Pub Date : 2026-03-01Epub Date: 2025-11-01DOI: 10.1016/j.na.2025.113986
Umberto Guarnotta , Patrick Winkert
In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the -Laplacian, the -Laplacian, the double phase operator, and the logarithmic double phase operator.
{"title":"Degenerate singular Kirchhoff problems in Musielak–Orlicz spaces","authors":"Umberto Guarnotta , Patrick Winkert","doi":"10.1016/j.na.2025.113986","DOIUrl":"10.1016/j.na.2025.113986","url":null,"abstract":"<div><div>In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the <span><math><mi>p</mi></math></span>-Laplacian, the <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>-Laplacian, the double phase operator, and the logarithmic double phase operator.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113986"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419858","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 : 2026-03-01Epub Date: 2025-11-06DOI: 10.1016/j.na.2025.113997
Daniel Spector , Dmitriy Stolyarov
In this paper, we define spaces of measures with dimensional stability . These spaces bridge between , the space of finite Radon measures, and , the real Hardy space. We show the spaces support Sobolev inequalities for , while for any we show that the lower Hausdorff dimension of an element of is at least .
{"title":"On dimension stable spaces of measures","authors":"Daniel Spector , Dmitriy Stolyarov","doi":"10.1016/j.na.2025.113997","DOIUrl":"10.1016/j.na.2025.113997","url":null,"abstract":"<div><div>In this paper, we define spaces of measures <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with dimensional stability <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. These spaces bridge between <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the space of finite Radon measures, and <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the real Hardy space. We show the spaces <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> support Sobolev inequalities for <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>]</mo></mrow></mrow></math></span>, while for any <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>]</mo></mrow></mrow></math></span> we show that the lower Hausdorff dimension of an element of <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> is at least <span><math><mi>β</mi></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113997"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467932","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 : 2026-03-01Epub Date: 2025-11-05DOI: 10.1016/j.na.2025.114004
David Jesus , María Soria-Carro
We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is , in the parabolic sense, then viscosity solutions are piecewise up to the interface. As byproducts, we obtain a new ABP–Krylov–Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.
{"title":"Fully nonlinear parabolic fixed transmission problems","authors":"David Jesus , María Soria-Carro","doi":"10.1016/j.na.2025.114004","DOIUrl":"10.1016/j.na.2025.114004","url":null,"abstract":"<div><div>We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>, in the parabolic sense, then viscosity solutions are piecewise <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> up to the interface. As byproducts, we obtain a new ABP–Krylov–Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114004"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467931","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 : 2026-03-01Epub Date: 2025-11-05DOI: 10.1016/j.na.2025.113998
Stefano Almi , Dario Reggiani , Francesco Solombrino
We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the -distance of a possibly incompatible strain field from a single well is controlled in terms of the -distance from a finite set of wells, of , and of . As a consequence, we derive a strain-gradient plasticity model as -limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.
{"title":"Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity","authors":"Stefano Almi , Dario Reggiani , Francesco Solombrino","doi":"10.1016/j.na.2025.113998","DOIUrl":"10.1016/j.na.2025.113998","url":null,"abstract":"<div><div>We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>∗</mo></mrow></msup></mrow></msup></math></span>-distance of a possibly incompatible strain field <span><math><mi>β</mi></math></span> from a single well is controlled in terms of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>∗</mo></mrow></msup></mrow></msup></math></span>-distance from a finite set of wells, of <span><math><mrow><mi>curl</mi><mi>β</mi></mrow></math></span>, and of <span><math><mrow><mi>div</mi><mi>β</mi></mrow></math></span>. As a consequence, we derive a strain-gradient plasticity model as <span><math><mi>Γ</mi></math></span>-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113998"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467934","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 : 2026-03-01Epub Date: 2025-11-05DOI: 10.1016/j.na.2025.114003
Willian Cintra , Cristian Morales-Rodrigo , Antonio Suárez
In this work, we analyze a stationary degenerate logistic equation with both local and non-local diffusion. Primarily employing bifurcation results, sub- and supersolution methods, and maximum principles, we establish results regarding the existence, non-existence, and uniqueness of positive solutions. Additionally, using appropriate large solutions, we conduct a detailed study of the asymptotic behavior of the solutions with respect to one of the equation’s parameters, showing that the presence of the non-local diffusion can drastically change this pointwise behavior when compared with the local case.
{"title":"Asymptotic behavior of positive solutions for a degenerate logistic equation with mixed local and non-local diffusion","authors":"Willian Cintra , Cristian Morales-Rodrigo , Antonio Suárez","doi":"10.1016/j.na.2025.114003","DOIUrl":"10.1016/j.na.2025.114003","url":null,"abstract":"<div><div>In this work, we analyze a stationary degenerate logistic equation with both local and non-local diffusion. Primarily employing bifurcation results, sub- and supersolution methods, and maximum principles, we establish results regarding the existence, non-existence, and uniqueness of positive solutions. Additionally, using appropriate large solutions, we conduct a detailed study of the asymptotic behavior of the solutions with respect to one of the equation’s parameters, showing that the presence of the non-local diffusion can drastically change this pointwise behavior when compared with the local case.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114003"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467933","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 : 2026-03-01Epub Date: 2025-10-29DOI: 10.1016/j.na.2025.113985
Hirokazu Saito , Yoshihiro Shibata
This paper is concerned with the global solvability for the Navier–Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. In the finite-depth case, Poincaré inequalities play a crucial role to handle lower order terms in previous works such as Beale (1984), Guo and Tice (2013) in -based Sobolev spaces. Our situation of the present paper, however, cannot use them due to lack of boundedness in the vertical direction. To overcome this difficultly, we employ time decay estimates of a -analytic semigroup generated by the Stokes operator with a free boundary condition. More precisely, we first prove a time weighted estimate of solutions to a linearized system of the Navier–Stokes equations by using the time decay estimates as above and maximal regularity estimates in an -in-time and -in-space setting with suitable , . The time weighted estimate then enables us to show the global solvability of the Navier–Stokes equations for small initial data by the contraction mapping principle. Although this approach is based on our previous paper studying the three-dimensional case, we introduce a new time weighted estimate to deal with higher dimensions and simplify the proof of the three-dimensional case. Furthermore, we establish a new estimate of Duhamel’s integral based on Shibata (2022) in our linear theory, which is of independent interest.
{"title":"Global solvability for viscous free surface flows of infinite depth in three and higher dimensions","authors":"Hirokazu Saito , Yoshihiro Shibata","doi":"10.1016/j.na.2025.113985","DOIUrl":"10.1016/j.na.2025.113985","url":null,"abstract":"<div><div>This paper is concerned with the global solvability for the Navier–Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. In the finite-depth case, Poincaré inequalities play a crucial role to handle lower order terms in previous works such as Beale (1984), Guo and Tice (2013) in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-based Sobolev spaces. Our situation of the present paper, however, cannot use them due to lack of boundedness in the vertical direction. To overcome this difficultly, we employ time decay estimates of a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-analytic semigroup generated by the Stokes operator with a free boundary condition. More precisely, we first prove a time weighted estimate of solutions to a linearized system of the Navier–Stokes equations by using the time decay estimates as above and maximal regularity estimates in an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-in-time and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-in-space setting with suitable <span><math><mi>p</mi></math></span>, <span><math><mi>q</mi></math></span>. The time weighted estimate then enables us to show the global solvability of the Navier–Stokes equations for small initial data by the contraction mapping principle. Although this approach is based on our previous paper studying the three-dimensional case, we introduce a new time weighted estimate to deal with higher dimensions and simplify the proof of the three-dimensional case. Furthermore, we establish a new estimate of Duhamel’s integral based on Shibata (2022) in our linear theory, which is of independent interest.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113985"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419859","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 : 2026-03-01Epub Date: 2025-11-12DOI: 10.1016/j.na.2025.113981
Jaume Giné , Dmitry I. Sinelshchikov
It has been conjectured that the only mechanisms capable of producing a center – whether degenerate or not – at a singular point of a polynomial differential system are algebraic reducibility and Liouvillian integrability. In this work, we present an example that is algebraically reducible but neither orbitally reversible nor Liouvillian integrable. The construction of this example is based on a recently developed mechanism that establishes a necessary and sufficient condition for the existence of a center.
{"title":"A new mechanism for producing degenerate centers in polynomial differential systems","authors":"Jaume Giné , Dmitry I. Sinelshchikov","doi":"10.1016/j.na.2025.113981","DOIUrl":"10.1016/j.na.2025.113981","url":null,"abstract":"<div><div>It has been conjectured that the only mechanisms capable of producing a center – whether degenerate or not – at a singular point of a polynomial differential system are algebraic reducibility and Liouvillian integrability. In this work, we present an example that is algebraically reducible but neither orbitally reversible nor Liouvillian integrable. The construction of this example is based on a recently developed mechanism that establishes a necessary and sufficient condition for the existence of a center.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113981"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520545","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 : 2026-03-01Epub Date: 2025-11-04DOI: 10.1016/j.na.2025.114002
Stefano Marò , Rafael Ortega
We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result is applied to the discrete Newton equation, the model of a bouncing ball and the Fermi–Ulam ping-pong model.
{"title":"Stability of some periodic configurations of discrete Lagrangian equations","authors":"Stefano Marò , Rafael Ortega","doi":"10.1016/j.na.2025.114002","DOIUrl":"10.1016/j.na.2025.114002","url":null,"abstract":"<div><div>We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result is applied to the discrete Newton equation, the model of a bouncing ball and the Fermi–Ulam ping-pong model.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114002"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467440","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 : 2026-02-01Epub Date: 2025-10-03DOI: 10.1016/j.na.2025.113967
Giulia Meglioli , Francescantonio Oliva , Francesco Petitta
We show a global existence result for a doubly nonlinear porous medium type equation of the form on a complete and non-compact Riemannian manifold of infinite volume. Here, for , we assume , and . In particular, under the assumptions that supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided and the initial datum is small enough; namely, we establish an explicit bound on the norm of the solution at all positive times, in terms of the norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in , we can establish the same result in the larger interval, i.e. . This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that is non-compact and has infinite measure.
在无限体积的完全非紧黎曼流形M上，给出了形式为ut=Δpum+uq的双非线性多孔介质型方程的整体存在性结果。这里，对于1<；p<；N，我们假设m(p−1)≥1,m>；1和q>；m（p−1）。特别地，在M支持Sobolev不等式的假设下，我们证明了在q>； M (p−1)+pN且初始基准足够小的情况下，该问题的解在时间上全局存在；也就是说，我们根据数据的L1范数，在所有正时刻的解的L∞范数上建立一个显式的界。在附加的假设下，一个poincar型不等式在M中也成立，我们可以在更大的区间，即q>； M （p−1）中建立同样的结果。这个结果没有欧几里得对应物，因为它完全不同于有界欧几里得定义域的情况，因为M是非紧致的并且具有无限的度量。
{"title":"Global existence for a Leibenson type equation with reaction on Riemannian manifolds","authors":"Giulia Meglioli , Francescantonio Oliva , Francesco Petitta","doi":"10.1016/j.na.2025.113967","DOIUrl":"10.1016/j.na.2025.113967","url":null,"abstract":"<div><div>We show a global existence result for a doubly nonlinear porous medium type equation of the form <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></math></span> on a complete and non-compact Riemannian manifold <span><math><mi>M</mi></math></span> of infinite volume. Here, for <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span>, we assume <span><math><mrow><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>m</mi><mo>></mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. In particular, under the assumptions that <span><math><mi>M</mi></math></span> supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>N</mi></mrow></mfrac></mrow></math></span> and the initial datum is small enough; namely, we establish an explicit bound on the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> norm of the solution at all positive times, in terms of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in <span><math><mi>M</mi></math></span>, we can establish the same result in the larger interval, i.e. <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that <span><math><mi>M</mi></math></span> is non-compact and has infinite measure.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113967"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222823","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 : 2026-02-01Epub Date: 2025-10-03DOI: 10.1016/j.na.2025.113964
Alysson Cunha
We prove that the Benjamin–Ono equation is globally well-posed in for . Our approach does not rely on the global gauge transformation introduced by Tao (Tao, 2004). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin–Ono (DGBO) equation.
{"title":"Improvement of the parabolic regularization method and applications to dispersive models","authors":"Alysson Cunha","doi":"10.1016/j.na.2025.113964","DOIUrl":"10.1016/j.na.2025.113964","url":null,"abstract":"<div><div>We prove that the Benjamin–Ono equation is globally well-posed in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Our approach does not rely on the global gauge transformation introduced by Tao (Tao, 2004). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin–Ono (DGBO) equation.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113964"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223263","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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