This paper is devoted to the study of a reaction–diffusion malaria model with a spatially and temporally heterogeneous structure. In the case of a bounded domain, we first establish the threshold type results for the mosquito-free system and disease-free system, and then establish the global dynamics for the model system in terms of the vector reproduction ratio Rv and the basic reproduction ratio

This paper deals with a general age-structured model with diffusion. The existence and uniqueness of solutions of the equivalent integral equation are obtained in light of the contraction mapping theorem. By taking the mosquito population growth as a motivating example, we derive a periodic stage-structured model with diffusion, intra-specific competition and periodic delay. Next, we show that the

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy’s law to capture

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity Cα,α>1∕2 in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager’s conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses

We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and

In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions.

In this paper, we first study the stability of the positive constant equilibrium and Turing instability induced by prey-taxis for phytoplankton–zooplankton system with prey-taxis. Then by applying Crandall–Rabinowitz bifurcation theory, the local existence of the nonconstant positive steady state bifurcating from the positive constant equilibrium E∗ is proven. In one dimension case, when ϕ(Z)=Z, we

This work studies an outstanding reaction–diffusion system modeling tumor invasion, with interactions among tumor tissue, acid concentration and normal tissue. This model has very different features from the models extensively studied in the mathematics literature. The most challenge issue for mathematical analysis of the present model is the existence of classical solution, since the diffusion of

The main purpose of this paper is to study the existence of traveling waves for a discrete diffusive SIR epidemic model with treatment. Compared to the work in Zhang and Wang (2014), more accurate results about the existence and nonexistence of nontrivial traveling wave solutions are obtained. We prove that when the basic reproduction number R0>1, there exists a critical number c∗>0 such that for each

In this paper, we establish a homogenization result for a nonlinear degenerate system arising from two-phase flow through fractured porous media with periodic microstructure taking into account the temperature effects. The mathematical model is given by a coupled system of two-phase flow equations, and an energy balance equation. The microscopic model consists of the usual equations derived from the

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially

This paper is devoted to the study of the blow-up solutions of the following weakly coupled degenerate parabolic systems with nonlinear boundary conditions: ut=div|∇u|p∇u+f(x,u,v,t),vt=div|∇v|q∇v+g(x,u,v,t),(x,t)∈D×(0,T∗),∂u∂n=b(u),∂v∂n=d(v),(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),v(x,0)=v0(x),x∈D¯.Here p>0,q>0, D is a bounded spatial region in Rn(n≥2), and the boundary ∂D is smooth. We mainly combine the maximum

This paper investigates a multi-dimensional attraction–repulsion system ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−μ1(t)+f1(u),x∈Ω,t>0,0=Δw−μ2(t)+f2(u),x∈Ω,t>0,where μ1(t)=1|Ω|∫Ωf1(u)dx, μ2(t)=1|Ω|∫Ωf2(u)dx, Ω=BR(0)⊂Rn(n≥2) and f1 and f2 are suitably regular functions generalizing the prototype determined by f1(s)=sγ1 and f2(s)=sγ2, s≥0, with γ1,γ2>0. Under homogeneous boundary conditions of Neumann type

This paper considers the planar magnetohydrodynamic system with the influence of radiation on the dynamics at high temperature regimes. When the viscosity μ depending on the specific volume of the gas (μ=μ̃1+μ̃2v−α with μ̃1>0,μ̃2≥0) and the heat conductivity κ being a power function of the temperature (κ=κ̃θβ with κ̃>0), the global existence of strong solution with large initial data to the magnetohydrodynamic

In the complex and competitive world of oceans, different size of plants and animals exist. All of them compete for the limited resources; e.g., nutrients, sunlight, minerals etc. Size-specific and intraspecific predation is common among zooplankton. We design a model food chain and explore dynamics, and patterns of species abundance in ocean. The proposed mathematical model is based on a parameter;

Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the k-Hessian operator acting on Φ0,radk(B), the space of radially symmetric k-admissible functions on the unit ball B⊂RN. We also prove both the existence of admissible extremal functions for the associated variational problem and the solvability of a related k-Hessian equation with supercritical growth.

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence

This paper is concerned with the lower and upper bounds of rates of decay for n-dimensional Navier–Stokes equations with fractional hyperviscosity (−Δ)α when 0<α

This paper deals with the Cauchy problem of a nonlocal bistable reaction–diffusion equation ∂u∂t=Δu+μu2(1−κJ∗u)−γu,(x,t)∈RN×(0,∞)with N≤2, μ,κ,γ>0 and u(x,0)=u0(x). Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, this problem admits a global bounded classical solution for N=1, while for N=2, global bounded classical solution exists for large

In this paper, we show the global boundedness and stability of solutions for prey-taxis model with handling and searching predators in a two-dimensional bounded domain with smooth boundary. First, entropy-like equations and boundedness criteria are derived, and it is proved that the system has a unique uniformly bounded global classical solution. In addition, we show that prey-only steady state is

In this paper we consider a class of diffusive ecological models with two free boundaries and with cross-diffusion and self-diffusion in one space dimension. The systems under consideration are strongly coupled, and the position of each free boundary is determined by the Stefan condition. We first show local existence of the solutions for the ecological models under some assumptions, and then prove

In this paper, we study the asymptotic properties of singular solutions to some nonstandard parabolic equation involving p(x,t)-Laplacian and nonlocal anisotropic sources. First, we show the existence and uniqueness of weak solutions by using the Galerkin’s approximations in the anisotropic Orlicz–Sobolev spaces. Second, in order to determine extinction rates and time of solutions, we prove some ordinary

In this paper, we formulate and study a compartmental model for Lassa fever transmission dynamics considering human-to-human, rodent-to-human transmission and the vertical transmission of the virus in rodents. To incorporate the impact of periodicity of weather on the spread of Lassa, we introduce a non-autonomous model with time-dependent parameters for rodent birth rate and carrying capacity of the

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either

This paper is concerned with the nonlocal dispersal logistic equation J∗u(x)−u(x)=−λu(x)+a(x)up in Ω̄,u(x)=0 in RN∖Ω̄,where Ω⊂RN (N≥1) is a bounded domain, λ and p>1 are constants, the dispersal kernel J and the coefficient a(x) are nonnegative. The work of García-Melián and Rossi (2009) reveals that the nonlocality makes a key change of positive solutions. In this paper, we investigate the sharp effects

We prove that the quadratic approximation of the capillarity-driven free-boundary Darcy flow, derived in Granero-Belinchón and Scrobogna (2019), is well posed in Ḣ3∕2S1, and globally well-posed if the initial datum is small in Ḣ3∕2S1.

We present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the

In this paper, we consider the following double phase problem with a nonlinear boundary condition −div|∇u|p−2∇u+μ(x)|∇u|q−2∇u=f(x,u)−|u|p−2u−μ(x)|u|q−2uinΩ,|∇u|p−2∇u+μ(x)|∇u|q−2∇u⋅ν=g(x,u)on∂Ω.First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on f and g

Let F=(f,g):R2→R2 be a polynomial map such that detDF(x,y) is different from zero for all (x,y)∈R2. We provide some new sufficient conditions for the injectivity of F. The proofs are based on the qualitative theory of differential equations.

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness

In this paper, we study the spatial dynamics of the lattice Lotka–Volterra competition system in a shifting habitat. First, we discuss the corresponding ranges of the environmental worsening speed c in the following three cases: (a) both species are extinct; (b) one species is extinct, the other is persistent; (c) both species are persistent. Moreover, for species persistence, it is achieved by diffusing

We study a free boundary problem modeling tumor growth with a T-periodic supply Φ(t) of external nutrients. The model contains two parameters μ and σ˜. We first show that (i) zero radially symmetric solution is globally stable if and only if σ˜⩾1T∫0TΦ(t)dt; (ii) If σ˜<1T∫0TΦ(t)dt, then there exists a unique radially symmetric positive solution σ∗(r,t),p∗(r,t),R∗(t) with period T and it is a global

In this paper, we consider the modified Ostrovsky, Stepanyams and Tsimring equation ut+uxxx−η(Hux+Huxxx)+u2ux=0. We prove that the associated initial value problem is locally well-posed in Sobolev spaces Hs(R) for s>−1∕2. We also prove that our result is sharp in the sense that the flow map of this equation fails to be C3 in Hs(R) for s<−1∕2. Moreover, we prove that for any s>1∕2 and T>0, its solution

Compared with the existing HIV/AIDS host model that considers only age-since-infection or only spacial diffusion, we propose a new age–space structured model that incorporating both two infection ages, spacial diffusion, and HAART (highly active antiretroviral therapy) to analyze the global dynamics of HIV/AIDS epidemic and study the incident of its transmission among MSM (men who have sex with men)

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