Pub Date : 2019-11-19DOI: 10.4310/dpde.2021.v18.n3.a1
R. Chen, D. Pelinovsky
It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under $W^{1,infty}$-perturbations. Moreover, we show that small initial $W^{1,infty}$-perturbations of the Novikov peakons can lead to the finite time blow-up of the corresponding solutions.
{"title":"$W^{1,infty}$ instability of $H^1$-stable peakons in the Novikov equation","authors":"R. Chen, D. Pelinovsky","doi":"10.4310/dpde.2021.v18.n3.a1","DOIUrl":"https://doi.org/10.4310/dpde.2021.v18.n3.a1","url":null,"abstract":"It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under $W^{1,infty}$-perturbations. Moreover, we show that small initial $W^{1,infty}$-perturbations of the Novikov peakons can lead to the finite time blow-up of the corresponding solutions.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47718680","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 : 2019-06-23DOI: 10.1887/0750306521/b803b2
Rowan Garnier, John Taylor
{"title":"Hints and Solutions to Selected Exercises","authors":"Rowan Garnier, John Taylor","doi":"10.1887/0750306521/b803b2","DOIUrl":"https://doi.org/10.1887/0750306521/b803b2","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88378399","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 : 2019-01-01DOI: 10.4310/dpde.2019.v16.n3.a4
Renhai Wang, Yangrong Li
{"title":"Asymptotic autonomy of kernel sections for Newton–Boussinesq equations on unbounded zonary domains","authors":"Renhai Wang, Yangrong Li","doi":"10.4310/dpde.2019.v16.n3.a4","DOIUrl":"https://doi.org/10.4310/dpde.2019.v16.n3.a4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426828","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 : 2019-01-01DOI: 10.4310/dpde.2019.v16.n3.a2
Huashui Zhan, Zhaosheng Feng
{"title":"Stability of hyperbolic-parabolic mixed type equations","authors":"Huashui Zhan, Zhaosheng Feng","doi":"10.4310/dpde.2019.v16.n3.a2","DOIUrl":"https://doi.org/10.4310/dpde.2019.v16.n3.a2","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426779","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 : 2019-01-01DOI: 10.4310/dpde.2019.v16.n4.a2
Ruoci Sun
. We are interested in the influence of filtering the positive Fourier modes to the integrable non linear Schr¨odinger equation. Equivalently, we want to study the effect of dispersion added to the cubic Szeg˝o equation, leading to the NLS-Szeg˝o equation on the circle S 1 There are two sets of results in this paper. The first result concerns the long time Sobolev estimates for small data. The second set of results concerns the orbital stability of plane wave solutions. Some instability results are also obtained, leading to the wave turbulence phenomenon.
{"title":"Long time behavior of the NLS-Szegő equation","authors":"Ruoci Sun","doi":"10.4310/dpde.2019.v16.n4.a2","DOIUrl":"https://doi.org/10.4310/dpde.2019.v16.n4.a2","url":null,"abstract":". We are interested in the influence of filtering the positive Fourier modes to the integrable non linear Schr¨odinger equation. Equivalently, we want to study the effect of dispersion added to the cubic Szeg˝o equation, leading to the NLS-Szeg˝o equation on the circle S 1 There are two sets of results in this paper. The first result concerns the long time Sobolev estimates for small data. The second set of results concerns the orbital stability of plane wave solutions. Some instability results are also obtained, leading to the wave turbulence phenomenon.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426358","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 : 2019-01-01DOI: 10.4310/DPDE.2019.V16.N2.A1
E. Kopylova, A. Komech
. The global attraction is proved for solutions to 3D Klein-Gordon equation coupled to several nonlinear point oscillators. Our main result is a convergence of each finite energy solution to the set of all solitary waves as t → ±∞ . This attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity . We show that any omega-limit trajectory has the time-spectrum in the spectral gap [ − m,m ] and satisfies the original equation. Then the application of the Titchmarsh convolution theorem reduces the time-spectrum to a single harmonic ω ∈ [ − m,m ].
{"title":"On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities","authors":"E. Kopylova, A. Komech","doi":"10.4310/DPDE.2019.V16.N2.A1","DOIUrl":"https://doi.org/10.4310/DPDE.2019.V16.N2.A1","url":null,"abstract":". The global attraction is proved for solutions to 3D Klein-Gordon equation coupled to several nonlinear point oscillators. Our main result is a convergence of each finite energy solution to the set of all solitary waves as t → ±∞ . This attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity . We show that any omega-limit trajectory has the time-spectrum in the spectral gap [ − m,m ] and satisfies the original equation. Then the application of the Titchmarsh convolution theorem reduces the time-spectrum to a single harmonic ω ∈ [ − m,m ].","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426719","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 : 2019-01-01DOI: 10.4310/dpde.2019.v16.n4.a3
J. Xiao, Junjie Zhang
{"title":"Predual forms, harmonic maps and liquid crystals of $(BMO-Q)$ and $(BMO-Q)^{-1}$","authors":"J. Xiao, Junjie Zhang","doi":"10.4310/dpde.2019.v16.n4.a3","DOIUrl":"https://doi.org/10.4310/dpde.2019.v16.n4.a3","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426372","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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