Pub Date : 2025-12-22DOI: 10.1016/j.nonrwa.2025.104585
Fábio Natali
In this paper, we consider the problem of well-posedness and orbital stability of odd periodic traveling waves for the sine-Gordon equation. We first establish novel results concerning the local well-posedness in smoother periodic Sobolev spaces to guarantee the existence of a local time where the associated Cauchy problem has a unique solution with the zero mean property. Afterwards, we prove the orbital stability of odd periodic waves using a convenient index theorem applied to the constrained linearized operator defined in the Sobolev space with the zero mean property.
{"title":"Remarks on the orbital stability for the sine-Gordon equation","authors":"Fábio Natali","doi":"10.1016/j.nonrwa.2025.104585","DOIUrl":"10.1016/j.nonrwa.2025.104585","url":null,"abstract":"<div><div>In this paper, we consider the problem of well-posedness and orbital stability of odd periodic traveling waves for the sine-Gordon equation. We first establish novel results concerning the local well-posedness in smoother periodic Sobolev spaces to guarantee the existence of a local time where the associated Cauchy problem has a unique solution with the zero mean property. Afterwards, we prove the orbital stability of odd periodic waves using a convenient index theorem applied to the constrained linearized operator defined in the Sobolev space with the zero mean property.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104585"},"PeriodicalIF":1.8,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-22DOI: 10.1016/j.nonrwa.2025.104583
Qian Jiang , Zhijun Liu , Lianwen Wang
This work proposes a novel age-structured tuberculosis (TB) model to evaluate the impact of pre- and post-exposure vaccinations on TB control, explicitly incorporating the waning of vaccine-induced efficacy over time. Mathematically, we establish a broadly applicable analytical framework for rigorously proving the global stability of steady states in the age-structured model. Furthermore, we formulate an optimal control problem with time-dependent vaccination schedules, identifying cost-effective vaccination strategies under realistic resource constraints. Epidemiologically, our analysis indicates that a combined vaccination strategy is most effective overall. While pre-exposure vaccination proves superior for long-term outbreak control, high-intensity post-exposure vaccination is crucial in high-exposure scenarios to rapidly reduce infected cases. This offers valuable insights for evaluating vaccination-based prevention and control strategies.
{"title":"Global stability analysis and optimal vaccination strategy for an age-structured tuberculosis model with general incidence","authors":"Qian Jiang , Zhijun Liu , Lianwen Wang","doi":"10.1016/j.nonrwa.2025.104583","DOIUrl":"10.1016/j.nonrwa.2025.104583","url":null,"abstract":"<div><div>This work proposes a novel age-structured tuberculosis (TB) model to evaluate the impact of pre- and post-exposure vaccinations on TB control, explicitly incorporating the waning of vaccine-induced efficacy over time. Mathematically, we establish a broadly applicable analytical framework for rigorously proving the global stability of steady states in the age-structured model. Furthermore, we formulate an optimal control problem with time-dependent vaccination schedules, identifying cost-effective vaccination strategies under realistic resource constraints. Epidemiologically, our analysis indicates that a combined vaccination strategy is most effective overall. While pre-exposure vaccination proves superior for long-term outbreak control, high-intensity post-exposure vaccination is crucial in high-exposure scenarios to rapidly reduce infected cases. This offers valuable insights for evaluating vaccination-based prevention and control strategies.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104583"},"PeriodicalIF":1.8,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.nonrwa.2025.104578
Jilei Huang, Peidong Lei, Junquan Zhou
In this paper, we prove controllability results for non-dissipative heat equations under natural unilateral constraints on the control. When the controlled parabolic system is non-dissipative, the controllability under nonnegative control constraints may fail in large time for general L2-initial data and final target trajectories. We establish the controllability of the general target trajectory when the difference between the initial states of the controlled system and the target trajectory lies within a specified subspace of L2(Ω). Conversely, if the difference lies outside this subspace, we prove that there exist infinitely many initial states causing system uncontrollability. We also prove that under nonnegative control constraints, there exists a minimum positive time required to achieve general target trajectory controllability, showing a waiting time phenomenon.
{"title":"Controllability of non-dissipative heat equations under unilateral control constraints","authors":"Jilei Huang, Peidong Lei, Junquan Zhou","doi":"10.1016/j.nonrwa.2025.104578","DOIUrl":"10.1016/j.nonrwa.2025.104578","url":null,"abstract":"<div><div>In this paper, we prove controllability results for non-dissipative heat equations under natural unilateral constraints on the control. When the controlled parabolic system is non-dissipative, the controllability under nonnegative control constraints may fail in large time for general <em>L</em><sup>2</sup>-initial data and final target trajectories. We establish the controllability of the general target trajectory when the difference between the initial states of the controlled system and the target trajectory lies within a specified subspace of <em>L</em><sup>2</sup>(Ω). Conversely, if the difference lies outside this subspace, we prove that there exist infinitely many initial states causing system uncontrollability. We also prove that under nonnegative control constraints, there exists a minimum positive time required to achieve general target trajectory controllability, showing a waiting time phenomenon.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104578"},"PeriodicalIF":1.8,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.nonrwa.2025.104521
Luiz Fernando Gonçalves, Bruno Rodrigues Freitas, Ronaldo Alves Garcia
In this paper, we investigate the existence of limit cycles in a class of planar piecewise smooth differential systems having the unit circle as their switching manifold. The vector field inside the circle is assumed to be linear and Hamiltonian, while the vector field outside is given by . We provide an upper bound for the number of crossing limit cycles such systems can possess, as well as for some of their perturbations.
{"title":"Limit cycles in a class of piecewise polynomial differential systems having the unit circle as their switching manifold","authors":"Luiz Fernando Gonçalves, Bruno Rodrigues Freitas, Ronaldo Alves Garcia","doi":"10.1016/j.nonrwa.2025.104521","DOIUrl":"10.1016/j.nonrwa.2025.104521","url":null,"abstract":"<div><div>In this paper, we investigate the existence of limit cycles in a class of planar piecewise smooth differential systems having the unit circle as their switching manifold. The vector field inside the circle is assumed to be linear and Hamiltonian, while the vector field outside is given by <span><math><mrow><mover><mi>z</mi><mo>˙</mo></mover><mo>=</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></math></span>. We provide an upper bound for the number of crossing limit cycles such systems can possess, as well as for some of their perturbations.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104521"},"PeriodicalIF":1.8,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-18DOI: 10.1016/j.nonrwa.2025.104562
Lin Zhao, Yini Liu
In this paper, we focus on a Zika virus model with diffusion and constant recruitment and analyze the existence and non-existence of traveling wave solutions of the model, which are determined by the basic reproduction number R0 and the minimal wave speed c*. Precisely speaking, if R0 > 1, then there exists a minimal wave speed c* > 0 such that the model admits traveling wave solutions with the wave speed c ≥ c*, and there are no non-trivial traveling wave solutions of this model with 0 < c < c*. If R0 ≤ 1, we prove that there are no non-trivial traveling wave solutions of the model. Finally, numerical simulations are carried out to verify and demonstrate some of the conclusions obtained in this study.
本文研究了一种具有扩散和不断招募的Zika病毒模型，分析了该模型的行波解的存在性和不存在性，其存在性由基本繁殖数R0和最小波速c*决定。准确地讲,如果R0 祝辞 1,那么存在一个最小波速c * 祝辞 0这样的模型承认行波解和波速c ≥ c *,并且没有不平凡的这个模型的行波解与0 & lt; c & lt; c *。当R0 ≤ 1时，我们证明了模型不存在非平凡行波解。最后，通过数值模拟验证和论证了本文的部分结论。
{"title":"Propagation dynamics of a Zika virus model with diffusion and constant recruitment","authors":"Lin Zhao, Yini Liu","doi":"10.1016/j.nonrwa.2025.104562","DOIUrl":"10.1016/j.nonrwa.2025.104562","url":null,"abstract":"<div><div>In this paper, we focus on a Zika virus model with diffusion and constant recruitment and analyze the existence and non-existence of traveling wave solutions of the model, which are determined by the basic reproduction number <em>R</em><sub>0</sub> and the minimal wave speed <em>c</em>*. Precisely speaking, if <em>R</em><sub>0</sub> > 1, then there exists a minimal wave speed <em>c</em>* > 0 such that the model admits traveling wave solutions with the wave speed <em>c</em> ≥ <em>c</em>*, and there are no non-trivial traveling wave solutions of this model with 0 < <em>c</em> < <em>c</em>*. If <em>R</em><sub>0</sub> ≤ 1, we prove that there are no non-trivial traveling wave solutions of the model. Finally, numerical simulations are carried out to verify and demonstrate some of the conclusions obtained in this study.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104562"},"PeriodicalIF":1.8,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-18DOI: 10.1016/j.nonrwa.2025.104570
Paolo Piersanti
In this paper we establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.
{"title":"Existence of solutions for time-dependent Signorini-type problems in linearised viscoelasticity","authors":"Paolo Piersanti","doi":"10.1016/j.nonrwa.2025.104570","DOIUrl":"10.1016/j.nonrwa.2025.104570","url":null,"abstract":"<div><div>In this paper we establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104570"},"PeriodicalIF":1.8,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study simultaneous homogenization and dimensional reduction of integral functionals for maps in manifold-valued Sobolev spaces. Due to the superlinear growth regime, we prove that the density of the Γ-limit is a tangential quasiconvex integrand represented by a cell formula.
{"title":"Homogenization and 3D-2D dimension reduction of a functional on manifold valued Sobolev spaces","authors":"Michela Eleuteri , Luca Lussardi , Andrea Torricelli , Elvira Zappale","doi":"10.1016/j.nonrwa.2025.104579","DOIUrl":"10.1016/j.nonrwa.2025.104579","url":null,"abstract":"<div><div>We study simultaneous homogenization and dimensional reduction of integral functionals for maps in manifold-valued Sobolev spaces. Due to the superlinear growth regime, we prove that the density of the Γ-limit is a tangential quasiconvex integrand represented by a cell formula.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104579"},"PeriodicalIF":1.8,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-16DOI: 10.1016/j.nonrwa.2025.104571
Giuseppe Cardone , Reine Gladys Noucheun , Carmen Perugia , Jean Louis Woukeng
We derive, through the periodic homogenization theory in thin heterogeneous domains, a 2D model consisting of Hele-Shaw equation coupled with the convective Cahn-Hilliard equation with non-constant mobility. The upscaled set of equations, which models in particular tumor growth, is then analyzed and we prove some regularity results. We heavily rely on the two-scale convergence concept in thin heterogeneous media associated to some Sobolev inequalities such as the Gagliardo-Nirenberg and Agmon inequalities to achieve our goal.
{"title":"Mathematical derivation and analysis of a mixture model of tumor growth","authors":"Giuseppe Cardone , Reine Gladys Noucheun , Carmen Perugia , Jean Louis Woukeng","doi":"10.1016/j.nonrwa.2025.104571","DOIUrl":"10.1016/j.nonrwa.2025.104571","url":null,"abstract":"<div><div>We derive, through the periodic homogenization theory in thin heterogeneous domains, a 2<em>D</em> model consisting of Hele-Shaw equation coupled with the convective Cahn-Hilliard equation with non-constant mobility. The upscaled set of equations, which models in particular tumor growth, is then analyzed and we prove some regularity results. We heavily rely on the two-scale convergence concept in thin heterogeneous media associated to some Sobolev inequalities such as the Gagliardo-Nirenberg and Agmon inequalities to achieve our goal.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104571"},"PeriodicalIF":1.8,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-15DOI: 10.1016/j.nonrwa.2025.104561
Xinshan Dong , Ben Niu , Lin Wang
We investigate a diffusive Rosenzweig-MacArthur system that includes nonlocal prey competition and prey-taxis under Neumann boundary conditions. Initially, we establish the global existence and boundedness of solutions for arbitrary spatial dimensions and small prey-taxis sensitivity coefficient. Subsequently, we analyze the local stability of the constant steady-state solution. Using the Lyapunov-Schmidt reduction method, we explore several bifurcations near the positive constant steady-state: steady-state bifurcation, Hopf bifurcation, and their interaction. Finally, numerical simulations are performed to validate our theoretical findings and illustrate complex spatiotemporal patterns. By selecting appropriate parameters and initial conditions, our simulations reveal the coexistence of a pair of stable spatially nonhomogeneous steady-states and stable spatially homogeneous periodic solutions, which indicates the system exhibits tristability, that is, the coexistence of three distinct stable states. Moreover, our results demonstrate that transient patterns transition from spatially nonhomogeneous periodic solutions to spatially nonhomogeneous steady-state and spatially homogeneous periodic solutions.
{"title":"Spatiotemporal patterns induced by nonlocal prey competition and prey-taxis in a diffusive Rosenzweig-MacArthur system","authors":"Xinshan Dong , Ben Niu , Lin Wang","doi":"10.1016/j.nonrwa.2025.104561","DOIUrl":"10.1016/j.nonrwa.2025.104561","url":null,"abstract":"<div><div>We investigate a diffusive Rosenzweig-MacArthur system that includes nonlocal prey competition and prey-taxis under Neumann boundary conditions. Initially, we establish the global existence and boundedness of solutions for arbitrary spatial dimensions and small prey-taxis sensitivity coefficient. Subsequently, we analyze the local stability of the constant steady-state solution. Using the Lyapunov-Schmidt reduction method, we explore several bifurcations near the positive constant steady-state: steady-state bifurcation, Hopf bifurcation, and their interaction. Finally, numerical simulations are performed to validate our theoretical findings and illustrate complex spatiotemporal patterns. By selecting appropriate parameters and initial conditions, our simulations reveal the coexistence of a pair of stable spatially nonhomogeneous steady-states and stable spatially homogeneous periodic solutions, which indicates the system exhibits tristability, that is, the coexistence of three distinct stable states. Moreover, our results demonstrate that transient patterns transition from spatially nonhomogeneous periodic solutions to spatially nonhomogeneous steady-state and spatially homogeneous periodic solutions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104561"},"PeriodicalIF":1.8,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-15DOI: 10.1016/j.nonrwa.2025.104569
Nguyen Van Y , Le Cong Nhan , Le Xuan Truong
In the paper, we consider a fractional thermo-viscoelastic system with nonlinear sources and study some of its qualitative properties based on the interaction of the fractional viscoelastic and thermal damping with the external forces. By using the theory of linear Volterra differential-integral equations of convolution type and the Banach fixed point theorem, we first prove the local well-posedness and maximal regularity of the weak solution. Then by using the variational and potential well methods, we give a sufficient condition for the continuity in time of the local weak solution when it starts in the potential wells. Besides that the asymptotic behavior of global solution is also concerned, unlike the classical thermoelasticity where the total energy does not decays uniformly, since the effect of the fractional viscoelastic damping, we show that the total energy shall decay uniformly. In addition, its decay rate is given explicitly and optimally in the sense of Lasiecka et. al.[1]. Finally, since the presence of the nonlinear sources, we show that the blow-up phenomenon may occur in finite time provided that the solution starts outside the potential wells and the relaxation function is small in some sense. Also notice that the effect of the thermal damping is not enough to make the total energy decays to zero, but it could retards the blow-up phenomenon.
{"title":"Some qualitative properties of solution to a fractional thermo-viscoelastic system with nonlinear sources","authors":"Nguyen Van Y , Le Cong Nhan , Le Xuan Truong","doi":"10.1016/j.nonrwa.2025.104569","DOIUrl":"10.1016/j.nonrwa.2025.104569","url":null,"abstract":"<div><div>In the paper, we consider a fractional thermo-viscoelastic system with nonlinear sources and study some of its qualitative properties based on the interaction of the fractional viscoelastic and thermal damping with the external forces. By using the theory of linear Volterra differential-integral equations of convolution type and the Banach fixed point theorem, we first prove the local well-posedness and maximal regularity of the weak solution. Then by using the variational and potential well methods, we give a sufficient condition for the continuity in time of the local weak solution when it starts in the potential wells. Besides that the asymptotic behavior of global solution is also concerned, unlike the classical thermoelasticity where the total energy does not decays uniformly, since the effect of the fractional viscoelastic damping, we show that the total energy shall decay uniformly. In addition, its decay rate is given explicitly and optimally in the sense of Lasiecka et. al.[1]. Finally, since the presence of the nonlinear sources, we show that the blow-up phenomenon may occur in finite time provided that the solution starts outside the potential wells and the relaxation function is small in some sense. Also notice that the effect of the thermal damping is not enough to make the total energy decays to zero, but it could retards the blow-up phenomenon.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104569"},"PeriodicalIF":1.8,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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