Pub Date : 2024-10-17DOI: 10.1016/j.jde.2024.10.009
Yating Niu , Yingshu Lü
Let be a locally finite tree, Δ be the normalized Laplacian. In this paper, we consider the following semilinear equation on G(0.1) We first establish the existence and nonexistence of positive solutions to (0.1) with a general assumption on f, and then find the critical exponent for (0.1) 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 f. Finally, the nonexistence results can be generalized to the elliptic system on a weighted tree.
设 G=(V,E) 是局部有限树，Δ 是归一化拉普拉卡。在本文中，我们考虑 G 上的以下半线性方程 (0.1)Δu+f(u)=0。我们首先在 f 的一般假设下建立了 (0.1) 正解的存在性和不存在性，然后找到了规则树上 (0.1) 的临界指数。此外，我们还证明了径向解的一些有趣性质，以及在更一般的 f 条件下径向解的渐近行为。最后，不存在结果可以推广到加权树上的椭圆系统。
{"title":"Existence and asymptotic behaviors of positive solutions for a semilinear elliptic equation on trees","authors":"Yating Niu , 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}
Pub Date : 2024-10-17DOI: 10.1016/j.jde.2024.10.014
Baoquan Yuan, Changhao Li
This paper studies the 2D Boussinesq equations with velocity supercritical 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 , the velocity dissipation term becomes the velocity damping term, and the solution has an exponential decay.
{"title":"Stability and large time behavior of the 2D Boussinesq equations with velocity supercritical dissipation","authors":"Baoquan Yuan, 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><</mo><mi>α</mi><mo><</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}
Pub Date : 2024-10-17DOI: 10.1016/j.jde.2024.10.015
Xueru Liu, Wei Wang
An -valued stochastic N-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 and some estimation of the solution for the mean field limit equation. We also show that the limits and commute.
{"title":"Small mass limit for stochastic N-interacting particles system in L2(Rd) in mean field limit","authors":"Xueru Liu, 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}
Pub Date : 2024-10-16DOI: 10.1016/j.jde.2024.10.010
Xin Li
We study the Cauchy problem for the full compressible Navier-Stokes-Maxwell system with a nonconstant background density in . 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.
{"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}
Pub Date : 2024-10-15DOI: 10.1016/j.jde.2024.10.004
Maciej J. Capiński , Bernd Krauskopf , Hinke M. Osinga , Piotr Zgliczyński
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 -persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.
{"title":"Characterising blenders via covering relations and cone conditions","authors":"Maciej J. Capiński , Bernd Krauskopf , Hinke M. Osinga , 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}
Pub Date : 2024-10-15DOI: 10.1016/j.jde.2024.10.008
Dengjun Guo, Lifeng Zhao
We consider the three-dimensional incompressible Euler equation in the whole space . Under the assumption that is helical and in the absence of vorticity stretching, we prove the global well-posedness of weak solutions in . 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.
{"title":"Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in R3","authors":"Dengjun Guo, 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}
Pub Date : 2024-10-14DOI: 10.1016/j.jde.2024.10.006
Deng-Shan Wang, Cheng Zhu, Xiaodong Zhu
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.
{"title":"Miura transformations and large-time behaviors of the Hirota-Satsuma equation","authors":"Deng-Shan Wang, Cheng Zhu, 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}
Pub Date : 2024-10-14DOI: 10.1016/j.jde.2024.10.007
Qiaoqiao Zhang , Wei Yan , Jinqiao Duan , Meihua Yang
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 , furthermore, we show that the maximal function estimate related to the Airy equation can fail with data in . Then, for the gKdV equation, we establish the pointwise convergence results with the data in , in particular, we establish the pointwise convergence results with small data in , 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.
{"title":"The convergence problem of the generalized Korteweg-de Vries equation in Fourier-Lebesgue space","authors":"Qiaoqiao Zhang , Wei Yan , Jinqiao Duan , 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><</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><</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><</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}
Pub Date : 2024-10-14DOI: 10.1016/j.jde.2024.09.043
Christoph Hamburger
We study the nonlinear Hodge system for an exterior form ω on a compact oriented Riemannian manifold M. Its solutions are called ρ-harmonic forms. Here the ρ-codifferential of ω is defined as with a given positive function .
We evolve a given closed form by the nonlinear heat flow system for a time dependent exterior form on M. Under an ellipticity condition on the function ρ, we show that the nonlinear heat flow system with initial condition has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of . This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.
我们研究紧凑定向黎曼流形 M 上外部形式 ω 的非线性霍奇系统 dω=δρω=0。这里，ω 的ρ-微分被定义为δρω=ρ-1δ(ρω)，其中有给定的正函数ρ=ρ(|ω|)。我们通过非线性热流系统ω˙=dδρω来演化一个给定的封闭形式ω0，该系统为 M 上与时间相关的外部形式ω(x,t)。在函数ρ的椭圆性条件下，我们证明了初始条件为ω(⋅,0)=ω0 的非线性热流系统在所有时间都有唯一解，该解收敛于ω0 的同调类中的ρ谐形式。这就产生了一个非线性霍奇定理，即 M 的每个共构类都有一个唯一的 ρ 谐波代表。
{"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}
Pub Date : 2024-10-14DOI: 10.1016/j.jde.2024.10.002
Shao-Yuan Huang , Shin-Hwa Wang
In this paper, we study the shapes of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operatorwhere is a bifurcation parameter and are evolution parameters. We determine the shapes of the bifurcation curves for different positive values a and L.
本文研究了具有闵科夫斯基曲率算子{-(u′(x)1-(u′(x))2)′=λexp(aua+u),-L<；x<L,u(-L)=u(L)=0，其中 λ>0 为分岔参数，a,L>0 为演化参数。我们确定了不同正值 a 和 L 的分岔曲线形状。
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