Pub Date : 2026-01-28DOI: 10.1016/j.jde.2026.114158
Jingyu Li , Xiaowen Li , Ming Mei
This paper is concerned with the large time behaviors of solutions to the Burgers equation of porous-media type in the form of , where the diffusion with possesses the strong singularity of fast-diffusion at . The main issue of the paper is to show the asymptotic stability of viscous shock profiles with the constant states , where the strong singularity exhibits for the equation when the viscous shock wave reaches the singular point . To overcome such a strong singularity for wave stability, we first need to analyze the rate of the viscous shock wave to , then we artfully choose some weight functions which are closely dependent on the decay rate of the viscous shock wave to the singular point , and further show the wave stability by the weighted-energy-method.
{"title":"Asymptotic stability of viscous shock profiles to Burgers equation with singular super-fast diffusion","authors":"Jingyu Li , Xiaowen Li , Ming Mei","doi":"10.1016/j.jde.2026.114158","DOIUrl":"10.1016/j.jde.2026.114158","url":null,"abstract":"<div><div>This paper is concerned with the large time behaviors of solutions to the Burgers equation of porous-media type in the form of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>f</mi><msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span>, where the diffusion <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mo>|</mo><mi>m</mi><mo>|</mo></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span> with <span><math><mi>m</mi><mo><</mo><mn>0</mn></math></span> possesses the strong singularity of fast-diffusion at <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>. The main issue of the paper is to show the asymptotic stability of viscous shock profiles with the constant states <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>></mo><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, where the strong singularity exhibits for the equation when the viscous shock wave reaches the singular point <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>. To overcome such a strong singularity for wave stability, we first need to analyze the rate of the viscous shock wave to <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, then we artfully choose some weight functions which are closely dependent on the decay rate of the viscous shock wave to the singular point <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, and further show the wave stability by the weighted-energy-method.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114158"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074031","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114140
Fan Bu, Dachun Yang, Wen Yuan, Mingdong Zhang
Using growth functions, we introduce generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces with matrix weights. We first characterize these spaces, respectively, in terms of the φ-transform, the Peetre-type maximal function, and the Littlewood–Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel–Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov–Triebel–Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov–Triebel–Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.
{"title":"Matrix-weighted Besov–Triebel–Lizorkin spaces of optimal scale: Real-variable characterizations, invariance on integrable index, and Sobolev-type embedding","authors":"Fan Bu, Dachun Yang, Wen Yuan, Mingdong Zhang","doi":"10.1016/j.jde.2026.114140","DOIUrl":"10.1016/j.jde.2026.114140","url":null,"abstract":"<div><div>Using growth functions, we introduce generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces with matrix <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> weights. We first characterize these spaces, respectively, in terms of the <em>φ</em>-transform, the Peetre-type maximal function, and the Littlewood–Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel–Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov–Triebel–Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov–Triebel–Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"463 ","pages":"Article 114140"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146057371","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114160
Kaibin Zhang , Xinhua Li , Chunyou Sun
In this paper, we focus on the stability and long-time behavior problem for the 2D Boussinesq equations near the hydrostatic equilibrium with partial dissipation in the velocity and horizontal thermal diffusion. The lack of dissipation in the first component of the velocity and vertical thermal diffusion leads to the main difficulties. We establish the stability in , and demonstrate the exponential decay of its oscillatory portion in the .
{"title":"Stability and exponential decay for the 2D anisotropic Boussinesq equations near the hydrostatic equilibrium","authors":"Kaibin Zhang , Xinhua Li , Chunyou Sun","doi":"10.1016/j.jde.2026.114160","DOIUrl":"10.1016/j.jde.2026.114160","url":null,"abstract":"<div><div>In this paper, we focus on the stability and long-time behavior problem for the 2D Boussinesq equations near the hydrostatic equilibrium with partial dissipation in the velocity and horizontal thermal diffusion. The lack of dissipation in the first component of the velocity and vertical thermal diffusion leads to the main difficulties. We establish the stability in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and demonstrate the exponential decay of its oscillatory portion in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114160"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074033","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 : 2026-01-28DOI: 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":"2026-01-28","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}
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":"2026-01-28","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114167
Qunyi Bie , Hui Fang , Shu Wang , Yanping Zhou
This paper aims to study the global stability and long-time behavior of the three-dimensional incompressible anisotropic Navier-Stokes equations with only fractional horizontal dissipation. The absence of vertical dissipation induces substantial analytical difficulties, rendering classical methods such as Schonbek's Fourier splitting technique inapplicable. By developing refined anisotropic energy estimates that exploit both the divergence-free condition and the structure of the dissipation, we establish the global existence and asymptotic stability of small solutions in Sobolev spaces under weaker dissipation conditions than previously known. Furthermore, for suitably regular initial data, we prove sharp decay rates for the solution and its first-order derivatives. Our results substantially enlarge the admissible parameter regime and provide robust analytical tools that may also be applied to other fractional anisotropic fluid models.
{"title":"Stability and sharp decay for the 3D incompressible anisotropic Navier-Stokes equations with fractional horizontal dissipation","authors":"Qunyi Bie , Hui Fang , Shu Wang , Yanping Zhou","doi":"10.1016/j.jde.2026.114167","DOIUrl":"10.1016/j.jde.2026.114167","url":null,"abstract":"<div><div>This paper aims to study the global stability and long-time behavior of the three-dimensional incompressible anisotropic Navier-Stokes equations with only fractional horizontal dissipation. The absence of vertical dissipation induces substantial analytical difficulties, rendering classical methods such as Schonbek's Fourier splitting technique inapplicable. By developing refined anisotropic energy estimates that exploit both the divergence-free condition and the structure of the dissipation, we establish the global existence and asymptotic stability of small solutions in Sobolev spaces under weaker dissipation conditions than previously known. Furthermore, for suitably regular initial data, we prove sharp decay rates for the solution and its first-order derivatives. Our results substantially enlarge the admissible parameter regime and provide robust analytical tools that may also be applied to other fractional anisotropic fluid models.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"463 ","pages":"Article 114167"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146057372","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 : 2026-01-28DOI: 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":"2026-01-28","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114136
Émeric Bouin , Jérôme Coville , Xi Zhang
This paper is devoted to studying propagation phenomena in integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity and the standard weak Allee effect one. We study the effect of the tails of the dispersal kernel on the rate of expansion. When the tail of the kernel is sub-exponential, the exact separation between existence and non-existence of travelling waves is exhibited. This, in turn, provides the exact separation between finite speed propagation and acceleration in the Cauchy problem. Moreover, the exact rates of acceleration for dispersal kernels with sub-exponential and algebraic tails are provided. Our approach is generic and covers a large variety of dispersal kernels including those leading to convolution and fractional Laplace operators. Numerical simulations are provided to illustrate our results.
{"title":"Acceleration or finite speed propagation in integro-differential equations with logarithmic Allee effects","authors":"Émeric Bouin , Jérôme Coville , Xi Zhang","doi":"10.1016/j.jde.2026.114136","DOIUrl":"10.1016/j.jde.2026.114136","url":null,"abstract":"<div><div>This paper is devoted to studying propagation phenomena in integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity and the standard weak Allee effect one. We study the effect of the tails of the dispersal kernel on the rate of expansion. When the tail of the kernel is sub-exponential, the exact separation between existence and non-existence of travelling waves is exhibited. This, in turn, provides the exact separation between finite speed propagation and acceleration in the Cauchy problem. Moreover, the exact rates of acceleration for dispersal kernels with sub-exponential and algebraic tails are provided. Our approach is generic and covers a large variety of dispersal kernels including those leading to convolution and fractional Laplace operators. Numerical simulations are provided to illustrate our results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114136"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075455","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114135
Simone Mauro , Delia Schiera , Hugo Tavares
We study the following gradient elliptic system with Neumann boundary conditions where is a bounded domain with , and ν denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative () and the competitive () regimes, considering both the definite and the indefinite case, namely . We emphasize that our analysis includes both the subcritical case and the critical case .
Depending on the values of , the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions.
{"title":"Least energy solutions for cooperative and competitive Schrödinger systems with Neumann boundary conditions","authors":"Simone Mauro , Delia Schiera , Hugo Tavares","doi":"10.1016/j.jde.2026.114135","DOIUrl":"10.1016/j.jde.2026.114135","url":null,"abstract":"<div><div>We study the following gradient elliptic system with Neumann boundary conditions<span><span><span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>u</mi><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>v</mi><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>β</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>v</mi><mspace></mspace><mtext>in </mtext><mi>Ω</mi><mo>,</mo><mspace></mspace><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mspace></mspace><mtext>on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a bounded <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> domain with <span><math><mi>N</mi><mo>≤</mo><mn>4</mn></math></span>, and <em>ν</em> denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative (<span><math><mi>β</mi><mo>></mo><mn>0</mn></math></span>) and the competitive (<span><math><mi>β</mi><mo><</mo><mn>0</mn></math></span>) regimes, considering both the definite and the indefinite case, namely <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>R</mi></math></span>. We emphasize that our analysis includes both the subcritical case <span><math><mi>N</mi><mo>≤</mo><mn>3</mn></math></span> and the critical case <span><math><mi>N</mi><mo>=</mo><mn>4</mn></math></span>.</div><div>Depending on the values of <span><math><mi>β</mi><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114135"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074032","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 : 2026-01-28DOI: 10.1016/j.jde.2026.114142
Jie Chen , Fan Gu , Boling Guo
In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in . To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear terms with . Then, to overcome regularity problems, we introduce a series of approximation equations with localized nonlinear terms, which are also cutted-off in both the physical and the frequency space. By limitations, these approximation equations will help us get a priori estimate in the Bourgain space and finish the proof of the global well-posedness of the initial system.
{"title":"Stochastic Schrödinger-Korteweg de Vries systems driven by multiplicative noises","authors":"Jie Chen , Fan Gu , Boling Guo","doi":"10.1016/j.jde.2026.114142","DOIUrl":"10.1016/j.jde.2026.114142","url":null,"abstract":"<div><div>In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>×</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear terms with <span><math><mi>b</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></math></span>. Then, to overcome regularity problems, we introduce a series of approximation equations with localized nonlinear terms, which are also cutted-off in both the physical and the frequency space. By limitations, these approximation equations will help us get a priori estimate in the Bourgain space and finish the proof of the global well-posedness of the initial system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114142"},"PeriodicalIF":2.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074310","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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