Pub Date : 2024-10-01DOI: 10.1016/j.aml.2024.109329
Peng Jin, Xianhua Tang
This paper is concerned with the following -supercritical Schrödinger equation where , , is a given mass and will arise as a Lagrange multiplier depending on the solution . By introducing new weak -supercritical conditions on , we develop robust arguments to establish the existence of normalized solutions to the above equation. Our result complements the ones of Chen and Tang (2024).
{"title":"Normalized solutions for L2-supercritical Schrödinger equations","authors":"Peng Jin, Xianhua Tang","doi":"10.1016/j.aml.2024.109329","DOIUrl":"10.1016/j.aml.2024.109329","url":null,"abstract":"<div><div>This paper is concerned with the following <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-supercritical Schrödinger equation <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>λ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mi>c</mi><mo>,</mo><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>f</mi><mo>∈</mo><mi>C</mi><mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> is a given mass and <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span> will arise as a Lagrange multiplier depending on the solution <span><math><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. By introducing new weak <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-supercritical conditions on <span><math><mi>f</mi></math></span>, we develop robust arguments to establish the existence of normalized solutions to the above equation. Our result complements the ones of Chen and Tang (2024).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109329"},"PeriodicalIF":2.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.aml.2024.109327
Zehu Yu, Yuxiang Li
This paper deals with a no-flux initial–boundary value problem ()in a bounded domain with smooth boundary, where and the nonnegative motility function admits the degenerate case: . It is shown that for any appropriately regular initial data, global classical solutions of () can be constructed, if either or when .
{"title":"Global classical solutions of a degenerate migration system with indirect signal absorption in arbitrary dimensions","authors":"Zehu Yu, Yuxiang Li","doi":"10.1016/j.aml.2024.109327","DOIUrl":"10.1016/j.aml.2024.109327","url":null,"abstract":"<div><div>This paper deals with a no-flux initial–boundary value problem <span><span><span>(<span><math><mo>⋆</mo></math></span>)</span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>u</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>w</mi><mi>v</mi><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><mi>τ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>u</mi><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> with smooth boundary, where <span><math><mrow><mi>τ</mi><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> and the nonnegative motility function <span><math><mi>ϕ</mi></math></span> admits the degenerate case: <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>. It is shown that for any appropriately regular initial data, global classical solutions of <span><span>(<span><math><mo>⋆</mo></math></span>)</span></span> can be constructed, if either <span><math><mrow><mi>τ</mi><mo>=</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><msub><mrow><mo>sup</mo></mrow><mrow><mi>ξ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>]</mo></mrow></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo><</mo><mn>1</mn></mrow></math></span> when <span><math><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109327"},"PeriodicalIF":2.9,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.aml.2024.109328
Jiashu Lu, Lei Zhang, Xuncheng Guo, Qiong Qi
A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods.
针对非稳态非局部扩散模型，提出了一种基于降阶模型（ROM）的快速算法。它结合了适当正交分解（POD）方法和局部径向基函数（RBFs）的配位方法，从而使使用 ROM 求解非局部模型成为可能。一些数值实验表明，这种方法大大降低了非局部模型的计算成本，同时与 RBF 配准方法相比保持了相似的收敛性。
{"title":"A POD based reduced-order local RBF collocation approach for time-dependent nonlocal diffusion problems","authors":"Jiashu Lu, Lei Zhang, Xuncheng Guo, Qiong Qi","doi":"10.1016/j.aml.2024.109328","DOIUrl":"10.1016/j.aml.2024.109328","url":null,"abstract":"<div><div>A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109328"},"PeriodicalIF":2.9,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.aml.2024.109325
Feng Wang, Zhenyu Wang, Qiang Ma, Xiaohua Ding
In this paper, the evolution laws of global momentum and averaged global momentum for the damped nonlinear stochastic wave equation (DNSWE) influenced by multiplicative space-time noise are derived. We innovatively combine the second-order central finite difference method and discrete gradient method in space, and integrate the splitting method and Störmer-Verlet type method in time. Both the novel spatial semi-discrete scheme and fully-discrete scheme constructed in this way can successfully preserve the corresponding evolution laws for discrete global momentum and discrete averaged global momentum. Numerical experiments on DNSWE with cubic nonlinearity validate the theoretical results.
{"title":"A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise","authors":"Feng Wang, Zhenyu Wang, Qiang Ma, Xiaohua Ding","doi":"10.1016/j.aml.2024.109325","DOIUrl":"10.1016/j.aml.2024.109325","url":null,"abstract":"<div><div>In this paper, the evolution laws of global momentum and averaged global momentum for the damped nonlinear stochastic wave equation (DNSWE) influenced by multiplicative space-time noise are derived. We innovatively combine the second-order central finite difference method and discrete gradient method in space, and integrate the splitting method and Störmer-Verlet type method in time. Both the novel spatial semi-discrete scheme and fully-discrete scheme constructed in this way can successfully preserve the corresponding evolution laws for discrete global momentum and discrete averaged global momentum. Numerical experiments on DNSWE with cubic nonlinearity validate the theoretical results.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109325"},"PeriodicalIF":2.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.aml.2024.109324
Bingtao Han, Daqing Jiang
In this paper, we develop a stochastic smoking epidemic model, where Black-Karasinski process is for the first time introduced to describe the environmental fluctuations in smoking transmission. By constructing suitable Lyapunov functions and compact sets, we establish sufficient conditions for the exponential extinction of smoking populations and the existence of a stationary distribution (i.e., a reflection of smoking persistence). Our results show that stochastic noise will be conducive to smoking pandemic.
{"title":"Global dynamics of a stochastic smoking epidemic model driven by Black-Karasinski process","authors":"Bingtao Han, Daqing Jiang","doi":"10.1016/j.aml.2024.109324","DOIUrl":"10.1016/j.aml.2024.109324","url":null,"abstract":"<div><div>In this paper, we develop a stochastic smoking epidemic model, where Black-Karasinski process is for the first time introduced to describe the environmental fluctuations in smoking transmission. By constructing suitable Lyapunov functions and compact sets, we establish sufficient conditions for the exponential extinction of smoking populations and the existence of a stationary distribution (i.e., a reflection of smoking persistence). Our results show that stochastic noise will be conducive to smoking pandemic.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109324"},"PeriodicalIF":2.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.aml.2024.109326
Aiyong Chen, Xiaokai He
We consider a Hamiltonian PDE arising from a class of equations appearing in the study of magma dynamics in the Earth’s interior. Previously, it has been shown that the Hamiltonian PDE admits solitary wave solutions. Simpson et al. proved that the solitary wave solutions are orbitally stable for the case . We verify the stability criterion analytically for the case . Our results answer partially an open question proposed by Simpson et al. (2008).
{"title":"Orbital stability of solitary wave solutions of a Hamiltonian PDE arising in magma dynamics","authors":"Aiyong Chen, Xiaokai He","doi":"10.1016/j.aml.2024.109326","DOIUrl":"10.1016/j.aml.2024.109326","url":null,"abstract":"<div><div>We consider a Hamiltonian PDE arising from a class of equations appearing in the study of magma dynamics in the Earth’s interior. Previously, it has been shown that the Hamiltonian PDE admits solitary wave solutions. Simpson et al. proved that the solitary wave solutions are orbitally stable for the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math></span>. We verify the stability criterion analytically for the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow></math></span>. Our results answer partially an open question proposed by Simpson et al. (2008).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109326"},"PeriodicalIF":2.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.aml.2024.109319
Xuyan Jiang, Zhiyuan Li
The paper investigates an inverse problem of determining the orders and potential function of multi-term time fractional wave equations. By analyzing the asymptotic behavior of the multivariate Mittag–Leffler function, we show the uniqueness of the inverting the orders and potential term using interior domain measurement data. The article also provides numerical simulations, showing how to handle noisy data and reconstruct the parameters of the equation through Tikhonov regularization.
{"title":"Unique inversion of orders and potential for multi-term time fractional wave equations","authors":"Xuyan Jiang, Zhiyuan Li","doi":"10.1016/j.aml.2024.109319","DOIUrl":"10.1016/j.aml.2024.109319","url":null,"abstract":"<div><div>The paper investigates an inverse problem of determining the orders and potential function of multi-term time fractional wave equations. By analyzing the asymptotic behavior of the multivariate Mittag–Leffler function, we show the uniqueness of the inverting the orders and potential term using interior domain measurement data. The article also provides numerical simulations, showing how to handle noisy data and reconstruct the parameters of the equation through Tikhonov regularization.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109319"},"PeriodicalIF":2.9,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.aml.2024.109322
Xuemin Yao , Jinying Ma , Gaoqing Meng
In this paper, the state transition mechanism in the variable-coefficient (3+1)-dimensional Kadomtsev–Petviashvili equation with external force control is reported, which could be beneficial for providing potential theoretical insights in fluid mechanics or plasma. Utilizing the Hirota bilinear method, the non-autonomous two-soliton solution can be derived. Subsequently, the transformed waves under various external forces are modulated based on the state transition condition. The graphical representation indicates that the external force affects the modulation plane of the transformed waves. In conclusion, the discovery that wave solutions can still occur in non-autonomous systems through the addition of external forces is more conducive to providing practical guidance for applications.
{"title":"The state transition mechanism of nonlinear waves with external force control in the fluid or plasma","authors":"Xuemin Yao , Jinying Ma , Gaoqing Meng","doi":"10.1016/j.aml.2024.109322","DOIUrl":"10.1016/j.aml.2024.109322","url":null,"abstract":"<div><div>In this paper, the state transition mechanism in the variable-coefficient (3+1)-dimensional Kadomtsev–Petviashvili equation with external force control is reported, which could be beneficial for providing potential theoretical insights in fluid mechanics or plasma. Utilizing the Hirota bilinear method, the non-autonomous two-soliton solution can be derived. Subsequently, the transformed waves under various external forces are modulated based on the state transition condition. The graphical representation indicates that the external force affects the modulation plane of the transformed waves. In conclusion, the discovery that wave solutions can still occur in non-autonomous systems through the addition of external forces is more conducive to providing practical guidance for applications.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109322"},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.aml.2024.109320
Chun Shen
Two kinds of exact Riemann solutions for a hyperbolic system arising from the steady 2D Helmholtz equation under a paraxial assumption are constructively achieved by using either delta shock wave or contact-vacuum-contact composite wave, which depends on the ordering relation between the left and right initial velocities. Moreover, the interaction between delta shock wave and contact-vacuum-contact composite wave is carefully explored by considering the initial data in three pieces separated by two jump discontinuities.
{"title":"Riemann solutions and wave interactions for a hyperbolic system derived from the steady 2D Helmholtz equation under a paraxial assumption","authors":"Chun Shen","doi":"10.1016/j.aml.2024.109320","DOIUrl":"10.1016/j.aml.2024.109320","url":null,"abstract":"<div><div>Two kinds of exact Riemann solutions for a hyperbolic system arising from the steady 2D Helmholtz equation under a paraxial assumption are constructively achieved by using either delta shock wave or contact-vacuum-contact composite wave, which depends on the ordering relation between the left and right initial velocities. Moreover, the interaction between delta shock wave and contact-vacuum-contact composite wave is carefully explored by considering the initial data in three pieces separated by two jump discontinuities.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109320"},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.aml.2024.109323
Shu Li , Binxiang Dai , Hao Wang
This paper focuses on a reaction–diffusion equation with spatiotemporal memory and Dirichlet boundary condition. We prove the existence of positive steady-state solutions through local and global bifurcation theory and provide the conditions for the stability of positive steady-state solutions. Our general results are applied to a diffusive logistic population model with spatiotemporal memory.
{"title":"Existence and stability of steady states of reaction–diffusion equation with spatiotemporal memory","authors":"Shu Li , Binxiang Dai , Hao Wang","doi":"10.1016/j.aml.2024.109323","DOIUrl":"10.1016/j.aml.2024.109323","url":null,"abstract":"<div><div>This paper focuses on a reaction–diffusion equation with spatiotemporal memory and Dirichlet boundary condition. We prove the existence of positive steady-state solutions through local and global bifurcation theory and provide the conditions for the stability of positive steady-state solutions. Our general results are applied to a diffusive logistic population model with spatiotemporal memory.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109323"},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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