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

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small

This paper studies the blow-up solution and its blow-up rate near the traveling waves of the second-order Camassa–Holm equation. The sufficient condition for the existence of blow-up solution is obtained by a rather ingenious method. Applying the extended pseudo-conformal transformation, an equivalent proposition of the solution breaking in finite time near the traveling waves is constructed. The relation

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with

This paper is mainly concerned with the forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats. We first show that there exist two positive numbers c1∗ and c2∗ such that the system admits a forced wave provided that the forcing speed c∈(−c2∗,c1∗) by the iterative techniques combining with some known results for the forced moving KPP equations

We study the conditional regularity for the incompressible Navier–Stokes equations in terms of one directional derivative of the velocity field. We show that if u is a weak solution in the whole three dimensional space and ∂3u∈Lp(0,T;Lq(R3)), T>0, where 2∕p+3∕q=2 and q∈(3,19∕6], then u is regular on (0,T], crossing so for the first time the barrier q=3. The proof is based on a suitable combination

A solidification process for a semi-infinite material is presented through a non-linear two-phase unidimensional Stefan problem, where a convective boundary condition is imposed at the fixed face x=0. The volumetric heat capacity and the thermal conductivity are non-linear functions of the temperature in both solid and liquid phases and they verify a Storm-type relation. A certain inequality on the

We study the existence and large-time behavior of globally defined entropy solutions to one dimensional bipolar hydrodynamic model for semiconductors on bounded interval. By taking advantage of vanishing viscosity method and compensated compactness framework, we construct approximate solutions uniformly bounded in both space x and time t. Compared with the paper, Huang and Li (2009), a restrictive

A two-dimensional Kolmogorov system depending on two independent parameters and having a degenerate condition is studied in this work. We obtain local analytical properties of the system when the parameters vary in a sufficiently small neighborhood of the origin. The behavior of the system is described by bifurcation diagrams. Applications of Kolmogorov systems can be found particularly in modeling

The paper studies the existence and uniqueness of a positive periodic solution to the equation u′′=p(t)u−q(t,u), where p∈L([0,ω]) and q:[0,ω]×R→R is a Carathéodory function sub-linear in the second argument. The general results are applied to some particular cases such as the equation u′′=p(t)u−h(t)sinu with p,h∈L([0,ω]). This equation appears when approximating non-linearities in the equation of motion

We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci’s extremal operators. This is done through a dynamic programming principle derived from Denis et al. (2010). The formula can be seen as a nonlinear extension of the Feynman–Kac formula.

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate

We construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns

In this paper, we are concerned with the following prey-taxis system with liquid surrounding describing by the incompressible Navier–Stokes equations nt+u⋅∇n=Δn−∇⋅(χn∇c)+γnF(c)−θn−αn2,x∈Ω,t>0,ct+u⋅∇c=DΔc−nF(c)+f(c),x∈Ω,t>0,ut+u⋅∇u=Δu+∇P+n∇ϕ,∇⋅u=0,x∈Ω,t>0,∂n∂ν=∂c∂ν=u=0,x∈∂Ω,t>0,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,where Ω⊂R2 is a bounded domain with smooth boundary. Using the Lp-energy estimate

With successive coupling of two taxis mechanisms and with density-dependent taxis sensitivities, the producer–scrounger system ut=Δu−∇⋅(u(u+1)β−1∇w),vt=Δv−∇⋅(v(v+1)α−1∇u),wt=Δw−(u+v)w−λw+g(x,t),is considered in a bounded smooth domain Ω⊂Rn ( n≥2), where α≤1, β≤1 and λ>0 are prescribed parameters, and g is a given C1-smooth function. It is shown that whenever β<1andα

Insect-borne diseases are diseases carried by insects affecting humans, animals or plants. They have the potential to generate massive outbreaks such as the Zika epidemic in 2015–2016 mostly distributed in the Americas, the Pacific and Southeast Asia, and the multi-foci outbreak caused by the bacterium Xylella fastidiosa in Europe in the 2010s. In this article, we propose and analyze the behavior of

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper

We shall investigate the large-time behavior of solutions to an out-flow problem in one-dimensional half space for the Navier–Stokes–Korteweg equations which models compressible fluids with internal capillarity. Applying the center manifold theory, we prove the existence of the stationary solution under a smallness condition on the boundary data and a proper relation between the capillary coefficient

In this paper, we are concerned with the local-in-time well-posedness of a fluid-kinetic model in which the BGK model with density dependent collision frequency is coupled with the inhomogeneous Navier–Stokes equation through drag forces. To the best knowledge of authors, this is the first result on the existence of local-in-time smooth solution for particle–fluid model with nonlinear inter-particle

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads

In this paper we study the Degn–Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behavior of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincaré–Bendixson theorem and the direct

This paper deals with a two-species chemotaxis-consumption system involving nonlinear diffusion and chemotaxis ut=Δ(γ1(w)u)+μ1u(1−u−a1v),x∈Ω,t>0,vt=Δ(γ2(w)v)+μ2v(1−a2u−v),x∈Ω,t>0,wt=Δw−(αu+βv)w,x∈Ω,t>0,in an arbitrary smooth bounded domain Ω⊂Rn(n=2,3) under homogeneous Neumann boundary conditions, where μi,ai,α,β are positive constants and the motility functions γi(w)∈C3([0,∞)), γi(w)>0, γi′(w)<0 for

Considering multiple effects of spatial factors (spatial heterogeneity and mobility) and temporal heterogeneity (the periodic administration of the drug) in HIV infection and treatment, we formulate a time periodic reaction–diffusion equation model to study the HIV viral dynamics. The basic reproduction number R0 is derived and the global threshold dynamics is explored in terms of R0. When R0<1, the

We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong

This paper is devoted to the study of limit cycles that can bifurcate of a perturbation of piecewise non-Hamiltonian systems with nonlinear switching manifold. We derive the first order Melnikov function to these systems. As application, the sharp upper bound of the number of bifurcated limit cycles of two concrete systems, whose switching manifolds are algebraic curves, is presented.

This paper deals with the two-species aerotaxis-Navier–Stokes equations with Lotka–Volterra competitive kinetics in a two dimensional domain Ω⊂R2. We consider the general chemotactic sensitivity functions and oxygen(chemical) consumption rate functions. We obtain the stabilization of the solution to the system. For the specific conditions on the chemotactic sensitivity and initial data, we obtain the

In this paper, we are interested in the following attraction–repulsion chemotaxis system: ut−Δu=−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈R2,t>0,vt+βv−Δv=αu,x∈R2,t>0,δw−Δw=γu,x∈R2,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈R2,where the parameters χ,ξ,α,β,γ,δ are positive and the initial data u0,v0 are nonnegative and u0⁄≡0. By a modified free energy functional, we establish the global existence of classical solutions when the repulsion

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint)

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) iut+Δ2u+λ|x|−b|u|αu=0, where λ=±1 and α, b>0. We show local and global well-posedness in Hs(RN) in the Hs-subcritical case, with s=0,2. Moreover, we prove a stability result in H2(RN), in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates

Pyroptosis is a highly inflammatory form of cell death, which uses intracellularly generated pores to destroy electrolyte homeostasis and perform cell death. Gasdermin D, the pore-forming effector protein of pyroptosis, plays a critical role in coordinating membrane lysis and the release of highly inflammatory molecules. Recently, necrosulfonamide as a direct chemical inhibitor of gasdermin D has been

In this paper, we investigate a Lotka–Volterra competition model with Danckwerts boundary conditions in a one-dimensional habitat where one species assumes pure random diffusion while another one undergoes mixed movement (both random and directed movements). We focus on the joint influence of advection rate, intrinsic growth rate and interspecific competition coefficient on the competition outcomes

We investigate the stability of the Rayleigh–Taylor (RT) problem for stratified viscoelastic fluids with internal surface tension. More precisely, under the stability condition Dis(ϑ,κ)<1, we prove the existence of unique strong solution with exponential decay in time for the (stratified) viscoelastic RT (VRT) problem with proper initial data in Lagrangian coordinates. This shows that a sufficiently

This paper deals with the 2D incompressible Navier–Stokes equations with density-dependent viscosity over bounded domains. The global existence of strong solutions is established in the vacuum cases, provided the assumption that the bound of density is suitably small, which is in sharp contrast to the recent work (Huang and Wang, 2014), where the smallness assumption on ‖∇μ(ρ0)‖Lq(q>2) is needed. Furthermore

This article is concerned about the synchronization problem of the bidirectionally coupled Kuramoto model. In this model, each oscillator only interacts with the oscillators with adjacent labeling numbers. Namely, the oscillator θi only interacts with θi+1 and θi−1. In real applications, this is a typical setting of concatenation connection. We first prove the global convergence of frequency synchronization

We study the homogeneous Dirichlet problem for a class of nonlocal singular parabolic equations ut−div|∇u|p[u]−2∇u=fin Ω×(0,T),where Ω⊂Rd, d≥2, is a smooth bounded domain, p[u]=p(l(u)) is a given function with values in the interval [p−,p+]⊂(1,2), and l(u)=∫Ω|u(x,t)|αdx, α∈[1,2], is a functional of the unknown solution. We find sufficient conditions for global or local in time solvability of the problem

This paper is concerned about the S–K–T competition model with spatially heterogeneous interactions, in which the two competing species are identical in all aspects except for their interspecific competitions. Under small variations of interspecific competitions, a good understanding of the dynamics and the structure of the set of steady states is gained by the Lyapunov–Schmidt procedure and detailed

A new kind of model based on double degenerate parabolic equation in nondivergence form is proposed for multiplicative noise removal. We first utilize the regularization method to illustrate the existence of the weak solution for the problem and the non-expansion of support of the solution is also provided. In the numerical aspect, experiments and comparisons with other denoising models are presented

In this paper, we consider an age-structured population model with diffusion. We first establish a comparison principle. We then apply the comparison principle to show the existence and uniqueness of solutions by constructing monotone sequences of weak upper and lower solutions. We also use the comparison principle to study the long-time behavior of the solution and present extinction and boundedness

The zero dissipation limit of the one-dimensional non-isentropic micropolar equations is studied in this paper. If the given rarefaction wave which connects to vacuum at one side, a sequence of solution to the micropolar equations can be constructed which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence

We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely

In this paper we study the chemotaxis–Stokes system with slow p-Laplacian diffusion and rotation: nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)⋅∇c), ct+u⋅∇c=Δc−nc, ut+∇P=Δu+n∇ϕ+f(x,t) and ∇⋅u=0 in a bounded domain Ω⊂R3 with p>2, subject to the Neumann–Neumann–Dirichlet boundary conditions, where ϕ:Ω̄→R, f:Ω̄×[0,∞)→R3 and S:Ω̄×[0,∞)2→R3×3 are given sufficiently smooth functions with f bounded in Ω×(0,∞), |S(x

We provide sufficient conditions for the existence of periodic solutions of the Lorentz force equation, which models the motion of a charged particle under the action of an electromagnetic fields. The basic assumptions cover relevant models with singularities like Coulomb-like electric potentials or the magnetic dipole.

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.

Our main goal in the present work is to address an integro-differential model under localized viscoelastic and frictional effects arising in the Boltzmann theory of viscoelasticity. More precisely, we consider a general version in the history context of the pioneer localized viscoelastic problem approached by Cavalcanti and Oquendo (2003) in the null history scenario, and more recently by Cavalcanti

We study a class of Kolmogorov systems of dimension two depending on two independent parameters. The local behavior of the system is described in terms of bifurcation diagrams which contain the bifurcation curves separating a node from a focus.

In this paper, we study a chemotaxis-consumption system with signal-dependent motility and generalized logistic source in a smooth bounded domain. We show that if α>0, ρ∈R, μ>0 and l>maxn+22,2, then for all sufficiently smooth initial data the system ut=Δ(uv−α)+ρu−μul,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,possesses a unique global-in-time solution.

In the paper Koch et al. (2009), the authors make the following conjecture: any bounded ancient mild solution of the 3D axially symmetric Navier–Stokes equations is constant. And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin.

In this paper, we study a class of semilinear p(x)-Laplacian equations with variable exponent sources. By using potential well method, we first prove a threshold results on the existence and nonexistence of global solutions to the equations when initial energy is less than the mountain pass level d. In the former case we also show the decay properties of energy functional. We finally obtain the non-global

This paper deals with planar piecewise linear refracting systems with a straight line of separation. Using the Poincaré compactification, we provide the classification of the phase portraits in the Poincaré disc of piecewise linear refracting systems with focus-saddle dynamics.

This article is concerned with the mathematical analysis of models for Micro-Electro-Mechanical Systems (MEMS). These models arise in the form of coupled partial differential equations with a moving boundary. Although MEMS devices are often operated in non-isothermal environments, temperature is often neglected in the mathematical investigations. In light of this finding the focus of our modelling

In this study, we consider a directed–diffusion system describing the interactions between two organisms in heterogeneous environment. We first establish a linearly stability of the co-existence (positive) steady state. Then we further present a classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems.

Compressible Euler equations with Coriolis force are the fundamental model in the study of atmospheric dynamics. In this article, we study the lifespan of the projected 2-dimensional C2 non-vacuum solutions of the compressible Euler equations with Coriolis force. We first locate the vortex equations for the compressible Euler equations with Coriolis force and then use the integration method to prove

In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.