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Hardy inequalities for antisymmetric functions 反对称函数的哈代不等式
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-29 DOI: 10.1016/j.na.2024.113619
Shubham Gupta
{"title":"Hardy inequalities for antisymmetric functions","authors":"Shubham Gupta","doi":"10.1016/j.na.2024.113619","DOIUrl":"10.1016/j.na.2024.113619","url":null,"abstract":"<div><p>We study Hardy inequalities for antisymmetric functions in three different settings: Euclidean space, torus and the integer lattice. In particular, we show that under the antisymmetric condition the sharp constant in Hardy inequality increases substantially and grows as <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> as <span><math><mrow><mi>d</mi><mo>→</mo><mi>∞</mi></mrow></math></span> in all cases. As a side product, we prove Hardy inequality on a domain whose boundary forms a corner at the point of singularity <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868622","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}
引用次数: 0
Existence and asymptotic stability for the wave equation on compact manifolds with nonlinearities of arbitrary growth 具有任意增长非线性的紧凑流形上波方程的存在性和渐近稳定性
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-26 DOI: 10.1016/j.na.2024.113620
Marcelo M. Cavalcanti , Valéria N. Domingos Cavalcanti , José Guilherme Simion Antunes
{"title":"Existence and asymptotic stability for the wave equation on compact manifolds with nonlinearities of arbitrary growth","authors":"Marcelo M. Cavalcanti ,&nbsp;Valéria N. Domingos Cavalcanti ,&nbsp;José Guilherme Simion Antunes","doi":"10.1016/j.na.2024.113620","DOIUrl":"10.1016/j.na.2024.113620","url":null,"abstract":"<div><p>We study the wellposedness, stabilization and blow up of solutions of the wave equation with nonlinearities of arbitrary growth and locally distributed nonlinear dissipation posed in a 2-dimensional compact Riemannian manifold <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></math></span> without boundary. Differently of the previous literature we give a different proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof, even in the case of subcritical, critical or super critical growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than <span><math><mi>d</mi></math></span> of the ground state, where <span><math><mi>d</mi></math></span> is the level of the Mountain Pass Theorem.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868620","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}
引用次数: 0
Some rigidity results and asymptotic properties for solutions to semilinear elliptic P.D.E. 半线性椭圆 P.D.E. 解的一些刚性结果和渐近特性
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-25 DOI: 10.1016/j.na.2024.113610
Matteo Rizzi , Panayotis Smyrnelis
{"title":"Some rigidity results and asymptotic properties for solutions to semilinear elliptic P.D.E.","authors":"Matteo Rizzi ,&nbsp;Panayotis Smyrnelis","doi":"10.1016/j.na.2024.113610","DOIUrl":"10.1016/j.na.2024.113610","url":null,"abstract":"<div><p>We will present some rigidity results for solutions to semilinear elliptic equations of the form <span><math><mrow><mi>Δ</mi><mi>u</mi><mo>=</mo><msup><mrow><mi>W</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>W</mi></math></span> is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouville-type theorems and symmetry results, which generalise some known facts about the Cahn–Hilliard equation.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001299/pdfft?md5=a45788b8d984b525a97fb6104f87a266&pid=1-s2.0-S0362546X24001299-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868482","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}
引用次数: 0
On an L2 critical Boltzmann equation 关于[公式省略]临界玻尔兹曼方程
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-25 DOI: 10.1016/j.na.2024.113609
Thomas Chen, Ryan Denlinger, Nataša Pavlović
{"title":"On an L2 critical Boltzmann equation","authors":"Thomas Chen,&nbsp;Ryan Denlinger,&nbsp;Nataša Pavlović","doi":"10.1016/j.na.2024.113609","DOIUrl":"10.1016/j.na.2024.113609","url":null,"abstract":"<div><p>We prove the existence of a class of large global scattering solutions of Boltzmann’s equation with constant collision kernel in two dimensions. These solutions are found for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> perturbations of an underlying initial data which is Gaussian jointly in space and velocity. Additionally, the perturbation is required to satisfy natural physical constraints for the total mass and second moments, corresponding to conserved or controlled quantities. The space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a scaling critical space for the equation under consideration. If the initial data is Schwartz then the solution is unique and again Schwartz on any bounded time interval.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868552","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}
引用次数: 0
Global regularity of 2D Rayleigh–Bénard equations with logarithmic supercritical dissipation 具有对数超临界耗散的二维雷利-贝纳德方程的全局正则性
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-22 DOI: 10.1016/j.na.2024.113617
Baoquan Yuan, Xinyuan Xu, Changhao Li
{"title":"Global regularity of 2D Rayleigh–Bénard equations with logarithmic supercritical dissipation","authors":"Baoquan Yuan,&nbsp;Xinyuan Xu,&nbsp;Changhao Li","doi":"10.1016/j.na.2024.113617","DOIUrl":"10.1016/j.na.2024.113617","url":null,"abstract":"<div><p>In this paper, we study the global regularity problem for the 2D Rayleigh–Bénard equations with logarithmic supercritical dissipation. By exploiting a combined quantity of the system, the technique of Littlewood-Paley decomposition and Besov spaces, and some commutator estimates, we establish the global regularity of a strong solution to this equations in the Sobolev space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141783337","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}
引用次数: 0
Excess decay for minimizing hypercurrents mod 2Q 最小化超电流模式 2Q 的过量衰减
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-09 DOI: 10.1016/j.na.2024.113606
Camillo De Lellis , Jonas Hirsch , Andrea Marchese , Luca Spolaor , Salvatore Stuvard
{"title":"Excess decay for minimizing hypercurrents mod 2Q","authors":"Camillo De Lellis ,&nbsp;Jonas Hirsch ,&nbsp;Andrea Marchese ,&nbsp;Luca Spolaor ,&nbsp;Salvatore Stuvard","doi":"10.1016/j.na.2024.113606","DOIUrl":"https://doi.org/10.1016/j.na.2024.113606","url":null,"abstract":"<div><p>We consider codimension 1 area-minimizing <span><math><mi>m</mi></math></span>-dimensional currents <span><math><mi>T</mi></math></span> mod an even integer <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mi>Q</mi></mrow></math></span> in a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> Riemannian submanifold <span><math><mi>Σ</mi></math></span> of Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point <span><math><mrow><mi>q</mi><mo>∈</mo><mi>spt</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∖</mo><msup><mrow><mi>spt</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>∂</mi><mi>T</mi><mo>)</mo></mrow></mrow></math></span> where at least one such tangent cone is <span><math><mi>Q</mi></math></span> copies of a single plane. While an analogous decay statement was proved in Minter and Wickramasekera (2024) as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of <span><math><mi>Σ</mi></math></span>. This improvement is in fact crucial in De Lellis et al., (2022) to prove that the singular set of <span><math><mi>T</mi></math></span> can be decomposed into a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> <span><math><mrow><mo>(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most <span><math><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582863","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}
引用次数: 0
Global smooth solutions for hyperbolic systems with time-dependent damping 具有随时间变化的阻尼的双曲系统的全局平稳解
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-08 DOI: 10.1016/j.na.2024.113608
Cunming Liu , Han Sheng , Ning-An Lai
{"title":"Global smooth solutions for hyperbolic systems with time-dependent damping","authors":"Cunming Liu ,&nbsp;Han Sheng ,&nbsp;Ning-An Lai","doi":"10.1016/j.na.2024.113608","DOIUrl":"https://doi.org/10.1016/j.na.2024.113608","url":null,"abstract":"<div><p>The Cauchy problem for hyperbolic systems of balance laws admits global smooth solutions near the constant states under stability condition. This was widely studied in previous works. In this paper, we concern hyperbolic systems with time-dependent damping <span><math><mrow><mi>μ</mi><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mi>λ</mi></mrow></msup><mi>G</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>μ</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. In the following two cases, <span><math><mrow><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mspace></mspace><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>μ</mi><mo>&gt;</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mrow></math></span> where <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow></math></span> is a constant depending only on the coefficients of the system; <span><math><mrow><mrow><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow><mspace></mspace><mn>0</mn><mo>&lt;</mo><mi>λ</mi><mo>&lt;</mo><mn>1</mn><mo>,</mo><mi>μ</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo></mrow></math></span> we prove that the smooth solutions exist globally when the initial data is small. To obtain these stability results, we establish uniform energy estimates and various dissipative estimates for all time and employ an induction argument on the order of derivatives of smooth solutions. Finally, we apply these results to some physical models.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582859","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}
引用次数: 0
Carathéodory theory and a priori estimates for continuity inclusions in the space of probability measures 概率测度空间中连续性夹杂的卡拉瑟多理论和先验估计
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-08 DOI: 10.1016/j.na.2024.113595
Benoît Bonnet-Weill , Hélène Frankowska
{"title":"Carathéodory theory and a priori estimates for continuity inclusions in the space of probability measures","authors":"Benoît Bonnet-Weill ,&nbsp;Hélène Frankowska","doi":"10.1016/j.na.2024.113595","DOIUrl":"https://doi.org/10.1016/j.na.2024.113595","url":null,"abstract":"<div><p>In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result à la Peano for this class of dynamics under mere Carathéodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy–Lipschitz framework.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582862","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}
引用次数: 0
Uniqueness and stability of forced waves for the Fisher–KPP equation in a shifting environment 移动环境中 Fisher-KPP 方程受迫波的唯一性和稳定性
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-08 DOI: 10.1016/j.na.2024.113607
Jong-Shenq Guo , Karen Guo , Masahiko Shimojo
{"title":"Uniqueness and stability of forced waves for the Fisher–KPP equation in a shifting environment","authors":"Jong-Shenq Guo ,&nbsp;Karen Guo ,&nbsp;Masahiko Shimojo","doi":"10.1016/j.na.2024.113607","DOIUrl":"https://doi.org/10.1016/j.na.2024.113607","url":null,"abstract":"<div><p>In this paper, we investigate the existence, uniqueness and stability of forced waves for the Fisher–KPP equation in a shifting environment without imposing the monotonicity condition on the shifting intrinsic growth term. First, the existence of forced waves for some range of shifting speeds is proved. Then we prove the uniqueness of saturation forced waves. Moreover, a new method is introduced to derive the non-existence of forced waves. Finally, we derive the stability of forced waves under certain perturbation of a class of initial data.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582860","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}
引用次数: 0
Large time asymptotics for the modified Korteweg–de Vries-Benjamin–Ono equation 修正的科特维格-德弗里斯-本杰明-奥诺方程的大时间渐近线
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-07-07 DOI: 10.1016/j.na.2024.113604
Nakao Hayashi , Jesus A. Mendez-Navarro , Pavel I. Naumkin
{"title":"Large time asymptotics for the modified Korteweg–de Vries-Benjamin–Ono equation","authors":"Nakao Hayashi ,&nbsp;Jesus A. Mendez-Navarro ,&nbsp;Pavel I. Naumkin","doi":"10.1016/j.na.2024.113604","DOIUrl":"https://doi.org/10.1016/j.na.2024.113604","url":null,"abstract":"<div><p>We study the large time asymptotics of solutions to the Cauchy problem for the modified Korteweg–de Vries-Benjamin–Ono equation <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mi>H</mi><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>−</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></mfrac><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>u</mi><mo>=</mo><msub><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow></msub><mfenced><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfenced><mo>,</mo><mi>t</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>∈</mo><mi>R</mi><mi>,</mi><mi>u</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mi>x</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mfenced><mrow><mi>x</mi></mrow></mfenced><mo>,</mo><mi>x</mi><mo>∈</mo><mi>R</mi><mi>,</mi></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo></mrow></math></span> <span><math><mrow><mi>H</mi><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>π</mi></mrow></mfrac></mrow></math></span>p.v.<span><math><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><mfrac><mrow><mi>ϕ</mi><mfenced><mrow><mi>y</mi></mrow></mfenced></mrow><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></mfrac><mi>d</mi><mi>y</mi></mrow></math></span> is the Hilbert transform. We develop the factorization technique to obtain the sharp time decay estimate for solutions and to prove the modified scattering.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582861","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}
引用次数: 0
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