Pub Date : 2025-03-01DOI: 10.1016/j.na.2025.113780
Gregory J. Galloway , Abraão Mendes
In this note, we consider some initial data rigidity results concerning marginally outer trapped surfaces (MOTS). As is well known, MOTS play an important role in the theory of black holes and, at the same time, are interesting spacetime analogues of minimal surfaces in Riemannian geometry. The main results presented here expand upon earlier works by the authors, specifically addressing initial data sets incorporating charge.
{"title":"Some rigidity results for charged initial data sets","authors":"Gregory J. Galloway , Abraão Mendes","doi":"10.1016/j.na.2025.113780","DOIUrl":"10.1016/j.na.2025.113780","url":null,"abstract":"<div><div>In this note, we consider some initial data rigidity results concerning marginally outer trapped surfaces (MOTS). As is well known, MOTS play an important role in the theory of black holes and, at the same time, are interesting spacetime analogues of minimal surfaces in Riemannian geometry. The main results presented here expand upon earlier works by the authors, specifically addressing initial data sets incorporating charge.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"256 ","pages":"Article 113780"},"PeriodicalIF":1.3,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143519275","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 : 2025-02-26DOI: 10.1016/j.na.2025.113781
Mengting Fan , Ning-An Lai , Hiroyuki Takamura
The singular Cauchy problem for the semilinear Euler–Poisson–Darboux equation in with power type nonlinearity is studied in this paper. We show that the blow up power is related to the Strauss exponent, which generalizes the blow up result from the regular semilinear wave equation with scale invariant damping to the corresponding singular problem, and hence give some affirmative answer partially to the open problem posed by D’Abbicco in a recent paper.
{"title":"Nonexistence of global solutions to the Euler–Poisson–Darboux equation in Rn: Subcritical case","authors":"Mengting Fan , Ning-An Lai , Hiroyuki Takamura","doi":"10.1016/j.na.2025.113781","DOIUrl":"10.1016/j.na.2025.113781","url":null,"abstract":"<div><div>The singular Cauchy problem for the semilinear Euler–Poisson–Darboux equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with power type nonlinearity is studied in this paper. We show that the blow up power is related to the Strauss exponent, which generalizes the blow up result from the regular semilinear wave equation with scale invariant damping to the corresponding singular problem, and hence give some affirmative answer partially to the open problem posed by D’Abbicco in a recent paper.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"256 ","pages":"Article 113781"},"PeriodicalIF":1.3,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509684","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 : 2025-02-24DOI: 10.1016/j.na.2025.113778
Masaki Kawamoto , Hayato Miyazaki
This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In Kawamoto and Muramatsu (2021) and Kawamoto (2021), it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in Hayashi and Naumkin (2006). In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in Kawamoto (2021).
{"title":"Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials","authors":"Masaki Kawamoto , Hayato Miyazaki","doi":"10.1016/j.na.2025.113778","DOIUrl":"10.1016/j.na.2025.113778","url":null,"abstract":"<div><div>This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In Kawamoto and Muramatsu (2021) and Kawamoto (2021), it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in Hayashi and Naumkin (2006). In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in Kawamoto (2021).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"256 ","pages":"Article 113778"},"PeriodicalIF":1.3,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474344","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 : 2025-02-22DOI: 10.1016/j.na.2025.113754
Dukman Ri, Sungil Kwon
We study a class of double phase elliptic equations with variable exponents and critical growth. In the present paper we establish the boundedness, Holder continuity and higher integrability of weak solutions for these equations. Our results partially generalize those obtained by Winkert and his collaborators (2023)
{"title":"Holder continuity and higher integrability of weak solutions to. double phase elliptic equations involving variable exponents and. critical growth","authors":"Dukman Ri, Sungil Kwon","doi":"10.1016/j.na.2025.113754","DOIUrl":"10.1016/j.na.2025.113754","url":null,"abstract":"<div><div>We study a class of double phase elliptic equations with variable exponents and critical growth. In the present paper we establish the boundedness, Holder continuity and higher integrability of weak solutions for these equations. Our results partially generalize those obtained by Winkert and his collaborators (2023)</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113754"},"PeriodicalIF":1.3,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463542","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 : 2025-02-22DOI: 10.1016/j.na.2025.113777
Pablo Álvarez-Caudevilla , Cristina Brändle , Mónica Molina-Becerra , Antonio Suárez
In this work we consider an interface logistic problem where two populations live in two different regions, separated by a membrane or interface where it happens an interchange of flux. Thus, the two populations only interact or are coupled through such a membrane where we impose the so-called Kedem–Katchalsky boundary conditions. For this particular scenario we analyse the existence and uniqueness of positive solutions depending on the parameters involved in the system, obtaining interesting results where one can see for the first time the effect of the membrane under such boundary conditions. To do so, we first ascertain the asymptotic behaviour of several linear and nonlinear problems for which we include a diffusion coefficient and analyse the behaviour of the solutions when such a diffusion parameter goes to zero or infinity. Despite their own interest, since these asymptotic results have never been studied before, they will be crucial in analysing the existence and uniqueness for the main interface logistic problems under analysis. Finally, we apply such an asymptotic analysis to characterize the existence of solutions in terms of the growth rate of the populations, when both populations possess the same growth rate and, also, when they depend on different parameters.
{"title":"Interface logistic problems: Large diffusion and singular perturbation results","authors":"Pablo Álvarez-Caudevilla , Cristina Brändle , Mónica Molina-Becerra , Antonio Suárez","doi":"10.1016/j.na.2025.113777","DOIUrl":"10.1016/j.na.2025.113777","url":null,"abstract":"<div><div>In this work we consider an interface logistic problem where two populations live in two different regions, separated by a membrane or interface where it happens an interchange of flux. Thus, the two populations only interact or are coupled through such a membrane where we impose the so-called Kedem–Katchalsky boundary conditions. For this particular scenario we analyse the existence and uniqueness of positive solutions depending on the parameters involved in the system, obtaining interesting results where one can see for the first time the effect of the membrane under such boundary conditions. To do so, we first ascertain the asymptotic behaviour of several linear and nonlinear problems for which we include a diffusion coefficient and analyse the behaviour of the solutions when such a diffusion parameter goes to zero or infinity. Despite their own interest, since these asymptotic results have never been studied before, they will be crucial in analysing the existence and uniqueness for the main interface logistic problems under analysis. Finally, we apply such an asymptotic analysis to characterize the existence of solutions in terms of the growth rate of the populations, when both populations possess the same growth rate and, also, when they depend on different parameters.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113777"},"PeriodicalIF":1.3,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463543","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 : 2025-02-22DOI: 10.1016/j.na.2025.113779
Patricia Alonso Ruiz , Fabrice Baudoin
The paper studies properties of -limits of Korevaar–Schoen -energies on a Cheeger space. When , this kind of limit provides a natural -energy form that can be used to define a -Laplacian, and whose domain is the Newtonian Sobolev space . When , the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the -convergence of the -energies is improved to Mosco convergence for every .
{"title":"Korevaar–Schoen p-energies and their Γ-limits on Cheeger spaces","authors":"Patricia Alonso Ruiz , Fabrice Baudoin","doi":"10.1016/j.na.2025.113779","DOIUrl":"10.1016/j.na.2025.113779","url":null,"abstract":"<div><div>The paper studies properties of <span><math><mi>Γ</mi></math></span>-limits of Korevaar–Schoen <span><math><mi>p</mi></math></span>-energies on a Cheeger space. When <span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span>, this kind of limit provides a natural <span><math><mi>p</mi></math></span>-energy form that can be used to define a <span><math><mi>p</mi></math></span>-Laplacian, and whose domain is the Newtonian Sobolev space <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup></math></span>. When <span><math><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></math></span>, the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the <span><math><mi>Γ</mi></math></span>-convergence of the <span><math><mi>p</mi></math></span>-energies is improved to Mosco convergence for every <span><math><mrow><mi>p</mi><mo>≥</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"256 ","pages":"Article 113779"},"PeriodicalIF":1.3,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143471231","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}
Pub Date : 2025-02-19DOI: 10.1016/j.na.2025.113776
Alba Lia Masiello , Francesco Salerno
In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya–Szegő type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya–Szegő inequality for the -Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso (1989) for solutions to the -Hessian equation.
As an application of the first result, we prove a quantitative version of the Faber–Krahn and Saint-Venant inequalities for these equations.
{"title":"A quantitative result for the k-Hessian equation","authors":"Alba Lia Masiello , Francesco Salerno","doi":"10.1016/j.na.2025.113776","DOIUrl":"10.1016/j.na.2025.113776","url":null,"abstract":"<div><div>In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya–Szegő type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya–Szegő inequality for the <span><math><mi>k</mi></math></span>-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso (1989) for solutions to the <span><math><mi>k</mi></math></span>-Hessian equation.</div><div>As an application of the first result, we prove a quantitative version of the Faber–Krahn and Saint-Venant inequalities for these equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113776"},"PeriodicalIF":1.3,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437533","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 : 2025-02-18DOI: 10.1016/j.na.2025.113775
Lihua Dong
This paper is concerned with asymptotic stability of certain stationary solution to the 3D Boussinesq equations in the whole space with a damping term in the velocity equation. Precisely, the decay rates of solutions is optimal in sense that these rates coincide with that of the linearized equations.
{"title":"The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term in R3","authors":"Lihua Dong","doi":"10.1016/j.na.2025.113775","DOIUrl":"10.1016/j.na.2025.113775","url":null,"abstract":"<div><div>This paper is concerned with asymptotic stability of certain stationary solution to the 3D Boussinesq equations in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with a damping term in the velocity equation. Precisely, the decay rates of solutions is optimal in sense that these rates coincide with that of the linearized equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113775"},"PeriodicalIF":1.3,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429538","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 : 2025-02-16DOI: 10.1016/j.na.2025.113753
Y. Chitour , M. Kafnemer , P. Martinez , B. Mebkhout
In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the framework, with .
We begin by addressing the well-posedness problem, establishing the existence and uniqueness of weak and strong solutions for , under suitable assumptions on the damping function.
Next, we study the asymptotic behaviour of the associated energy when , and we provide decay estimates that appear to be almost optimal compared to similar problems with boundary damping.
Our work is motivated by earlier studies, particularly, those by Chitour, Marx and Prieur (2020), and Haraux (1978). The proofs combine arguments from Kafnemer, Mebkhout and Chitour (2022) for wave equation in the framework with a linear damping, techniques of weighted energy estimates introduced in Martinez (1999), new integral inequalities for , and convex analysis tools when .
{"title":"Lp asymptotic stability of 1D damped wave equation with nonlinear damping","authors":"Y. Chitour , M. Kafnemer , P. Martinez , B. Mebkhout","doi":"10.1016/j.na.2025.113753","DOIUrl":"10.1016/j.na.2025.113753","url":null,"abstract":"<div><div>In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> framework, with <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We begin by addressing the well-posedness problem, establishing the existence and uniqueness of weak and strong solutions for <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, under suitable assumptions on the damping function.</div><div>Next, we study the asymptotic behaviour of the associated energy when <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, and we provide decay estimates that appear to be almost optimal compared to similar problems with boundary damping.</div><div>Our work is motivated by earlier studies, particularly, those by Chitour, Marx and Prieur (2020), and Haraux (1978). The proofs combine arguments from Kafnemer, Mebkhout and Chitour (2022) for wave equation in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> framework with a linear damping, techniques of weighted energy estimates introduced in Martinez (1999), new integral inequalities for <span><math><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow></math></span>, and convex analysis tools when <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113753"},"PeriodicalIF":1.3,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421249","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}
Pub Date : 2025-02-12DOI: 10.1016/j.na.2025.113774
Cosmin Burtea , Maja Szlenk
We prove the existence of weak solutions for the steady Navier–Stokes system for compressible non-Newtonian fluids on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power and the pressure is given by , we construct a solution provided that and is sufficiently large, depending on the values of . Additionally, we also show the existence for time-discretized model for Herschel–Bulkley fluids, where the viscosity has a singular part.
{"title":"Weak solutions to the Navier–Stokes equations for steady compressible non-Newtonian fluids","authors":"Cosmin Burtea , Maja Szlenk","doi":"10.1016/j.na.2025.113774","DOIUrl":"10.1016/j.na.2025.113774","url":null,"abstract":"<div><div>We prove the existence of weak solutions for the steady Navier–Stokes system for compressible non-Newtonian fluids on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power <span><math><mi>r</mi></math></span> and the pressure is given by <span><math><msup><mrow><mi>ϱ</mi></mrow><mrow><mi>γ</mi></mrow></msup></math></span>, we construct a solution provided that <span><math><mrow><mi>r</mi><mo>></mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mi>γ</mi></math></span> is sufficiently large, depending on the values of <span><math><mi>r</mi></math></span>. Additionally, we also show the existence for time-discretized model for Herschel–Bulkley fluids, where the viscosity has a singular part.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113774"},"PeriodicalIF":1.3,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388285","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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