{"title":"Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption","authors":"Yingying Liu, Zhengce Zhang, Liping Zhu","doi":"10.57262/ade/1548212470","DOIUrl":"https://doi.org/10.57262/ade/1548212470","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42590374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-11DOI: 10.57262/ade027-0506-333
G. Citti, Gianmarco Giovannardi, Manuel Ritor'e
We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J. Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.
{"title":"Variational formulas for curves of fixed degree","authors":"G. Citti, Gianmarco Giovannardi, Manuel Ritor'e","doi":"10.57262/ade027-0506-333","DOIUrl":"https://doi.org/10.57262/ade027-0506-333","url":null,"abstract":"We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J. Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42966394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system","authors":"Y. Fukumoto, Xiaopeng Zhao","doi":"10.57262/ade/1544497234","DOIUrl":"https://doi.org/10.57262/ade/1544497234","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49420660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.
{"title":"Local well-posedness for third order Benjamin-Ono type equations on the torus","authors":"Tomoyuki Tanaka","doi":"10.57262/ade/1565661672","DOIUrl":"https://doi.org/10.57262/ade/1565661672","url":null,"abstract":"We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48929256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption.
{"title":"Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows","authors":"Takeyuki Nagasawa, Kohei Nakamura","doi":"10.57262/ade/1565661673","DOIUrl":"https://doi.org/10.57262/ade/1565661673","url":null,"abstract":"Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46998317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a generalized fractional Du Bois-Reymond lemma and its applications","authors":"R. Kamocki","doi":"10.57262/ade/1537840836","DOIUrl":"https://doi.org/10.57262/ade/1537840836","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45285583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A classification for wave models with time-dependent potential and speed of propagation","authors":"M. R. Ebert, Wanderley Nunes do Nascimento","doi":"10.57262/ade/1537840835","DOIUrl":"https://doi.org/10.57262/ade/1537840835","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46841041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marta Kowalczyk, Ernesto P'erez-Chavela, S. Rybicki
In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $ddot q(t)=-nabla U(q(t))$ in the presence of symmetries of a compact Lie group $Gamma$ acting linearly on $mathbb{R}^n.$ We look for non-stationary periodic solutions of this system in a~neighborhood of an orbit of critical points of the potential $U.$
{"title":"Symmetric Lyapunov center theorem for orbit with nontrivial isotropy group","authors":"Marta Kowalczyk, Ernesto P'erez-Chavela, S. Rybicki","doi":"10.57262/ade/1580958057","DOIUrl":"https://doi.org/10.57262/ade/1580958057","url":null,"abstract":"In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $ddot q(t)=-nabla U(q(t))$ in the presence of symmetries of a compact Lie group $Gamma$ acting linearly on $mathbb{R}^n.$ We look for non-stationary periodic solutions of this system in a~neighborhood of an orbit of critical points of the potential $U.$","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43732190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in five and higher dimensions","authors":"Isao Kato, S. Kinoshita","doi":"10.57262/ade/1528855477","DOIUrl":"https://doi.org/10.57262/ade/1528855477","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42300947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential for any given initial and boundary data.","authors":"Yonggeun Cho, Gyeongha Hwang, Changhun Yang","doi":"10.57262/ade/1528855474","DOIUrl":"https://doi.org/10.57262/ade/1528855474","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42057640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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