Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Zhihui Ma, Shufan Wang, Xiaohua LiAbstractThe aim of this paper is to theoretically study the effect of the contacting distance (CTD) between the susceptible and infectious individuals in controlling infectious diseases. This paper formulates a generalized SEIR model incorporating the effect of the contacting distance (CTD). The dynamical behaviors of the proposed model are investigated and the controlling measures of the infectious diseases are developed. The results show that the contacting distance (CTD) between the susceptible and infectious individuals plays an important role in controlling infectious diseases. Some diseases will be globally controlled when the contacting distance (CTD) is larger than the threshold value. That is to say, the long contacting distance (CTD) implies the corresponding diseases will be controlled. However, for other diseases, the long or short contacting distance (CTD) will induce them to spread and be endemic. The moderate contacting distance (CTD) may be beneficial to control these diseases. Therefore, the appropriate contacting distance (CTD) should be selected for the given diseases in order to control the corresponding infectious diseases. Finally, a special numerical experiment is given to test our results. These results give some theoretical and experimental guides for the disease control authorities.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): E. Freire, E. Ponce, J. Ros, E. Vela, A. AmadorAbstractIn this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chin-Chin WuAbstractWe study a predator–prey model with two alien predators and one aborigine prey in which the net growth rates of both predators are negative. We characterize the invading speed of these two predators by the minimal wave speed of traveling wave solutions connecting the predator-free state to the co-existence state. The proof of the existence of traveling waves is based on a standard method by constructing (generalized) upper-lower-solutions with the help of Schauder’s fixed point theorem. However, in this three species model, we are able to construct some suitable pairs of upper-lower-solutions not only for the super-critical speeds but also for the critical speed. Moreover, a new form of shrinking rectangles is introduced to derive the right-hand tail limit of wave profile.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): F. Hamel, F. Lavigne, G. Martin, L. RoquesAbstractWe study the dynamics of adaptation of a large asexual population in a n-dimensional phenotypic space, under anisotropic mutation and selection effects. When n=1 or under isotropy assumptions, the ‘replicator-mutator’ equation is a standard model to describe these dynamics. However, the n-dimensional anisotropic case remained largely unexplored.We prove here that the equation admits a unique solution, which is interpreted as the phenotype distribution, and we propose a new and general framework to the study of the quantitative behaviour of this solution. Our method builds upon a degenerate nonlocal parabolic equation satisfied by the distribution of the ‘fitness components’, and a nonlocal transport equation satisfied by the cumulant generating function of the joint distribution of these components. This last equation can be solved analytically and we then get a general formula for the trajectory of the mean fitness and all higher cumulants of the fitness distribution, over time. Such mean fitness trajectory is the typical outcome of empirical studies of adaptation by experimental evolution, and can thus be compared to empirical data.In sharp contrast with the known results based on isotropic models, our results show that the trajectory of mean fitness may exhibit (n−1) plateaus before it converges. It may thus appear ‘non-saturating’ for a transient but possibly long time, even though a phenotypic optimum exists. To illustrate the empirical relevance of these results, we show that the anisotropic model leads to a very good fit of Escherichia coli long-term evolution experiment, one of the most famous experimental dataset in experimental evolution. The two ‘evolutionary epochs’ that have been observed in this experiment have long puzzled the community: we propose that the pattern may simply stem from a climbing hill process, but in an anisotropic fitness landscape.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Patrick De Leenheer, Wenxian Shen, Aijun ZhangAbstractThis paper is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with moving speed c. It is shown that the species becomes extinct if the moving speed c is larger than the so called spreading speed c∗, where c∗ is determined by the maximum linearized growth rate function. If the moving speed c is smaller than c∗, it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some number L∗ and in this case, there is a traveling wave solution with speed c, and it becomes extinct if the patch size is smaller than L∗.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Kerui Jiang, Zuhan Liu, Ling ZhouAbstractIn this paper, we investigate the global existence and asymptotic dynamics of solutions to a fractional singular chemotaxis system in three dimensional whole space. We deal with the new difficulties arising from fractional diffusion by using Riesz transform and Kato-Ponce’s commutator estimates appropriately, and establish the local existence of solution. Then with the help of combining the local existence and the a priori estimates, the global existence and uniqueness of solution with small initial data is derived. Moreover, we obtain the asymptotic decay rates of solution by the method of energy estimates.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Guglielmo Feltrin, Paolo GidoniAbstractWe investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem p1′′+λ1w1(x,p2)f1(p1)=0,in Ω,p2′′+λ2w2(x,p1)f2(p2)=0,in Ω,p1′=p2′=0,on ∂Ω,where the coupling-weights wi are sign-changing in the first variable, and the nonlinearities fi:[0,1]→[0,+∞[ satisfy fi(0)=fi(1)=0, fi(s)>0 for all s∈]0,1[, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of 2N positive fully nontrivial solutions when the real positive parameters λ1 and λ2 are sufficiently large.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Yancong Xu, Yu Yang, Fanwei Meng, Pei YuAbstractIn this paper, we consider dynamics and bifurcations in two HIV models with cell-to-cell interaction. The difference between the two models lies in the inclusion or omission of the effect of involvement. Particular attention is focused on the effects due to the cell-to-cell transmission and the effect of the involvement. We investigate the local and global stability of equilibria of the two models and give a comparison. We derive the existence condition for Hopf bifurcation and prove no Bogdanov-Takens bifurcation in this system. In particular, we show that the system exhibits the recurrence phenomenon, yielding complex dynamical behavior. It is also shown that the effect of the involvement is the main cause of the periodic symptoms in HIV or malaria disease. Moreover, it is shown that the increase of cell-to-cell interaction may be the main factor causing Hopf bifurcation to disappear, and thus eliminating oscillation behavior. Finally, numerical simulations are present to demonstrate our theoretical results.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Danxia Song, Chao Li, Yongli SongAbstractThis paper presents a qualitative study of a diffusive predator–prey system with the hunting cooperation functional response. For the system without diffusion, the existence, stability and Hopf bifurcation of the positive equilibrium are explicitly determined. It is shown that the hunting cooperation affects not only the existence of the positive equilibrium but also the stability. For the diffusive system, the stability and cross-diffusion driven Turing instability are investigated according to the relationship of the self-diffusion and the cross-diffusion coefficients. Stability and cross-diffusion instability regions are theoretically determined in the plane of the cross-diffusion coefficients. The technique of multiple time scale is employed to deduce the amplitude equation of Turing bifurcation and then pattern dynamics driven by the cross-diffusion is also investigated by the corresponding amplitude equation.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): R. NegreaAbstractWe prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Lina ZhangAbstractIn this paper, we are concerned with the isentropic Euler equations with time-dependent damping like −μ1+tρu for physical parameter μ>0. By using the technical time-weighted energy method, the global existence is proved and the decay estimates are obtained for the solutions of Euler equations. It is interesting that the new decay estimates are dependent on the physical parameter μ. And the decay rates are much better than that obtained by Pan.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Xi-Chao Duan, Xue-Zhi Li, Maia MartchevaAbstractIn this paper, to understand the impact of spatial heterogeneity of treatment and movement of individuals on the persistence and extinction of heroin spread, we propose a new diffusive heroin transmission model with treatment-dependent age-structure. The basic reproduction number in heterogenous environment R0 of the system is defined, which is consistent with the one deduced from the next generation operator approach R(x). The threshold dynamics in terms of the basic reproduction number is established: if R0≤1, the drug-free steady state is globally asymptotically stable, if R0>1, heroin transmission is uniformly persistent if it is present initially. In particular, when the environment is homogeneous and R0>1, our system has a unique space-independent drug spread steady state and it is globally asymptotically stable. Finally, some numerical simulations are carried out to illustrate the main results.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Yingzhe Fan, Ping XuAbstractWe prove the global existence of smooth solutions near Maxwellians to the Cauchy problem of non-cutoff Vlasov–Poisson–Boltzmann equation for soft potentials, provided that the weak angular singularity assumption holds and the algebraic decay initial perturbation is sufficiently small. This extends the work of Duan and Liu (2013), in which the case of the strong angular singularity 12≤s

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Eric Kokomo, Bongor Danhrée, Yves EmvuduAbstractWe propound a deterministic, nonlinear model for the transmission dynamics of cholera in different human communities with individuals’ migration. The considered different human communities are crossed by a running water which is contaminated by the vibrio cholerae bacterium. The formulated model for each community which is an initial/boundary-value problem constituted of four parabolic partial differential equations, integrates antibiotic treatment, hydration therapy and contaminated water treatment as control mechanisms of the disease. Using semigroup theory, we prove that this model has a unique bounded positive solution. Also under a given condition, the existence of a trivial equilibrium and of a nontrivial equilibrium of each community is established and their local and global stabilities are studied. In analysis of Turing’s instability, we determine sufficient conditions allowing the formation of a spatially stationary and periodic heterogeneous pattern. Analytically the existence of a unique optimal control is established by the use of functional analysis techniques and an optimal control θ̄ is determined to eradicate the epidemic in each community. In order to confirm our theoretical results, we finish with a real-world application concerning the cholera epidemic that took place in Cameroon in 2011.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Keiichi WatanabeAbstractWe consider a free boundary problem of compressible–incompressible two-phase flows with phase transitions in general domains of N-dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The compressible fluid and the incompressible fluid are separated by either compact or non-compact sharp moving interface, and the surface tension is taken into account. In our model, the compressible fluid and incompressible fluid are occupied by the Navier–Stokes–Korteweg equations and the Navier–Stokes equations, respectively. This paper shows that for given T>0 the problem admits a unique strong solution on (0,T) in the maximal Lp−Lq regularity class provided the initial data are small in their natural norms.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Siyu Liu, Mingxin WangAbstractIn this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Lijun Pei, Yameng Chen, Shuo WangAbstractIn this paper, we consider the complex dynamics of a novel mathematical model for a feedback control system of the gut microbiota, which was proposed by Dong, et al. The main work of the present paper is to study the effect of antibiotics injection on the gut microbiota through some dynamical methods, such as double Hopf bifurcation analysis and so on. We first use DDE-BIFTOOL to find the non-resonant double Hopf bifurcation points of the system, and draw the bifurcation diagram with two bifurcation parameters, τ1 and τ2, i.e., respective measurement delays. Then we study small perturbations of two delay differential equations at these double Hopf bifurcation points, and the method of multiple scales is employed to obtain two common complex amplitude equations. By analyzing the amplitude equations, we can derive the classification and unfolding of these double Hopf bifurcation points. Finally, we verify the results by numerical simulations. We find more complicated dynamic behaviors of the system via analytical method. For example, there exists stable equilibrium, stable periodic solution or even the co-existing stable periodic solutions in respective region. And the numerical simulations are consistent with the analytic results, meanwhile it implies that the MMS is effective and accurate. All complex dynamical phenomena found in the present paper can be very helpful for the researchers to understand the mechanism of the system of gut microbiota. And it is also very significant to microbiology and engineering control. It reveals that the measurement delays can induce the complicated dynamics in this system and to the ends of excellent performance, we should take the proper values of these delays.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Dumitru Motreanu, Angela Sciammetta, Elisabetta TornatoreExistence and location of solutions to a Neumann problem driven by an nonhomogeneous differential operator and with gradient dependence are established developing a non-variational approach based on an adequate method of sub-supersolution. The abstract theorem is applied to prove the existence of finitely many positive solutions or even infinitely many positive solutions for a class of Neumann problems.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Zaitao Liang, Fangfang LiaoAbstractIn this paper, by using the third order approximation method, the averaging method and the theory of upper and lower solutions, we study the existence and radial stability of periodic orbits of damped Keplerian-like systems. Two different results are obtained: perturbative and global results. Our results are also applicable to the classical Keplerian-like systems.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Gurusamy Arumugam, André H. Erhardt, Indurekha Eswaramoorthy, Balachandran KrishnanAbstractIn this paper, we consider the Keller–Segel chemotaxis system with additional cross-diffusion term in the second equation. This system is consisting of a fully nonlinear reaction–diffusion equations with additional cross-diffusion. We establish the existence of weak solutions to the considered system by using Schauder’s fixed point theorem, a priori energy estimates and the compactness results.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): M. Djoukwe Tapi, L. Bagny-Beilhe, Y. DumontAbstractCocoa mirid, Sahlbergella singularis, is known to be one of the major pests of cocoa in West Africa. In this paper, we consider a biological control method, based on mating disrupting, using artificial sex pheromones, and trapping, to limit the impact of mirids in plots. We develop and study a piece-wise smooth delayed dynamical system. Based on previous results, a theoretical analysis is provided in order to derive all possible dynamics of the system. We show that two main threshold parameters exist that will be useful to derive long term successful control strategies for different level of infestation. We illustrate and discuss our results when cacao pods production is either constant along the year or seasonal. To conclude, we provide future perspectives based on this work.

Abstract: Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Shengmao Fu, Liangying MiaoAbstractIn this paper, we consider the global behavior of the fully parabolic predator–prey chemotaxis model u1t=d1Δu1+χ∇⋅(u1∇v)+μ1u1(1−u1−e1u2),x∈Ω,t>0,u2t=d2Δu2−ξ∇⋅(u2∇v)+μ2u2(1+e2u1−u2),x∈Ω,t>0,vt=d3Δv+αu1+βu2−γv,x∈Ω,t>0,∂u1∂ν=∂u2∂ν=∂v∂ν=0,x∈∂Ω,t>0,u1(x,0)=u1,0(x),u2(x,0)=u2,0(x),v(x,0)=v0(x),x∈Ωin a smooth bounded domain Ω⊂Rn, where d1,d2,

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Caidi Zhao, Yanjiao Li, Zhongchun SongAbstractIn this article, we construct the trajectory statistical solution for the 3D incompressible Navier–Stokes equations via the natural translation semigroup and trajectory attractor. In our construction, the trajectory statistical solution is an invariant space–time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space, and the trajectory statistical solution possesses the invariant property under the acting of the translation semigroup.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Chi Phan, Yuncheng YouAbstractGlobal dynamics of nonautonomous diffusive Hindmarsh–Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Naoki TsugeAbstractWe are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax–Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Min Zhu, Zhigui Lin, Lai ZhangAbstractThe nonlocal incidence and free boundaries are introduced into a classic SIR-SI model describing the transmission dynamics of dengue fever, where the nonlocal incidence allows for interactions of susceptible population at a given location with infected mosquitoes in the whole area, and free boundaries represent the expanding front of the area contaminated by dengue virus. We derive a spatial–temporal risk index in terms of the basic reproduction number, which depends on the nonlocal incidence and time variable. More importantly, we explore the relationships between different model variants regarding these risk indices. We additionally find sufficient conditions to ensure the vanishing and spreading of dengue fever, and demonstrate, for a special case, the asymptotic behavior of its solution when spreading occurs. Finally, we carry out numerical simulations to demonstrate our analytical findings and further provide their epidemiological explanations.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Yu Jin, Feng-Bin WangAbstractDiel vertical migration is a common movement pattern of zooplankton in marine and freshwater habitats. In this paper, we use a temporally periodic reaction–diffusion–advection system to describe the dynamics of zooplankton and fish in aquatic habitats. Zooplankton live in both the surface water and the deep water, while fish only live in the surface water. Zooplankton undertake diel vertical migration to avoid predation by fish during the day and to consume sufficient food in the surface water during the night. We establish the persistence theory for both species as well as the existence of a time-periodic positive solution to investigate how zooplankton manage to maintain a balance with their predators via vertical migration. Numerical simulations discover the effects of migration strategy, advection rates, domain boundary conditions, as well as spatially varying growth rates, on persistence of the system.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Yaru Dou, Gang MengAbstractIn this paper we study the continuity of periodic solutions of the Landesman–Lazer equations in coefficient functions. It will be proved that these periodic solutions are not only continuous in coefficient functions with respect to the usual Lp topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are the basis to study some quantities defined from solutions in a quantitative way.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Lu Cao, Zaihong Jiang, Mingxuan ZhuAbstractIn this paper, we study the Holm–Staley b-family equation with weakly dissipative term. Firstly, we show the infinite propagation speed that if the initial datum u0(x) has a compact support, the corresponding solution u(x,t) does not have a compact x-support any longer in its lifespan. Then, we obtain the large time behavior of the support of momentum density with the initial data compact supported.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Bakary Traoré, Ousmane Koutou, Boureima SangaréAbstractIn this paper, a mathematical model of malaria transmission which takes into account the four distinct mosquito metamorphic stages is presented. The model is formulated thanks to the coupling of two sub-models, namely the model of mosquito population and the model of malaria parasite transmission due to the interaction between mosquitoes and humans. Moreover, considering that climate factors have a great impact on the mosquito life cycle and parasite survival in mosquitoes, we incorporate seasonality in the model by considering some parameters which are periodic functions. Through a rigorous analysis via theories and methods of dynamical systems, we prove that the global behavior of the model depends strongly on two biological insightful quantities : the vector reproduction ratio Rv and the basic reproduction ratio R0. Indeed, if Rv1 and R01 and R0>1 the disease remains persistent. Finally, using the reported monthly mean temperature of Burkina Faso, we perform some numerical simulations to illustrate our theoretical results.

Abstract: Publication date: Available online 12 December 2019Source: Nonlinear Analysis: Real World ApplicationsAuthor(s): N. Duruk MutlubaşAbstractThe local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime was proved in Duruk Mutlubaş (2013). In this paper, we correct the mistake made in the proof of the main result and give the appropriate assumptions and corresponding results.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Gaocheng YueAbstractThe present paper is devoted to the well-posedness issue of solutions of a full system of the 3-D incompressible magnetohydrodynamic (MHD) equations with large initial velocity and magnetic field slowly varying in one space variable. By means of the anisotropic Littlewood–Paley analysis we prove the global well-posedness of solutions in the framework of anisotropic type Besov spaces for ϵ and σ sufficiently small. Toward this and due to the divergence-free property of magnetic field, the proof is based on unified energy estimates which is valid for the magnetic field satisfying the inhomogeneous damped wave equation.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Hui Sun, Ming Mei, Kaijun ZhangAbstractIn this paper, we study the asymptotic behaviors in time of solutions to the unipolar hydrodynamic model of semiconductors in the form of Euler–Poisson equations on the half line with the boundary effect, where the boundary conditions are proposed physically as the inflow/outflow/impermeable boundary or the insulating boundary. Different from the Cauchy problem, the boundary effect causes some essential difficulties in determining the asymptotic profiles for the solutions and occurs the boundary layers. First of all, by heuristically analyzing, we reasonably propose some additional boundary conditions at far field to the corresponding steady-state equations such that the steady-state systems are well-posed. Thus, we can determine the corresponding steady-states as the expected asymptotic profiles for the solutions of original IBVPs. Secondly, there are some L2-boundary-layers between the solutions of original IBVPs and their steady-states. After investigating the exact form of gaps, we technically construct some correction functions to delete these gaps. Finally, by using the energy estimates, we further prove that the original solutions of the inflow/outflow/impermeable problem (insulating problem) time-exponentially (time-exponentially/algebraically) converge to their asymptotic profiles. Finally, we carry out some numerical simulations, which show that, the graphs for the asymptotic profiles in different boundary cases are significantly distinct.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): David Greenhalgh, Qamar J.A. Khan, Fatma Ahmed Al-KharousiAbstractWe investigate a model consisting of a predator population and both susceptible and infected prey populations. The predator can feed on either prey species but instead of choosing individuals at random the predator feeds preferentially on the most abundant prey species. More specifically we assume that the likelihood of a predator catching a susceptible prey or an infected prey is proportional to the numbers of these two different types of prey species. This phenomenon, involving changing preference from susceptible to infected prey, is called switching. Mukhopadhyay studied a switching model and proposed that the interaction of predators with infected prey is beneficial for the growth of the predator. In this model, we assume that the predator will eventually die as a result of eating infected prey. We find a threshold parameter R0 and showed that the disease will be eradicated from the system if R0

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Xianbo Sun, Wentao Huang, Junning CaiAbstractThe existence of a solitary wave for the shallow water model in convecting circumstance was established in previous works. It is still unknown that whether there exist periodic waves. In this paper, we prove that the models possess periodic waves with a fixed range of wave speed. The amplitude and wave speed are explicitly given. Moreover, the coexistence of the solitary wave and one periodic wave is established.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Caochuan Ma, Zhaoyun ZhangAbstractIn this paper, we focus on the global regularity of 2D generalized magnetohydrodynamics equations with magnetic damping b β−1b. Basing on the Maximal regularity of parabolic equation, we are able to show that this system has global smooth solutions when the initial data is sufficiently smooth.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Hakho HongAbstractIn this paper, we are concerned with the outflow problem in the half line (0,∞) to the isothermal compressible Navier–Stokes–Korteweg system with a nonlinear boundary condition for vanishing capillary tensor at x=0. We first give some necessary and sufficient conditions for the existence of the stationary solutions with the aid of center manifold theory. We also show the stability of the stationary solutions under smallness assumptions on the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solutions is obtained, provided that the initial perturbations belong to the weighted Sobolev spaces.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Yu Yang, Lan Zou, Jinling Zhou, Cheng-Hsiung HsuAbstractThis paper is concerned with the dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate. We first establish the well-posedness of this model. Then we clarify the relationship between the local basic reproduction number R̃ and the basic reproduction number R0. It could be seen that R0 plays an important role in determining the global dynamics of this model. In fact, we show that the disease-free equilibrium is globally asymptotically stable when R01. We also consider the local and global stability of endemic equilibrium when all the parameters of this model are constant. In the case R0>1, we further establish the existence of traveling wave solutions of this model. Moreover, we provide an example and numerical simulations to support our theoretical results. Our model extended some known results.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Meina SunAbstractThe analytical solutions of the Riemann problem for the isentropic Euler system with the logarithmic equation of state are derived explicitly for all the five different cases. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is shown that the solution consisting of two shock waves converges to a delta shock wave solution as well as the solution consisting of two rarefaction waves converges to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Yongkai LiaoAbstractWe study the nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional viscous radiative and reactive gas when the viscosity and heat conductivity coefficients depend on both density and absolute temperature. Our main idea is to use the smallness of the strength of the rarefaction waves to control the possible growth of its solutions induced by the nonlinearity of the system and the interactions of rarefaction waves from different families. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and the absolute temperature.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Wenlong Sun, Tomás Caraballo, Xiaoying Han, Peter E. KloedenAbstractA non-autonomous free boundary model for tumor growth is studied. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First the global existence and uniqueness of a transient solution is established under some general conditions. Then with additional regularity assumptions on the consumption and proliferation rates, the existence and uniqueness of steady-state solutions is obtained. Furthermore the convergence of the transient solutions toward the steady-state solution is verified. Finally the long time behavior of the solutions is investigated by transforming the time-dependent domain to a fixed domain.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Samir Bendoukha, Salem Abdelmalek, Mokhtar KiraneAbstractThis paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Andrea Giacobbe, Giuseppe Mulone, Wendi WangAbstractA mathematical model is proposed to simulate the eating disorders of bulimia or anorexia. Earlier models are extended to incorporate the body mass index, which plays a key role in the eating attitude of self thinners. The global existence and ultimate boundedness of solutions of the nonlocal model are proved by using estimates of solutions. The basic reproduction number of eating disorder contagion is shown to be the invasion threshold. The testable linear and nonlinear stability conditions are established by Lyapunov functions. Further numerical simulations are given to reveal how self-forces and peer pressures to be thinner affect the emergence and distributions of eating disorders.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Stephen Baigent, Belgin SeymenoğluAbstractWe study the two-locus-two-allele (TLTA) Selection–Recombination model from population genetics and establish explicit bounds on the TLTA model parameters for an invariant manifold to exist. Our method for proving existence of the invariant manifold relies on two key ingredients: (i) monotone systems theory (backwards in time) and (ii) a phase space volume that decreases under the model dynamics. To demonstrate our results we consider the effect of a modifier gene β on a primary locus α and derive easily testable conditions for the existence of the invariant manifold.

Abstract: Publication date: June 2020Source: Nonlinear Analysis: Real World Applications, Volume 53Author(s): Y. Paulina Martínez, Claudia VallsAbstractWe study a Chemostat system of the form ẋ=−qx+R̃K+yxy,ẏ=(c̃−y)q−R̃ã(K+y)xy,where q>0, R̃>0, K>0, c̃>0 and ã≠0. This system appears in competition modelling in biology. We describe its global dynamics on the Poincaré disc and study its Liouvillian integrability. For the first topic we use the well-known Poincaré compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves.