{"title":"The derivative structure for a quadratic nonlinearity and uniqueness for SQG","authors":"Tsukasa Iwabuchi","doi":"10.1016/j.jde.2025.02.057","DOIUrl":"10.1016/j.jde.2025.02.057","url":null,"abstract":"<div><div>We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, we demonstrate that the uniqueness of the mild solution holds in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivatives for the nonlinear term involving fractional derivatives of the Dirichlet Laplacian.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 802-825"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508301","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":"The local well-posedness of analytic solution to the boundary layer system for compressible flow in three dimensions","authors":"Yufeng Chen , Lizhi Ruan , Anita Yang","doi":"10.1016/j.jde.2025.02.056","DOIUrl":"10.1016/j.jde.2025.02.056","url":null,"abstract":"<div><div>In this paper, we consider three dimensional boundary layer equations for compressible isentropic flow with no-slip boundary condition. The local well-posedness of the compressible boundary layer system is established when the initial datum is real-analytic in the tangential direction and has Sobolev regularity in the normal direction. The proof is based on the introduction of a change of variables to eliminate the linear growth in normal direction and subtle energy estimates with algebraic weights.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 716-746"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487185","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":"New type of solutions for the critical polyharmonic equation","authors":"Wenjing Chen, Zexi Wang","doi":"10.1016/j.jde.2025.02.058","DOIUrl":"10.1016/j.jde.2025.02.058","url":null,"abstract":"<div><div>In this paper, we consider the following critical polyharmonic equation<span><span><span><math><mrow><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><msup><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>y</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><msup><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></math></span>, <span><math><mi>N</mi><mo>></mo><mn>4</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, and <span><math><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo></math></span> is a bounded nonnegative function in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>. By using the reduction argument and local Pohoz̆aev identities, we prove that if <span><math><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo></math></span> has a stable critical point <span><math><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>)</mo></math></span> with <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>)</mo><mo>></mo><mn>0</mn></math></span>, then the above problem has a new type of solutions, which concentrate at points lying on the top and the bottom circles of a cylinder.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 678-715"},"PeriodicalIF":2.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479842","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":"On the cyclicity of hyperbolic polycycles","authors":"Claudio Buzzi , Armengol Gasull , Paulo Santana","doi":"10.1016/j.jde.2025.02.061","DOIUrl":"10.1016/j.jde.2025.02.061","url":null,"abstract":"<div><div>Let <em>X</em> be a planar smooth vector field with a polycycle <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <em>n</em> sides and all its corners, that are at most <em>n</em> singularities, being hyperbolic saddles. In this paper we study the cyclicity of <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least <em>k</em>, for any <span><math><mi>k</mi><mo>⩽</mo><mi>n</mi></math></span>. Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when <em>X</em> is polynomial there is a polynomial perturbation (in general with degree much higher that the one of <em>X</em>) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 646-677"},"PeriodicalIF":2.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479841","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}
Gleiciane S. Aragão , José M. Arrieta , Simone M. Bruschi
{"title":"Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation","authors":"Gleiciane S. Aragão , José M. Arrieta , Simone M. Bruschi","doi":"10.1016/j.jde.2025.02.041","DOIUrl":"10.1016/j.jde.2025.02.041","url":null,"abstract":"<div><div>In this paper we obtain the continuity of attractors for nonlinear parabolic equations with nonlinear boundary conditions when the boundary of the domain varies rapidly as a parameter <em>ϵ</em> goes to zero. We consider the case where the boundary of the domain presents a highly oscillatory behavior as the parameter <em>ϵ</em> goes to zero. For the case where we have a Lipschitz deformation of the boundary with the Lipschitz constant uniformly bounded in <em>ϵ</em> but the boundaries do not approach in a Lipschitz sense, the solutions of these equations converge in certain sense to the solution of a limit parabolic equation of the same type but where the boundary condition has a factor that captures the oscillations of the boundary. To address this problem, it is necessary to consider the notion of convergence of functions defined in varying domains and the convergence of a family of operators defined in different Banach spaces. Moreover, since we consider problems with nonlinear boundary conditions, it is necessary to extend these concepts to the case of spaces with negative exponents and to operators defined between these spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 460-502"},"PeriodicalIF":2.4,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455045","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":"Recovering initial states in semilinear parabolic problems from time-averages","authors":"Lina Sophie Schmitz, Christoph Walker","doi":"10.1016/j.jde.2025.02.049","DOIUrl":"10.1016/j.jde.2025.02.049","url":null,"abstract":"<div><div>Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of superlinear behavior near zero from small time-averages over arbitrary time periods.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 503-528"},"PeriodicalIF":2.4,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455043","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":"Polyhedral entire solutions in reaction-diffusion equations","authors":"Masaharu Taniguchi","doi":"10.1016/j.jde.2025.02.034","DOIUrl":"10.1016/j.jde.2025.02.034","url":null,"abstract":"<div><div>This paper studies polyhedral entire solutions to a bistable reaction-diffusion equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We consider a pyramidal traveling front solution to the same equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. As the speed goes to infinity, its projection converges to an <em>n</em>-dimensional polyhedral entire solution. Conversely, as the time goes to −∞, an <em>n</em>-dimensional polyhedral entire solution gives <em>n</em>-dimensional pyramidal traveling front solutions. The result in this paper suggests a correlation between traveling front solutions and entire solutions in general reaction-diffusion equations or systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 529-565"},"PeriodicalIF":2.4,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455046","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":"Dynamic boundary flux-driven shallow waters: Insights from a dissipative-dispersive system","authors":"Neng Zhu , Kun Zhao","doi":"10.1016/j.jde.2025.02.055","DOIUrl":"10.1016/j.jde.2025.02.055","url":null,"abstract":"<div><div>This paper is concerned with a shallow water system under dynamic boundary conditions:<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>u</mi><mi>w</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mi>ϵ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi>w</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mi>μ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>δ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>α</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>β</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>w</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>w</mi><mo>(</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>.</mo></math></span></span></span> By constructing suitable relative entropy functionals, it is shown that under certain conditions on <span><math><mi>α</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>β</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, classical solutions with potentially large energy exist globally in time, and the solutions converge to the equilibria determined by the initial and boundary conditions. The results hold for all values of <span><math><mi>m</mi><mo>⩾</mo><mn>1</mn></math></span> when <span><math><mi>μ</mi><mo>></mo><mn>0</mn></math></span>, and for <span><math><mi>m</mi><mo>⩾<","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 566-611"},"PeriodicalIF":2.4,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455055","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":"Nondegeneracy of positive solutions for a biharmonic Hartree equation and its application","authors":"Minbo Yang , Weiwei Ye , Xinyun Zhang","doi":"10.1016/j.jde.2025.02.024","DOIUrl":"10.1016/j.jde.2025.02.024","url":null,"abstract":"<div><div>We study the nondegeneracy of positive solutions of the following biharmonic Hartree equation<span><span><span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⁎</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>p</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>4</mn></mrow></mfrac></math></span>, <span><math><mi>N</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>N</mi></math></span>. Our method relies on the spherical harmonic decomposition and the Funk-Heck formula of the spherical harmonic functions. Then as an application, by applying a finite dimension reduction and local Pohožaev identity, we can construct multi-bubble solutions for the following equation with potential<span><span><span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mi>u</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⁎</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>N</mi><mo>≥</mo><mn>9</mn></math></span>, <span><math><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span> and <span><math><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo></math></span> is a bounded and nonnegative function. We prove that the existence result is restricted to the range <span><math><mn>6</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>≤</mo><mi>α</mi><mo><</mo><mi>N</mi></math></span> which shows the influence of the order of Riesz potential.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 796-849"},"PeriodicalIF":2.4,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444437","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":"A uniform C1 connecting lemma for singular flows","authors":"Ming Li , Xingzhong Liu","doi":"10.1016/j.jde.2025.02.043","DOIUrl":"10.1016/j.jde.2025.02.043","url":null,"abstract":"<div><div>We extend the uniform <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> connecting lemma <span><span>[28]</span></span> to singular flows. The perturbation parameters are valid in a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-neighborhood of the given vector field. Moreover, they are also uniform on the points except singularities and some periodic ones with small period.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 247-261"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437021","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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