Pub Date : 2022-09-01DOI: 10.57262/ade027-0910-571
Giovanni Molica Bisci, Nguyen Van Thin, L. Vilasi
{"title":"On a class of nonlocal Schrödinger equations with exponential growth","authors":"Giovanni Molica Bisci, Nguyen Van Thin, L. Vilasi","doi":"10.57262/ade027-0910-571","DOIUrl":"https://doi.org/10.57262/ade027-0910-571","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46006033","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 : 2022-07-01DOI: 10.57262/ade027-0708-407
Eudes Barboza, O. Miyagaki, F. Pereira, Cláudia Santana
{"title":"Nonlocal Hénon equation with nonlinearities involving Sobolev critical and supercritical growth","authors":"Eudes Barboza, O. Miyagaki, F. Pereira, Cláudia Santana","doi":"10.57262/ade027-0708-407","DOIUrl":"https://doi.org/10.57262/ade027-0708-407","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45959664","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 : 2022-05-01DOI: 10.57262/ade027-0506-253
Emmanuel Hebey
{"title":"Blowing-up solutions to Bopp-Podolsky-Schrödinger-Proca and Schrödinger-Poisson-Proca systems in the electro-magneto-static case","authors":"Emmanuel Hebey","doi":"10.57262/ade027-0506-253","DOIUrl":"https://doi.org/10.57262/ade027-0506-253","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44962982","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 : 2022-05-01DOI: 10.57262/ade027-0506-385
Tacksun Jung, Q. Choi
{"title":"On the fractional elliptic problems with difference in the Orlicz-Sobolev spaces","authors":"Tacksun Jung, Q. Choi","doi":"10.57262/ade027-0506-385","DOIUrl":"https://doi.org/10.57262/ade027-0506-385","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48375416","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 : 2022-04-05DOI: 10.57262/ade027-1112-781
Alysson Cunha
. We show that uniqueness results of the kind those obtained for KdV and Schr¨odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces for appropriated s and r . In particular, we obtain that the uniqueness result proved for the dispersion generalized- Benjamin-Ono equation ([13]), is not true for all pairs of solutions u 1 6 = 0 and u 2 6 = 0. To achieve these results we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized-Benjamin-Ono equation and for the Benjamin-Ono equation ([13], [12]).
{"title":"On decay of the solutions for the dispersion generalized Benjamin--Ono and Benjamin--Ono equations","authors":"Alysson Cunha","doi":"10.57262/ade027-1112-781","DOIUrl":"https://doi.org/10.57262/ade027-1112-781","url":null,"abstract":". We show that uniqueness results of the kind those obtained for KdV and Schr¨odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces for appropriated s and r . In particular, we obtain that the uniqueness result proved for the dispersion generalized- Benjamin-Ono equation ([13]), is not true for all pairs of solutions u 1 6 = 0 and u 2 6 = 0. To achieve these results we employ the techniques present in our recent work [6]. We also improve some Theorems established for the dispersion generalized-Benjamin-Ono equation and for the Benjamin-Ono equation ([13], [12]).","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47204418","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 : 2022-03-28DOI: 10.57262/ade028-0506-413
A. Faminskii
The initial value problem is considered for a higher order nonlinear Schr"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.
{"title":"The higher order nonlinear Schrödinger equation with quadratic nonlinearity on the real axis","authors":"A. Faminskii","doi":"10.57262/ade028-0506-413","DOIUrl":"https://doi.org/10.57262/ade028-0506-413","url":null,"abstract":"The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46913976","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}
. The objective of this paper is to construct geometrically Riemann k -wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann k -waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the k -wave problem and discuss in detail the conditions for the existence of k -wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the k -wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.
{"title":"Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems","authors":"A. Grundland, J. Lucas","doi":"10.57262/ade028-0102-73","DOIUrl":"https://doi.org/10.57262/ade028-0102-73","url":null,"abstract":". The objective of this paper is to construct geometrically Riemann k -wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann k -waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the k -wave problem and discuss in detail the conditions for the existence of k -wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the k -wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44578514","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 : 2022-03-27DOI: 10.57262/ade028-0506-373
G. Gie, J. Kelliher, A. Mazzucato
In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized Euler equations in vorticity form. We extend their result on the linearized problem to multiply connected domains and establish compatibility conditions on the initial data that allow higher regularity solutions.
{"title":"The linearized 3d Euler equations with inflow, outflow","authors":"G. Gie, J. Kelliher, A. Mazzucato","doi":"10.57262/ade028-0506-373","DOIUrl":"https://doi.org/10.57262/ade028-0506-373","url":null,"abstract":"In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized Euler equations in vorticity form. We extend their result on the linearized problem to multiply connected domains and establish compatibility conditions on the initial data that allow higher regularity solutions.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44695304","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 : 2022-03-16DOI: 10.57262/ade028-0708-613
U. Maio, R. Hadiji, C. Lefter, C. Perugia
We consider a Ginzburg-Landau type equation in $R^2$ of the form $-Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H.Brezis, F.Merle, T.Rivi`ere from cite{BMR} who treat the case when $J$ behaves polinomially near 0, as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.
{"title":"A Liouville type result and quantization effects on the system $-Delta u = u J'(1-|u|^{2})$ for a potential convex near zero","authors":"U. Maio, R. Hadiji, C. Lefter, C. Perugia","doi":"10.57262/ade028-0708-613","DOIUrl":"https://doi.org/10.57262/ade028-0708-613","url":null,"abstract":"We consider a Ginzburg-Landau type equation in $R^2$ of the form $-Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H.Brezis, F.Merle, T.Rivi`ere from cite{BMR} who treat the case when $J$ behaves polinomially near 0, as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47099264","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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