Journal of Differential Equations最新文献

{"title":"Existence and asymptotic behaviors of positive solutions for a semilinear elliptic equation on trees","authors":"Yating Niu ,&nbsp;Yingshu Lü","doi":"10.1016/j.jde.2024.10.009","DOIUrl":"10.1016/j.jde.2024.10.009","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a locally finite tree, Δ be the normalized Laplacian. In this paper, we consider the following semilinear equation on <em>G</em><span><span><span>(0.1)</span><span><math><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>.</mo></math></span></span></span> We first establish the existence and nonexistence of positive solutions to <span><span>(0.1)</span></span> with a general assumption on <em>f</em>, and then find the critical exponent for <span><span>(0.1)</span></span> on a regular tree. Moreover, we prove some interesting properties of radial solutions and the asymptotic behaviors of radial solutions under a more general condition on <em>f</em>. Finally, the nonexistence results can be generalized to the elliptic system on a weighted tree.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 953-972"},"PeriodicalIF":2.4,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445049","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}

{"title":"Stability and large time behavior of the 2D Boussinesq equations with velocity supercritical dissipation","authors":"Baoquan Yuan,&nbsp;Changhao Li","doi":"10.1016/j.jde.2024.10.014","DOIUrl":"10.1016/j.jde.2024.10.014","url":null,"abstract":"<div><div>This paper studies the 2D Boussinesq equations with velocity supercritical <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>(</mo><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn><mo>)</mo></math></span> dissipation and temperature damping near the hydrostatic equilibrium. We are able to establish the global stability and the large time behavior of the solution. By introducing a diagonalization process to eliminate the linear terms, the temporal decay rate of the global solution is obtained. Furthermore, when <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span>, the velocity dissipation term becomes the velocity damping term, and the solution has an exponential decay.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 927-952"},"PeriodicalIF":2.4,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445048","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}

{"title":"Small mass limit for stochastic N-interacting particles system in L2(Rd) in mean field limit","authors":"Xueru Liu,&nbsp;Wei Wang","doi":"10.1016/j.jde.2024.10.015","DOIUrl":"10.1016/j.jde.2024.10.015","url":null,"abstract":"<div><div>An <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>-valued stochastic <em>N</em>-interacting particles system with small mass is investigated. Mean field limit and the propagation of chaos are derived. Moreover the small mass limit of the solution is also built, which can be seen as a Smoluchowski–Kramers approximation on unbounded domain. Here a key step is the asymptotic compactness of the distribution of the solution, which is derived via a splitting technique of the domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and some estimation of the solution for the mean field limit equation. We also show that the limits <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span> and <span><math><mi>ϵ</mi><mo>→</mo><mn>0</mn></math></span> commute.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 897-926"},"PeriodicalIF":2.4,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445047","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}

{"title":"Large time behavior of the full compressible Navier-Stokes-Maxwell system with a nonconstant background density","authors":"Xin Li","doi":"10.1016/j.jde.2024.10.010","DOIUrl":"10.1016/j.jde.2024.10.010","url":null,"abstract":"<div><div>We study the Cauchy problem for the full compressible Navier-Stokes-Maxwell system with a nonconstant background density in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. By means of suitable choosing of symmetrizers and weighted energy estimates with some new developments, we establish the global existence and uniqueness of the classical solution provided that the initial data are near this equilibrium. Furthermore, by using the spectrum analysis on the linearized homogeneous system of the full compressible Navier-Stokes-Maxwell equations and refining the convergence property, we obtain the time-algebraic convergence rates of the perturbed solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 869-896"},"PeriodicalIF":2.4,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442175","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}

{"title":"Characterising blenders via covering relations and cone conditions","authors":"Maciej J. Capiński ,&nbsp;Bernd Krauskopf ,&nbsp;Hinke M. Osinga ,&nbsp;Piotr Zgliczyński","doi":"10.1016/j.jde.2024.10.004","DOIUrl":"10.1016/j.jde.2024.10.004","url":null,"abstract":"<div><div>We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by finding finite series of sets that form suitable sequences of alignments. This characterisation is applicable in arbitrary dimension. Moreover, the approach naturally extends to establishing <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 768-805"},"PeriodicalIF":2.4,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432937","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}

{"title":"Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in R3","authors":"Dengjun Guo,&nbsp;Lifeng Zhao","doi":"10.1016/j.jde.2024.10.008","DOIUrl":"10.1016/j.jde.2024.10.008","url":null,"abstract":"<div><div>We consider the three-dimensional incompressible Euler equation<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>Ω</mi><mo>+</mo><mi>U</mi><mo>⋅</mo><mi>∇</mi><mi>Ω</mi><mo>−</mo><mi>Ω</mi><mo>⋅</mo><mi>∇</mi><mi>U</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mrow></math></span></span></span> in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Under the assumption that <span><math><msup><mrow><mi>Ω</mi></mrow><mrow><mi>z</mi></mrow></msup></math></span> is helical and in the absence of vorticity stretching, we prove the global well-posedness of weak solutions in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mo>⋂</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>. Moreover, the vortex transport formula and the conservation of the energy and the second momentum are also obtained in our article, which will serve as valuable tools in our subsequent exploration of the dynamics of helical vortex filaments.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 806-868"},"PeriodicalIF":2.4,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442589","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}

{"title":"Miura transformations and large-time behaviors of the Hirota-Satsuma equation","authors":"Deng-Shan Wang,&nbsp;Cheng Zhu,&nbsp;Xiaodong Zhu","doi":"10.1016/j.jde.2024.10.006","DOIUrl":"10.1016/j.jde.2024.10.006","url":null,"abstract":"<div><div>The good Boussinesq equation has several modified versions, such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has a simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value are studied by Deift-Zhou nonlinear steepest descent analysis. In such initial conditions, the asymptotic expressions away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well with the direct numerical simulations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 642-699"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432843","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}

{"title":"The convergence problem of the generalized Korteweg-de Vries equation in Fourier-Lebesgue space","authors":"Qiaoqiao Zhang ,&nbsp;Wei Yan ,&nbsp;Jinqiao Duan ,&nbsp;Meihua Yang","doi":"10.1016/j.jde.2024.10.007","DOIUrl":"10.1016/j.jde.2024.10.007","url":null,"abstract":"<div><div>In this paper, we investigate the pointwise convergence problem of the generalized Korteweg-de Vries (gKdV) equation with data in the Fourier-Lebesgue space. Firstly, for the Airy equation, we show the almost everywhere pointwise convergence with data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>s</mi><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>s</mi><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mn>5</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mo>∞</mo><mo>)</mo></math></span>, furthermore, we show that the maximal function estimate related to the Airy equation can fail with data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>s</mi><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>(</mo><mi>s</mi><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo></math></span>. Then, for the gKdV equation, we establish the pointwise convergence results with the data in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>(</mo><mn>5</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mrow><mn>23</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></math></span>, in particular, we establish the pointwise convergence results with small data in <span><math><msup><mrow><mover><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, which implies that the pointwise convergence of generalized KdV equation is closely related to the pointwise convergence of linear KdV equation in the Fourier-Lebesgue spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 614-641"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432842","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}

{"title":"The heat flow in nonlinear Hodge theory under general growth","authors":"Christoph Hamburger","doi":"10.1016/j.jde.2024.09.043","DOIUrl":"10.1016/j.jde.2024.09.043","url":null,"abstract":"<div><div>We study the <em>nonlinear Hodge system</em> <span><math><mi>d</mi><mi>ω</mi><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi><mo>=</mo><mn>0</mn></math></span> for an exterior form <em>ω</em> on a compact oriented Riemannian manifold <em>M</em>. Its solutions are called <em>ρ-harmonic forms</em>. Here the <em>ρ</em>-codifferential of <em>ω</em> is defined as <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>δ</mi><mo>(</mo><mi>ρ</mi><mi>ω</mi><mo>)</mo></math></span> with a given positive function <span><math><mi>ρ</mi><mo>=</mo><mi>ρ</mi><mo>(</mo><mo>|</mo><mi>ω</mi><mo>|</mo><mo>)</mo></math></span>.</div><div>We evolve a given closed form <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> by the <em>nonlinear heat flow system</em> <span><math><mover><mrow><mi>ω</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>d</mi><msub><mrow><mi>δ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ω</mi></math></span> for a time dependent exterior form <span><math><mi>ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> on <em>M</em>. Under an ellipticity condition on the function <em>ρ</em>, we show that the nonlinear heat flow system with initial condition <span><math><mi>ω</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> has a unique solution for all times, which converges to a <em>ρ</em>-harmonic form in the cohomology class of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. This yields a <em>nonlinear Hodge theorem</em> that every cohomology class of <em>M</em> has a unique <em>ρ</em>-harmonic representative.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 531-575"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bifurcation curves for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator","authors":"Shao-Yuan Huang ,&nbsp;Shin-Hwa Wang","doi":"10.1016/j.jde.2024.10.002","DOIUrl":"10.1016/j.jde.2024.10.002","url":null,"abstract":"<div><div>In this paper, we study the shapes of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>λ</mi><mi>exp</mi><mo>⁡</mo><mrow><mo>(</mo><mfrac><mrow><mi>a</mi><mi>u</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mtext></mtext><mspace></mspace></mrow><mo>−</mo><mi>L</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mi>L</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mo>−</mo><mi>L</mi><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span>where <span><math><mi>λ</mi><mo>&gt;</mo><mn>0</mn></math></span> is a bifurcation parameter and <span><math><mi>a</mi><mo>,</mo><mi>L</mi><mo>&gt;</mo><mn>0</mn></math></span> are evolution parameters. We determine the shapes of the bifurcation curves for different positive values <em>a</em> and <em>L</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 700-726"},"PeriodicalIF":2.4,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432938","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}