Pub Date : 2026-04-25Epub Date: 2026-01-28DOI: 10.1016/j.jde.2026.114138
Paolo Acampora , Emanuele Cristoforoni , Carlo Nitsch , Cristina Trombetti
A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's original estimate—which is not sharp—can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.
{"title":"An improved version of a spectral inequality by Payne","authors":"Paolo Acampora , Emanuele Cristoforoni , Carlo Nitsch , Cristina Trombetti","doi":"10.1016/j.jde.2026.114138","DOIUrl":"10.1016/j.jde.2026.114138","url":null,"abstract":"<div><div>A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's original estimate—which is not sharp—can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114138"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074311","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 : 2026-04-25Epub Date: 2026-01-30DOI: 10.1016/j.jde.2026.114161
Guanghui Wang , Lingda Xu , Mingying Zhong
In this paper, we study the hydrodynamic limit for rarefaction wave from the Boltzmann equation to Euler equations. We obtain the convergence rate of ϵ in norm on finite time interval , where is the Knudsen number and is any fixed constant. This convergence rate coincides with Caflisch 1980, cf. [1], which studied the hydrodynamic limit for smooth Euler solutions. This rate improves the result of Xin-Zeng 2010, where the convergence rate is in norm, cf. [25]. The result is obtained by a refined energy estimate and the better rates are obtained for the higher-order derivatives.
{"title":"Hydrodynamic limit to the rarefaction wave for the Boltzmann equation","authors":"Guanghui Wang , Lingda Xu , Mingying Zhong","doi":"10.1016/j.jde.2026.114161","DOIUrl":"10.1016/j.jde.2026.114161","url":null,"abstract":"<div><div>In this paper, we study the hydrodynamic limit for rarefaction wave from the Boltzmann equation to Euler equations. We obtain the convergence rate of <em>ϵ</em> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> norm on finite time interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></math></span>, where <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> is the Knudsen number and <span><math><mi>T</mi><mo>></mo><mn>0</mn></math></span> is any fixed constant. This convergence rate coincides with Caflisch 1980, cf. <span><span>[1]</span></span>, which studied the hydrodynamic limit for smooth Euler solutions. This rate improves the result of Xin-Zeng 2010, where the convergence rate is <span><math><msup><mrow><mi>ϵ</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> norm, cf. <span><span>[25]</span></span>. The result is obtained by a refined energy estimate and the better rates are obtained for the higher-order derivatives.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114161"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074315","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 : 2026-04-25Epub Date: 2026-01-21DOI: 10.1016/j.jde.2026.114126
Pedro C.C.R. Pereira , Mike R. Jeffrey , Douglas D. Novaes
When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order ‘guiding system’, placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that -universal bifurcations in the guiding system ‘persist’ in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.
{"title":"Averaging theory and catastrophes","authors":"Pedro C.C.R. Pereira , Mike R. Jeffrey , Douglas D. Novaes","doi":"10.1016/j.jde.2026.114126","DOIUrl":"10.1016/j.jde.2026.114126","url":null,"abstract":"<div><div>When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order ‘guiding system’, placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that <span><math><mi>K</mi></math></span>-universal bifurcations in the guiding system ‘persist’ in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114126"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036010","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 : 2026-04-25Epub Date: 2026-01-26DOI: 10.1016/j.jde.2026.114143
Hai Zhou, Tao Zhou
<div><div>In this paper, we investigate the properties of the spreading speeds for the following Fisher-KPP lattice system in the almost periodic media:<span><span><span>(⁎)</span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>i</mi><mo>)</mo><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mspace></mspace><mtext>is nonzero with compact support</mtext><mo>.</mo></mtd></mtr></mtable></mrow></math></span></span></span> First, we prove the existence of spreading speeds <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> of <span><span>(⁎)</span></span> in the positive and negative directions, respectively, without the “small drift” assumption. Moreover, the difference between the speeds on both sides (i.e., which is larger) is determined by a certain average of the left and right fluxes. Specifically,<span><span><span><math><mtext>sgn</mtext><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>−</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo><mo>=</mo><mtext>sgn</mtext><mo>(</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>ln</mi><mo></mo><mfrac><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>)</mo><mo>.</mo></math></span></span></span> We also prove the convergence of the average in the discrete case to that in the continuous case. Additionally, we demonstrate that, in the homogeneous case, any small perturbation of the 2-periodic drift reduces the expanding spread of the level set, i.e., the value <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>+</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo
{"title":"Propagation properties of Fisher-KPP lattice equations with almost periodic coefficients","authors":"Hai Zhou, Tao Zhou","doi":"10.1016/j.jde.2026.114143","DOIUrl":"10.1016/j.jde.2026.114143","url":null,"abstract":"<div><div>In this paper, we investigate the properties of the spreading speeds for the following Fisher-KPP lattice system in the almost periodic media:<span><span><span>(⁎)</span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>i</mi><mo>)</mo><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>i</mi><mo>)</mo><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mspace></mspace><mtext>is nonzero with compact support</mtext><mo>.</mo></mtd></mtr></mtable></mrow></math></span></span></span> First, we prove the existence of spreading speeds <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> of <span><span>(⁎)</span></span> in the positive and negative directions, respectively, without the “small drift” assumption. Moreover, the difference between the speeds on both sides (i.e., which is larger) is determined by a certain average of the left and right fluxes. Specifically,<span><span><span><math><mtext>sgn</mtext><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>−</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo><mo>=</mo><mtext>sgn</mtext><mo>(</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>ln</mi><mo></mo><mfrac><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>)</mo><mo>.</mo></math></span></span></span> We also prove the convergence of the average in the discrete case to that in the continuous case. Additionally, we demonstrate that, in the homogeneous case, any small perturbation of the 2-periodic drift reduces the expanding spread of the level set, i.e., the value <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>+</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>−</mo","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114143"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074313","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}
The Tropical Climate Model (TCM) is a simplified system that captures key aspects of equatorial atmospheric dynamics through the interaction of barotropic and baroclinic velocity modes with temperature fields. This study focuses on the nonlinear stability of Couette flow in a two-dimensional TCM with only partial dissipation. Two main difficulties arise: the absence of full dissipation, and the lack of a divergence-free condition for the baroclinic velocity. To address these challenges, we develop a refined Fourier multiplier approach that captures enhanced dissipation via the interaction between the shear-induced mixing term and vertical viscosity. Furthermore, this paper introduces new techniques to handle terms involving non-divergence-free components and exploits key couplings within the system to control potentially unstable linear terms. Under appropriate smallness conditions on the initial perturbations in anisotropic Sobolev spaces, we rigorously establish the nonlinear stability of the Couette flow and identify a possible precise transition threshold for stability.
{"title":"Stability of 2D tropical climate system with partial dissipations near Couette flow","authors":"Dongjuan Niu , Huiru Wu , Jiahong Wu , Xiaojing Xu","doi":"10.1016/j.jde.2026.114148","DOIUrl":"10.1016/j.jde.2026.114148","url":null,"abstract":"<div><div>The Tropical Climate Model (TCM) is a simplified system that captures key aspects of equatorial atmospheric dynamics through the interaction of barotropic and baroclinic velocity modes with temperature fields. This study focuses on the nonlinear stability of Couette flow in a two-dimensional TCM with only partial dissipation. Two main difficulties arise: the absence of full dissipation, and the lack of a divergence-free condition for the baroclinic velocity. To address these challenges, we develop a refined Fourier multiplier approach that captures enhanced dissipation via the interaction between the shear-induced mixing term and vertical viscosity. Furthermore, this paper introduces new techniques to handle terms involving non-divergence-free components and exploits key couplings within the system to control potentially unstable linear terms. Under appropriate smallness conditions on the initial perturbations in anisotropic Sobolev spaces, we rigorously establish the nonlinear stability of the Couette flow and identify a possible precise transition threshold for stability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114148"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074314","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 : 2026-04-25Epub Date: 2026-01-16DOI: 10.1016/j.jde.2026.114107
Wanxiao Xu , Hongying Shu , Lin Wang , Xiang-Sheng Wang , Jianshe Yu
Incorporating spatial diffusion and digestion delay into an intraguild predation (IGP) model, this work demonstrates rich spatiotemporal dynamics governing biological invasions. We derive criteria for the successful invasion of the intraguild predator and identify a critical diffusion threshold that eliminates spatially heterogeneous steady states. The digestion delay induces stability switches, resulting in a finite number of stability intervals, and causing abrupt shifts in coexistence patterns as the delay crosses critical thresholds. Through steady state bifurcation analysis, we rigorously establish the emergence of spatially heterogeneous coexistence states. We further derive Turing instability conditions for Hopf-bifurcating periodic solutions in a general three-dimensional delayed diffusive system. Our results reveal multiple coexistence mechanisms, including homogeneous steady states, periodic oscillations, and complex spatiotemporal patterns, highlighting the intricate interplay between time delay and spatial heterogeneity in biological invasions.
{"title":"Complex spatiotemporal dynamics in a diffusive intraguild predation model with digestion delay","authors":"Wanxiao Xu , Hongying Shu , Lin Wang , Xiang-Sheng Wang , Jianshe Yu","doi":"10.1016/j.jde.2026.114107","DOIUrl":"10.1016/j.jde.2026.114107","url":null,"abstract":"<div><div>Incorporating spatial diffusion and digestion delay into an intraguild predation (IGP) model, this work demonstrates rich spatiotemporal dynamics governing biological invasions. We derive criteria for the successful invasion of the intraguild predator and identify a critical diffusion threshold that eliminates spatially heterogeneous steady states. The digestion delay induces stability switches, resulting in a finite number of stability intervals, and causing abrupt shifts in coexistence patterns as the delay crosses critical thresholds. Through steady state bifurcation analysis, we rigorously establish the emergence of spatially heterogeneous coexistence states. We further derive Turing instability conditions for Hopf-bifurcating periodic solutions in a general three-dimensional delayed diffusive system. Our results reveal multiple coexistence mechanisms, including homogeneous steady states, periodic oscillations, and complex spatiotemporal patterns, highlighting the intricate interplay between time delay and spatial heterogeneity in biological invasions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114107"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980541","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 : 2026-04-25Epub Date: 2026-01-16DOI: 10.1016/j.jde.2026.114112
Andrew Yang , Xu Zhao , Wenshu Zhou
We study free boundary problem of the one dimensional compressible isentropic Navier–Stokes equations with density–dependent viscosity when the initial density connects to the vacuum states continuously and is either of compact or infinite support. Precisely, the pressure and the viscosity coefficient are assumed to be proportional to and respectively, where ρ is the density, and γ and θ are positive constants. We prove the global existence of smooth solutions with large initial data when and . Since the power θ of the previous results on this topic does not exceed 2, the result of this paper fills at least the gap for large θ. The result includes also the case of the infinite support of the initial density, which just corresponds to the one when . Notice that two key estimates of the proof are the uniform lower bound of the density and the uniform bound of the velocity with respect to the construction of the approximate solutions. In contrast to the traditional techniques relying on weighted energy estimates, they are proved independently by the comparison principle and the maximal principle, respectively. Moreover, we obtain some results on regularity up to boundary and uniqueness of solutions. The results of this paper cover some important models, for instance, the viscous Saint–Venant model for the motion of shallow water, i.e., and .
{"title":"Global smooth solutions of compressible Navier–Stokes equations with degenerate viscosity and vacuum","authors":"Andrew Yang , Xu Zhao , Wenshu Zhou","doi":"10.1016/j.jde.2026.114112","DOIUrl":"10.1016/j.jde.2026.114112","url":null,"abstract":"<div><div>We study free boundary problem of the one dimensional compressible isentropic Navier–Stokes equations with density–dependent viscosity when the initial density connects to the vacuum states continuously and is either of compact or infinite support. Precisely, the pressure and the viscosity coefficient are assumed to be proportional to <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mi>γ</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mi>θ</mi></mrow></msup></math></span> respectively, where <em>ρ</em> is the density, and <em>γ</em> and <em>θ</em> are positive constants. We prove the global existence of smooth solutions with large initial data when <span><math><mi>θ</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>γ</mi><mo>≥</mo><mn>1</mn><mo>+</mo><mi>θ</mi></math></span>. Since the power <em>θ</em> of the previous results on this topic does not exceed 2, the result of this paper fills at least the gap for large <em>θ</em>. The result includes also the case of the infinite support of the initial density, which just corresponds to the one when <span><math><mn>0</mn><mo><</mo><mi>θ</mi><mo>≤</mo><mn>1</mn></math></span>. Notice that two key estimates of the proof are the uniform lower bound of the density and the uniform <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> bound of the velocity with respect to the construction of the approximate solutions. In contrast to the traditional techniques relying on weighted energy estimates, they are proved independently by the comparison principle and the maximal principle, respectively. Moreover, we obtain some results on regularity up to boundary and uniqueness of solutions. The results of this paper cover some important models, for instance, the viscous Saint–Venant model for the motion of shallow water, i.e., <span><math><mi>θ</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>γ</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114112"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980607","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 : 2026-04-25Epub Date: 2026-01-19DOI: 10.1016/j.jde.2026.114105
Mohameden Ahmedou , Mohamed Ben Ayed , Khalil El Mehdi
Given a smooth positive function K on the standard sphere , we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function K, there are arbitrarily many metrics g conformally equivalent to and whose scalar curvature is given by the function K provided that the function is sufficiently close to the scalar curvature of . Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function K.
{"title":"On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting","authors":"Mohameden Ahmedou , Mohamed Ben Ayed , Khalil El Mehdi","doi":"10.1016/j.jde.2026.114105","DOIUrl":"10.1016/j.jde.2026.114105","url":null,"abstract":"<div><div>Given a smooth positive function <em>K</em> on the standard sphere <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function <em>K</em>, there are arbitrarily many metrics <em>g</em> conformally equivalent to <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and whose scalar curvature is given by the function <em>K</em> provided that the function is sufficiently close to the scalar curvature of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function <em>K</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114105"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036009","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 : 2026-04-25Epub Date: 2026-01-28DOI: 10.1016/j.jde.2026.114145
Zhengni Hu, Miaomiao Zhu
In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the “k-symmetric” condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the Toda system. Furthermore, the blow-up points are precisely located at the “k-symmetric” centers of the surface.
{"title":"Blow-up solutions for general Toda systems on Riemann surfaces","authors":"Zhengni Hu, Miaomiao Zhu","doi":"10.1016/j.jde.2026.114145","DOIUrl":"10.1016/j.jde.2026.114145","url":null,"abstract":"<div><div>In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the “<em>k</em>-symmetric” condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the <span><math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> Toda system. Furthermore, the blow-up points are precisely located at the “<em>k</em>-symmetric” centers of the surface.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114145"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074312","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 : 2026-04-25Epub Date: 2026-01-20DOI: 10.1016/j.jde.2026.114129
Lassi Paunonen , David Seifert
We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler–Bernoulli beam with a tip mass subject to non-linear damping.
{"title":"Polynomial stability of non-linearly damped contraction semigroups","authors":"Lassi Paunonen , David Seifert","doi":"10.1016/j.jde.2026.114129","DOIUrl":"10.1016/j.jde.2026.114129","url":null,"abstract":"<div><div>We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler–Bernoulli beam with a tip mass subject to non-linear damping.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114129"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146034392","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号