchy are studied. The bi-Hamiltonian structure of the multi-component GI hierarchy is obtained by the trace identity which shows that the multi-component GI hierarchy is integrable. In order to solve the multi-component GI system, a class

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, Mpc(n)

This work studies the global existence of weak solutions to a doubly haptotactic cross-diffusion system modeling oncolytic viral therapy. The model was proposed to illustrate the coupled dynamics of oncolytic virus (OV) particles, extracellular matrix (ECM), uninfected cancer cells and OV-infected cancer cells in the process of the oncolytic viral therapy. It is shown that the model has a global weak

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L2(0,T;BMO−1). We also prove that

We consider in this article the damped wave equation in the scale-invariant case with combined two nonlinearities as source term, namely |ut|p+|u|q, and with small initial data. Owing to a better understanding of the influence of the damping term (μ1+tut) in the global dynamics of the solution, we obtain a new interval for μ that we conjecture to be closer to optimality, or probably optimal, and, thus

In this paper we study an identification problem related to a nonlinear parabolic system. This system, called the nonlinear single phase mud filtrate invasion model, arises in the study of the mud filtrate invasion phenomenon and is presented in Boaca and Boaca (2018). Our objective is to determine the minimal value of the oil reservoir permeability starting from the observed values of the mud filtrate

This paper is devoted to the unique global admissible conservative solutions of the periodic dispersive Hunter–Saxton equation. Using the standard ordinary differential equation theory, we first get the global admissible conservative solutions of the periodic dispersive Hunter–Saxton equation. Then, given an admissible conservative solution u(t,x), an equation is introduced which singles out a unique

Seasonality is pervasive in nature and WNV is a complex disease which appears to be transmitted periodically. Lacking of vaccine and anti-virus treatment brings culling to be an effective strategy of controlling the spread of WNV. This paper proposes a piecewise smooth model of WNV transmission between mosquitoes and birds with periodic forcing by employing the threshold policy of culling mosquitoes

This paper presents a second-order asymptotic solution in the Lagrangian description for nonlinear partial standing wave in the finite water depth. The asymptotic solution that is uniformly valid satisfies the irrotationality condition and zero pressure at the free surface. In the Lagrangian approximation, the explicit nonlinear parametric equations for the particle trajectories are obtained. In particular

This paper is concerned with a nonlinear free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, set with a Robin boundary condition, in which, both nonnecrotic tumors and necrotic tumors are taken into consideration. The well-posedness and asymptotic behavior of solutions are studied. It is shown that there exist two thresholds, denoted by σ̃ and σ∗, on the surrounding

We analyze some configurations of the general chemotaxis predator–prey model with pursuit-evasion dynamics ∂tu−∇⋅(Fu(u)∇u)+∇⋅(Fp(u)∇p)=uF1(w)−F2(u)∂tw−∇⋅(Fw(w)∇w)−∇⋅(Fq(w)∇q)=wF3(w)−δuwin Ω×(0,T) with Neumann boundary condition and non-negative initial data, where p and q are the predator’s and the prey’s pheromone, respectively, modeled by parabolic or elliptic equations, and Ω⊂Rd, with d≥1, is a

The article is devoted to the mathematical analysis of a fluid–structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier–Stokes–Fourier system and the structure displacement is described by a structurally damped plate equation

We study the local well-posedness of the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation iut+Δu=x−bf(u),u(0)=u0∈Hs(Rn),where b>0 and f(u) is a nonlinear function that behaves like λ uσu with λ ∈ ℂ and σ>0. First, we obtain the local well-posedness result in Hs with 0≤s

In this paper we study quasi-neutral limit and the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck–Nernst–Poisson and Navier–Stokes equations. Different from other studies, we consider the physical case that the mobilities of the charges are different. For the generally smooth doping profile and for the ill-prepared initial data, under

In this paper, we consider the Cauchy problem for semilinear σ-evolution models with an exponential decay memory term. Concerning the corresponding linear Cauchy problem, we derive some regularity-loss-type estimates of solutions and generalized diffusion phenomena. Particularly, the obtained estimates for solutions are sharper than those in the previous paper (Liu and Ueda, 2020). Then, we determine

In this paper we consider a nonlinear class viscoelastic plate equation with a lower order by perturbation of p⃗(x,t)-Laplace operator of the form utt+Δ2u−Δp⃗(x,t)u+∫0tg(t−s)Δu(s)ds−ϵΔut+f(u)=0,(x,t)∈QT=Ω×(0,T), associated with initial and Dirichlet–Neumann boundary conditions. Under suitable conditions on g, f and the variable exponent of the p⃗(x,t)-Laplace operator, we prove a blow up in finite

In this paper we study a simplified transient energy-transport model in semiconductors with a general conductivity and the Dirichlet boundary conditions on an interval. By using a new iterative scheme, we prove the global existence and uniqueness of strong solutions provided that the variation of the temperature is small. Also, the existence and stability of stationary solutions are proved if the temperature

This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, we prove that the system has a unique

We study the existence of entire solutions of two-species diffusive monostable Lotka–Volterra competitive systems. Our results cover both cases of monostable with stable coexistence equilibrium and monostable with competition–extinction equilibria. We prove the existence of entire solutions describing different types of invasion scenarios of an empty territory by two competitors as the time variable

The global-in-time existence of bounded weak solutions to the Maxwell–Stefan–Fourier equations in Fick–Onsager form is proved. The model consists of the mass balance equations for the partial mass densities and the energy balance equation for the total energy. The diffusion and heat fluxes depend linearly on the gradients of the thermo-chemical potentials and the gradient of the temperature and include

In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal

In a bounded domain Ω⊂R3, we are concerned with the evolution system (⋆)nt+u⋅∇n=Δn−∇⋅(nf(|∇c|2)∇c),ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0,coupling the incompressible Navier–Stokes equations to a class of flux-limited Keller–Segel systems which has received noticeable attention in the recent biomathematical literature. When considered without such fluid interaction, no-flux boundary value problems

In this paper, we prove global existence of smooth solutions to the two dimensional FENE (Finite Extensible Nonlinear Elastic) systems with an extra concentrated center-of-mass diffusion.

Recently, Zhu et al. (2020) proposed a kind of rotation-Camassa–Holm equation. In this paper, we study the question of nonexistence of periodic peaked traveling wave solution for rotation-Camassa–Holm equation. Indeed, rotation-Camassa–Holm equation has no nontrivial periodic Camassa–Holm peaked solution unlike Camassa–Holm equation, modified Camassa–Holm equation, Novikov equation.

This paper is concerned with a diffusive predator–prey system with multiple Allee effects induced by fear factors subject to Neumann boundary conditions. Firstly, we introduce a priori estimates of non-negative and non-trivial solutions. Moreover, nonexistence of non-constant positive solutions are shown for certain parameters ranges. Secondly, we study the stability of non-negative constant solutions

Most diseases have multiple pathogenic strains, which may impose difficulty in combatting the disease and lead to rich dynamics. However, their dynamical properties are not well understood. For this purpose, we formulate and analyze a two-strain SIS epidemic model with a competing mechanism and general infection force on complex networks. We derive the basic reproduction number and introduce the invasion

Let the three-dimensional differential system defined by the jerk equation x⃛=−aẍ+xẋ2−x3−bx+cẋ, with a,b,c∈R. When a=b=0 and c<0 the equilibrium point localized at the origin of coordinates is a zero-Hopf equilibrium. We analyse the zero-Hopf bifurcation occurring at this singular point after persuading a quadratic perturbation of the coefficients. Particularly, by using averaging theory of second

The purpose of this paper is to study radially symmetric solutions of a parabolic–elliptic chemotaxis system (0.1)ut=Δu−∇⋅(f(u)∇v)inBR×(0,+∞),0=Δv−μ(t)+g(u)inBR×(0,+∞)subject to homogeneous Neumann boundary conditions, where BR=BR(0)⊂Rn with n⩾1, R>0 and μ(t)=1|BR|∫BRg(u(x,t))dx denotes the time-dependent spatial mean of g(u). The chemosensitivity f and nonlinear signal production g are suitably regular

In this paper, we consider an angiogenesis free boundary tumor model with external periodic nutrient supply. The model is in the form of a reaction–diffusion equation describing the concentration of nutrients σ and an elliptic equation describing the distribution of the internal pressure p. The vasculature provides a periodic supply of nutrients to the tumor at a rate proportional to β, so that ∂σ∂n+β(σ−ϕ(t))=0

Using olivine LiFePO4 as a model system, we study the existence of global solutions to a phase-field model with elasticity energy for Lithium-Ion batteries, which consists of a linear elasticity sub-system and nonlinear evolution equations for the order parameter and the lithium concentration. This model can be described the evolving microstructure for electrochemically induced phase transitions in

We study the large-time asymptotic behavior of solutions toward the weak rarefaction wave of Navier–Stokes equations coupling with the Maxwell equations under small perturbations of initial data and also under the dielectric constant suitably small. The result is proved based on elementary L2 energy methods.

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions

We study a parabolic Lotka–Volterra type equation that describes the evolution of a population structured by a phenotypic trait, under the effects of mutations and competition for resources modelled by a nonlocal feedback. The limit of small mutations is characterized by a Hamilton–Jacobi equation with constraint that describes the concentration of the population on some traits. This result was already

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier–Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn–Hilliard equations. We show global in time existence of weak solutions

The stability of the null solution of different systems of differential equations describing the motion of 1-D coupled nonlinear oscillators is discussed. Under certain assumptions we derive some stability results. Specifically, in the case of coupled damped oscillators we obtain asymptotic stability of the null solution (see Theorem 3.1, Example 3.1, and Fig. 2), while in the case of partial lack

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.

A single species population model subject to additive Allee effects, due to predation satiation is proposed and analyzed by considering a continuous strategy evolutionary game theory (EGT). Moreover, we consider the aposematic behavior of the population. We assume a single trait that affects aposematic behavior and saturation constant, following the Gaussian distribution. First, we analyze our single

In this paper, a cholera infection model with vaccination is investigated, in which hyperinfectious and hypoinfectious vibrios, both human-to-human and environment-to-human transmission pathways, and waning vaccine-induced immunity are considered. The basic reproduction number is calculated and verified to be a threshold determining the global dynamics of the model. In addition, an application is demonstrated

In this paper, we investigate some nonlocal diffusion problems with free boundaries. We first give the existence and uniqueness of local solution by the ODE basic theory and the contraction mapping principle. Then we provide a complete classification for the global existence and finite time blow-up of solutions. Moreover, estimates of blow-up rate and blow-up time are also obtained for the blow-up

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As

This paper is concerned with the existence, monotonicity, asymptotic behavior and uniqueness of traveling wave solutions for a three-species competitive–cooperative system with nonlocal dispersal and bistable dynamics. By considering a related truncated problem, we first establish the existence and strict monotonicity of traveling waves by means of a limiting argument and a comparative lemma. Then

We study the Γ-convergence of the functionals Fn(u):=||f(⋅,u(⋅),Du(⋅))||pn(⋅) and Fn(u):=∫Ω1pn(x)fpn(x)(x,u(x),Du(x))dx defined on X∈{L1(Ω,Rd),L∞(Ω,Rd),C(Ω,Rd)} (endowed with their usual norms) with effective domain the Sobolev space W1,pn(⋅)(Ω,Rd). Here Ω⊆RN is a bounded open set, N,d≥1 and the measurable functions pn:Ω→[1,+∞) satisfy the conditions ess supΩpn≤βess infΩpn<+∞ for a fixed constant β>1

In this paper, we investigate non-isothermal one-dimensional model of capillary compressible fluids as derived by Slemrod (1984) and Dunn and Serrin (1985). We establish the global existence, uniqueness and exponential stability of strong solutions in H+i(i=1,2,4) for the one-dimensional Navier–Stokes equations with capillarity, which implies the existence and exponential stability of the nonlinear

The limit cycle problem of planar piecewise linear refracting systems has been solved except the focus–focus (FF) type. Here we concern this remaining FF type, and prove that such systems have at most one limit cycle and have 14 topologically different global phase portraits.

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic

We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast–slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without

We consider a nonlinear Robin problem driven by the sum of p-Laplacian and q-Laplacian (i.e. the (p,q)-equation). In the reaction there are competing effects of a singular term and a parametric perturbation λf(z,x), which is Carathéodory and (p−1)-superlinear at x∈R, without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools, together with truncation and comparison techniques,

In this paper, we are concerned with existence/non-existence of traveling waves of a diffusive SIR epidemic model with general incidence rate of the form of f(S)g(I) and infinitely distributed latency but without demography. We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which

This paper presents a new nonlinear reaction–diffusion–convection system coupled with a system of ordinary differential equations that models a combustion front in a multilayer porous medium. The model includes heat transfer between the layers and heat loss to the external environment. A few assumptions are made to simplify the model, such as incompressibility; then, the unknowns are determined to

We are concerned with global well-posedness of strong solutions to the Cauchy problem for compressible non-isothermal nematic liquid crystal flows with vacuum as far field density in R3. By using energy method, we establish the global existence and uniqueness of strong solutions provided that the quantity‖ρ0‖L∞+‖∇d0‖L3 is suitably small and the viscosity coefficients verify 3μ>λ. In particular, the

This paper is dedicated to studying the periodic solutions of non-autonomous second order Hamiltonian systems. By virtue of control functions and the minimax methods in critical point theory, we propose a unified approach when the nonlinearity is unbounded while grows no more than |x| at infinity. We establish some new existence theorems on periodic solutions, which unify and generalize many previously

This paper is concerned with the global existence and large time behavior of strong solutions to the Cauchy problem of one-dimensional compressible isentropic micropolar fluid model with density-dependent viscosity and microviscosity coefficients, where the far-fields of the initial data are prescribed to be different. The pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α

In most fluid dynamics problems, the governing equations are nonlinear because of the presence of convective terms. Nevertheless, existence of solutions can be shown by direct sum provided one identifies, in the relevant Banach space of solutions, particular subspaces which are invariant under time evolution. As an example, we consider classical convection problems and show how symmetry arguments can