Journal of Differential Equations最新文献

{"title":"On certain degenerate and singular elliptic PDEs IV: Nondivergence-form operators with logarithmic degeneracies or singularities","authors":"Diego Maldonado","doi":"10.1016/j.jde.2024.10.017","DOIUrl":"10.1016/j.jde.2024.10.017","url":null,"abstract":"<div><div>Harnack inequalities for nonnegative strong solutions to nondivergence-form elliptic PDEs with degeneracies or singularities of logarithmic type are proved. The results are developed within the Monge-Ampère real-analytic and geometric tools associated to certain convex functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530875","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 for degenerate wave equations with drift under simultaneous degenerate damping","authors":"Mohammad Akil ,&nbsp;Genni Fragnelli ,&nbsp;Ibtissam Issa","doi":"10.1016/j.jde.2024.10.022","DOIUrl":"10.1016/j.jde.2024.10.022","url":null,"abstract":"<div><div>In this paper we study the stability of two different problems. The first one is a one-dimensional degenerate wave equation with degenerate damping, incorporating a drift term and a leading operator in non-divergence form. In the second problem we consider a system that couples degenerate and non-degenerate wave equations, connected through transmission, and subject to a single dissipation law at the boundary of the non-degenerate equation. In both scenarios, we derive exponential stability results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530865","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":"Dispersive estimates for Maxwell's equations in the exterior of a sphere","authors":"Yan-long Fang ,&nbsp;Alden Waters","doi":"10.1016/j.jde.2024.10.024","DOIUrl":"10.1016/j.jde.2024.10.024","url":null,"abstract":"<div><div>The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> operator norm in terms of time <em>t</em> and powers of <em>h</em>. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527658","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 growth mechanism of boundary layers for the 2D Navier-Stokes equations","authors":"Fei Wang ,&nbsp;Yichun Zhu","doi":"10.1016/j.jde.2024.10.012","DOIUrl":"10.1016/j.jde.2024.10.012","url":null,"abstract":"<div><div>We give a detailed description of formation of the boundary layers in the inviscid limit problem. To be more specific, we prove that the magnitude of the vorticity near the boundary is growing to the size of <span><math><mn>1</mn><mo>/</mo><msqrt><mrow><mi>ν</mi></mrow></msqrt></math></span> and the width of the layer is spreading out to be proportional the <span><math><msqrt><mrow><mi>ν</mi></mrow></msqrt></math></span> in a finite time period. In fact, the growth time scaling is almost <em>ν</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530869","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":"Asymptotic behavior for stationary Navier-Stokes equations","authors":"Yupei Li,&nbsp;Wei Luo","doi":"10.1016/j.jde.2024.10.011","DOIUrl":"10.1016/j.jde.2024.10.011","url":null,"abstract":"<div><div>In this paper, we investigate the asymptotic behavior of solutions to the axisymmetric stationary Navier-Stokes equations. We assume that the flow is periodic in <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-direction and has no swirl. Under the general integrability condition, we prove the pointwise decay estimate of the vorticity <em>ω</em> and obtain the Liouville-type theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530871","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":"More limit cycles for complex differential equations with three monomials","authors":"M.J. Álvarez ,&nbsp;B. Coll ,&nbsp;A. Gasull ,&nbsp;R. Prohens","doi":"10.1016/j.jde.2024.10.013","DOIUrl":"10.1016/j.jde.2024.10.013","url":null,"abstract":"<div><div>In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, <span><math><mover><mrow><mi>z</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>A</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>l</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup></math></span>, being <span><math><mi>k</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></math></span> non-negative integers and <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>C</mi></math></span>. More concretely, if <span><math><mi>N</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>(</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo>,</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> denotes the maximum number of limit cycles of the above equations, we show that for <span><math><mi>N</mi><mo>≥</mo><mn>4</mn></math></span>, <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>≥</mo><mi>N</mi><mo>−</mo><mn>3</mn></math></span> and that for some values of <em>N</em> this new lower bound is <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span>. We also present examples with many limit cycles and different configurations. Finally, we show that <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530874","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":"L1-theory for incompressible limit of reaction-diffusion porous medium flow with linear drift","authors":"Noureddine Igbida","doi":"10.1016/j.jde.2024.09.042","DOIUrl":"10.1016/j.jde.2024.09.042","url":null,"abstract":"<div><div>Our aim is to study existence, uniqueness and the limit, as <span><math><mi>m</mi><mo>→</mo><mo>∞</mo></math></span>, of the solution of the porous medium equation with linear drift <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mspace></mspace><mi>V</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span> in bounded domain with Dirichlet boundary condition. We treat the problem without any sign restriction on the solution with an outpointing vector field <em>V</em> on the boundary and a general source term <em>g</em> (including the continuous Lipschitz case). Under reasonably sharp Sobolev assumptions on <em>V</em>, we show uniform <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-convergence towards the solution of reaction-diffusion Hele-Shaw flow with linear drift.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530870","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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}