In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed

The aim of this paper is to analyze the long-time dynamical behavior of the solution for a degenerate wave equation with time-dependent damping term ∂ttu+β(t)∂tu=Lu(x,t)+f(u) on a bounded domain Ω⊂RN with Dirichlet boundary conditions. Under some restrictions on β(t) and critical growth restrictions on the nonlinear term f, we will prove the local and global well-posedness of the solution and derive

In this paper, we are concerned with the Turing instability of the spatially homogeneous Hopf bifurcating periodic solutions for the diffusive Sel’kov model with saturation effect. By using the center manifold theorem, normal form theory and the regularly perturbed theory, we derive a formula in terms of the diffusion rates to determine the Turing instability of the spatially homogeneous Hopf bifurcating

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous

We look for positive solutions for the singular equation −Δu−12x⋅∇u=μh(x)uq−1+λu+u(N+2)/(N−2),in RN, where N≥3, λ>0, μ>0 is a parameter, 00 is small.

In this paper, we investigate the dependence on initial data of solutions to higher dimensional Camassa–Holm equations. We show that the data-to-solution map is not uniformly continuous dependence in Besov spaces.

In this paper, we deal with the Kakutani–Matsuuchi model which describes the surface elevation η of the water-waves under the effect of viscosity. We first derive the decay rate of weak solutions. This can be used to obtain the decay rate of ‖η(t)‖Ḣ1 when initial data is sufficiently small in Ḣ1. We next show the existence, uniqueness, Gevrey regularity and decay rates of η with sufficiently small

We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state x̄=0 (at time t=−∞) into an arbitrary non-stationary state x (at time t=0). This is the opposite to what is commonly studied in the literature on null controllability

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces HN(Rx,v6) with N≥2. But if we want to obtain the optimal temporal decay estimates, we need to add

We obtain the global small solutions to the generalized Oldroyd-B model without damping on the stress tensor in Rn. Our result give positive answers partially to the question proposed by Elgindi and Liu (Remark 2 in Elgindi and Liu (2015)). The proof relies heavily on the trick of transferring dissipation from u to τ, and a new commutator estimate which may be of interest for future works. Moreover

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This

This paper is devoted to studying the inflow problem governed by the non-viscous and heat-conductive gas dynamic system in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact wave. The contact discontinuity in the linearly degenerate field is less stable, and the dissipative mechanism for non-viscous systems is also weaker

We are concerned with large-time behaviors of solutions for Vlasov–Navier–Stokes equations in two dimensions and Vlasov–Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into

The relativistic BGK model is a representative relaxation-time approximation of the relativistic Boltzmann equation. In this paper, we prove the global existence of bounded solutions to the relativistic BGK model proposed by Anderson and Witting when the initial data satisfies the excess conservations of mass and energy, and the excess entropy inequality from global equilibrium. The L∞ estimate established

We study the weak solvability of a macroscopic, quasilinear reaction–diffusion system posed in a 2D porous medium which undergoes microstructural problems. The solid matrix of this porous medium is assumed to be made out of circles of not-necessarily uniform radius. The growth or shrinkage of these circles, which are governed by an ODE, has direct feedback to the macroscopic diffusivity via an additional

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary conditions.

Combustion processes in porous media have been used by the petroleum engineering industry to extract heavy oil from reservoirs. This study focuses on a one-dimensional nonlinear hybrid system consisting of n reaction–diffusion–convection equations coupled with n ordinary differential equations, which models a combustion front moving through a porous medium with n parallel layers. The state variables

The doubly degenerate nutrient taxis model ut=∇⋅(uv∇u)−∇⋅(u2v∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,is considered in smoothly bounded convex subdomains of the plane, with ℓ≥0. It is shown that for any p>2 and each fixed nonnegative u0∈W1,∞(Ω), a smallness condition exclusively involving v0 can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with (u,v)|t=0=(u0

This paper deals with the quasilinear parabolic–elliptic Keller–Segel system with logistic source, ut=Δ(u+1)m−χ∇⋅(u(u+1)α−1∇v)+λ(|x|)u−μ(|x|)uκ,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,where Ω≔BR(0)⊂Rn(n≥3) is a ball with some R>0; m>0, χ>0, α>0 and κ≥1; λ and μ are continuous nonnegative functions. About this problem, Winkler (2018) found the condition for κ such that solutions blow up in finite time when m=α=1.

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation (ut−βuxxx−f(u)x)x=γu, where the nonlinearity f(u) satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for

In this paper, we prove the energy conservation for the weak solutions of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness in Lp and the continuity in Lpp−2

Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model ∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and by (U±,V±) its asymptotic states at ±∞, we investigate existence in the regimes d>1:(U−,V−)=(0,1)and(U+

This paper is to investigate the optimal Lp (p≥2) time decay rate of global solutions to the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm’s law. The result shows that the time decay rate of this system achieves the same as that of the incompressible Navier–Stokes equation, in other words, the coupling of the self-consistent Poisson equation does not change the decay rate of

Malaria is one of the most common mosquito-borne diseases widespread in tropical and subtropical regions, causing thousands of deaths every year in the world. Few models considering a multiple structure model formulation including (i) the chronological age of human and mosquito populations, (ii) the time since they are infected, and (iii) humans waning immunity (i.e. the progressive loss of protective

This paper presents a study of the long-time dynamics of the dynamical system generated by a nonlinear system modeling mixture of solids with nonlinear damping and Fourier’s law. By using the recent quasi-stability theory, we prove the existence of a smooth finite dimensional global attractor, which is characterized as an unstable manifold of the set of stationary solutions. The quasi-stability of

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation

The paper addresses nonlinear inverse Sturm–Liouville-type problems with constant delay. Since many processes in the real world possess nonlocal nature, operators with delay as well as other classes of nonlocal operators are continuously finding numerous applications in the natural sciences and engineering. However, in spite of a large number of works devoted to inverse problems for operators with

This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order α∈(12,1) nt+u⋅∇n=−(−Δ)αn−χ∇⋅(n∇c)+an−bn2,ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ+f,∇⋅u=0on three dimensional periodic torus T3. Since there is no classical solution in the three-dimensional full Navier–Stokes equations, our main purpose of this paper is to investigate the global existence

In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction–diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counter-examples

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.

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce

A dynamical model of echinococcosis transmission with optimal control strategies is first presented. The basic reproduction number of the model is determined and employed to study the global stability of the disease-free and endemic equilibrium points. The optimal control problem is formulated and solved analytically. Numerical simulations show that optimal control strategies could effectively reduce

We will be concerned with the problem of deformation of the lateral surface of a column that rotates with constant speed around its axis of symmetry. The column is filled by a gas and our goal is to investigate the deformation of the lateral surface depending on the pressure of the gas.

Our aim in this paper is to introduce and study a mathematical model for the description of traveling sand dunes. We use surface flow process of sand under the effect of wind and gravity. We model this phenomenon by a nonlinear diffusion–transport equation coupling the effect of transportation of sand due to the wind and the avalanches due to the gravity and the repose angle. The avalanche flow is

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially

This paper deal with the global dynamics of planar piecewise linear refracting systems of saddle–saddle type with a straight line of separation. We investigate the singularities, limit cycles, homoclinic orbits, heteroclinic orbits and make the classification of global phase portraits in the Poincaré disk for the refracting systems. We prove that these systems have 18 topologically different global

We establish some regularity criteria for the solutions to the Navier–Stokes equations in the full three-dimensional space in terms of one directional derivative of the velocity field. Revising the method used by Zujin Zhang (2018), we show that a weak solution u is regular on (0, T] provided that ∂u∂x3∈Lp(0,T;Lq(R3)) with s=2 for 3≤q≤6, 116

We study the asymptotic behavior of global solutions to forest kinetic model equations composed of young trees, old trees, and air-borne seeds. Under some parameter assumptions, we prove the asymptotic convergence of global solutions to a stationary solution. To this end, we show a non-smooth version of the Łojasiewicz–Simon gradient inequality on a suitable functional space and a certain norm estimate

In this paper, we are concerned with the low Mach number limit for the full compressible Hall-magnetohydrodynamic equations within the frame work of local smooth solution in R3. Under the assumption of large temperature variations, we first obtain the uniform estimates of the solutions in a ε-weighted Sobolev space, which establishes the local existence theorem of the full compressible Hall-magnetohydrodynamic

The aim of this paper is to study the asymptotic behavior of solutions for some reaction–diffusion systems in biology. First, we establish a Liouville type theorem for entire solutions of these reaction–diffusion systems. Based on this theorem, we derive the stabilization of the solutions of the reaction–diffusion system to the unique positive constant state, under the condition that this positive

In this paper we prove some regularity properties of solutions to variational inequalities of the form ∫Ω〈A(x,u,Du),D(φ−u)〉dx≥∫ΩB(x,u,Du)(φ−u)dx,∀φ∈Kψ(Ω).Here Ω is a bounded open set of Rn, n≥2, the function ψ:Ω→[−∞,+∞), called obstacle, belongs to the Sobolev class W1,p(Ω) and Kψ(Ω)={w∈W1,p(Ω):w≥ψq.o. inΩ} is the class of the admissible functions. First we establish a local Calderòn–Zygmund type estimate

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