In this paper, we study the local well-posedness of the Boussinesq equation for MHD convection with fractional thermal diffusion in Hs(Rn)×Hs+1−ϵ(Rn)×Hs+α−ϵ(Rn) with s>n2−1 and any small enough ϵ>0 such that s+1−ϵ>n2 and s+α−ϵ≥s+2−(ϵ+α)>n2. We present here the fractional operator (−Δ)αθ for α>1 which is estimated by using Littlewood–Paley projection.

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of

The life cycles of many species include separate dispersal and sedentary stages. To understand the population dynamics of such species, we study a hybrid model consisting of a reaction–diffusion equation that governs the random movement and settlement of dispersal individuals and an age-structured hyperbolic equation that describes the growth of sedentary individuals. We establish the existence and

This paper concerns the chemotaxis-Stokes system nt+u⋅∇n=Δnm−∇⋅(n∇c)+μn(1−n),ct+u⋅∇c=Δc−cnα,ut=Δu−∇π+n∇φ,∇⋅u=0in a three dimensional bounded domain under no-flux boundary conditions for n,c and no-slip boundary conditions for u. The purpose of this paper is to study the global solvability and large time asymptotic behavior of solutions. Here, it is worth mentioning that the nonlinear consumption term

The well-known Lane–Emden conjecture indicates that, for the elliptic Lane–Emden system −Δu=vp, −Δv=uq in RN, the Sobolev hyperbola 1p+1+1q+1=N−2N is expected as the critical curve for the existence and nonexistence of entire solutions. In this paper, we study the periodic Lane–Emden heat flow system ut−Δu=a(t)vp, vt−Δv=b(t)uq in a bounded domain Ω of RN, subject to homogeneous Dirichlet boundary condition

The Nilaparvata lugens is primary vector of rice diseases such as rice ragged stunt. A recent study reported the Wolbachia wStri can cause the cytoplasmic incompatibility of N.lugens, and inhibit the infection and transmission of rice ragged stunt virus in the laboratory. In this work, based on multi-strains infection mechanisms including incomplete cytoplasmic incompatibility and imperfect maternal

Dengue has grown dramatically in recent decades globally. In order to investigate the spread of dengue with vector control, especially, the impact of Wolbachia on dengue transmission, a mathematical model is established and analyzed to study dengue transmission between humans and mosquitoes. Firstly, model qualitative analysis including the existence and local asymptotic stability of dengue-free equilibria

Due to the strong ability of restoring textures and details in images, nonlocal equations have attracted an extensive interest for image denoising. However, the lack of regularity causes residual noise in the restored images. In this paper, we propose and study an evolution equation consisting of the weighted local and the weighted nonlocal p-Laplacian equations. The existence and uniqueness of solutions

In this paper, inspired by the work of Chen–Liang–Wang–Xu (Chen et al., 2020) on compressible Navier–Stokes equations, we obtain the energy conservation for weak solutions of the compressible non-resistive magnetohydrodynamic flows in a bounded domain Ω⊂R3. To ensure the energy conservation, we need the same regularity conditions of density and velocity as in Chen et al. (2020), moreover, H∈Lt4Lx4

We discuss the Cauchy problem for the following parabolic attraction–repulsion chemotaxis system: ∂tu=Δu−∇⋅(β1u∇v1)+∇⋅(β2u∇v2),t>0,x∈R2,∂tvj=Δvj−λjvj+u,t>0,x∈R2(j=1,2),u(0,x)=u0(x),vj(0,x)=vj0(x),x∈R2(j=1,2)with constants βj, λj>0 (j=1,2). In this paper we prove that the nonnegative solutions exist globally in time under the assumption (β1−β2)∫R2u0dx<8π in the attractive dominant case β1>β2.

In this paper, the blow-up of solutions to the Cauchy problems for semilinear Moore–Gibson–Thompson equations with power nonlinearity |u|p, derivative nonlinearity |ut|p, combined nonlinearity |ut|p+|u|q are investigated. Upper bound lifespan estimates of solutions to the problems in the sub-critical and critical cases are also deduced by applying the test function approach. It is worth noticing that

This paper deals with a semilinear weighted parabolic problem in general bounded domains, subject to zero Dirichlet boundary conditions, where the weighted functions depend not only on space variable but also on time variable. Fujita exponents for blow-up and global existence of solutions are determined by using semigroup methods and the comparison principle, which are composed by the dimensions of

In this paper, we deal with an SIRS reaction–diffusion epidemic model with saturation infection mechanism. Based on the uniform boundedness of the parabolic system, we investigate the extinction and persistence of the infectious disease in terms of the basic reproduction number. To better investigate the effects of infection mechanism and individual diffusion, we further analyze the asymptotic profiles

This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurrence of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap is needed. The main tool used here is a crucial Lemma which reveals to be needed because it allows us to move from the variational obstacle

In this work, we introduce an efficient second-order reaction–diffusion (R.D.) equation for noise removal and image super-resolution. The main idea is to decompose the image into two components: cartoon and texture parts using a new spatially controlled diffusion obtained by incorporating into R.D. equation an anti-diffusion effect modelled by a space dependent parameter γ. Hence, the proposed equation

In this paper, we mainly study the stability of Riemann problem for the improved Aw–Rascle–Zhang model which describes the formation and dynamics of traffic jams. First of all, we construct the classical Riemann solutions by elementary waves with the method of characteristic analysis. With the generalized Rankine–Hugoniot and entropy conditions, we prove the existence and uniqueness of δ-shock wave

We study the Laplacian equation with dynamical boundary condition involving Dirichlet-to-Neumann operator, critical growth, and Hardy potential. We first prove the existence and decay estimates of global solutions and finite time blowup of local solutions under certain assumptions. Then we focus on the asymptotic behavior of global solutions approaching a stationary solution in the long time series

In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the

New criteria are established for the existence of positive solutions of a semipositone elliptic problem on the exterior of a ball in which the Neumann boundary conditions are non local. These criteria are determined by the relationship between the behaviour of the nonlinearity as the variable tends to 0 or ∞ and the principal (positive) eigenvalue of a suitable associate linear Hammerstein integral

In this paper, we consider the fractional Camassa–Holm equation modeling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space B2,1s0 with s0=2ν−12 for ν>32 and s0=52 for 1<ν≤32. Then, with a given analytic initial data, we establish the analyticity of the solutions in both

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted p-Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities

In this work, we investigate the following nonlinear Timoshenko equation with variable exponents: utt+△2u−M▽uL2(Ω)2△u+utpx−2ut=uqx−2u.By using the Faedo–Galerkin method, we prove the local existence of the solution under suitable conditions. We also investigate the nonexistence of solutions with negative initial energy.

This paper deals with a two-species competition model in a homogeneous advective environment, where two species are subjected to a net loss of individuals at the downstream end. Under the assumption that the advection and diffusion rates of two species are proportional, we give a basic classification on the global dynamics by employing the theory of monotone dynamical system. It turns out that bistability

The time periodic problem of the Navier–Stokes equations on a non-cylindrical space–time domain is studied. Motivated by a recent result by Saal (2006) on maximal regularity for this kind of system we construct time periodic solutions in Lq-spaces provided the bounded domain moves periodically with small amplitude and the given periodic external force is small. The proof is based on new decay estimates

Bat-borne rabies is a viral infection caused by lyssaviruses. To explore the complex interaction between bats and cattle, we develop a periodic compartment system with seasonality, temperature-dependent incubation period and different control measures. We derive the basic reproduction number R0 and show it determines the global dynamics. In particular, we prove the non-existence of positive periodic

This paper is devoted to the global existence of weak solutions to the following degenerate kinetic model of chemotaxis (0.1)ut=Δ(γ(v)u)τvt=Δv−v+uin a smooth bounded domain with no-flux boundary conditions under the assumption 0≤τ<(sup(0,∞)γ)−1. The problem features a positive signal-dependent motility function γ(⋅) which may vanish as v becomes unbounded. In this paper, we first modify the comparison

We consider the Cauchy problem of the ideal MHD system with a constant non-zero background magnetic field e. We provide a new proof of global existence for small perturbations of an initial stable state (0,e). The proof is based on the null structure of the system and some new weighted energy estimates established along the characteristics.

In order to explore the impact of periodically evolving domain on the transmission of disease, we study an SIS reaction–diffusion model with logistic term on a periodically evolving domain. The basic reproduction number R0 is given by the next generation infection operator, and relies on the evolving rate of the periodically evolving domain, diffusion coefficient of infected individuals dI and size

In this paper we investigate a two-species competitive chemotaxis Keller–Segel system. We first obtain the local existence and global existence of classical solutions and then study the asymptotic behavior of the solutions.

Persistence and propagation of species are fundamental questions in spatial ecology. This paper focuses on the impact of Allee effect on the persistence and propagation of a population with birth pulse. We investigate the threshold dynamics of an impulsive reaction–diffusion model and provide the existence of bistable traveling waves connecting two stable equilibria. To prove the existence of bistable

We start with a mathematical model which describes the frictionless contact of an elastic body with an obstacle and prove that it leads to a stationary inclusion for the strain field. Then, inspired by this contact model, we consider a general stationary inclusion in a real Hilbert space, governed by three parameters. We prove the unique solvability of the inclusion as well as the continuous dependence

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