Pub Date : 2021-11-01DOI: 10.57262/die034-1112-641
N. Hayashi, E. Kaikina, T. Ogawa
{"title":"Inhomogeneous Neumann-boundary value problem for nonlinear Schrödinger equations in the upper half-space","authors":"N. Hayashi, E. Kaikina, T. Ogawa","doi":"10.57262/die034-1112-641","DOIUrl":"https://doi.org/10.57262/die034-1112-641","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48255081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.57262/die034-1112-595
Federico Cianci, G. Dal Maso
{"title":"Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks","authors":"Federico Cianci, G. Dal Maso","doi":"10.57262/die034-1112-595","DOIUrl":"https://doi.org/10.57262/die034-1112-595","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44692570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.57262/die034-1112-621
Zeyi Liu, Deli Zhang
{"title":"A new Kirchhoff-Schrödinger-Poisson type system on the Heisenberg group","authors":"Zeyi Liu, Deli Zhang","doi":"10.57262/die034-1112-621","DOIUrl":"https://doi.org/10.57262/die034-1112-621","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44552762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.57262/die034-1112-675
Yutaka Tamada
{"title":"Global existence for one-dimensional hyperbolic equation with power type nonlinearity","authors":"Yutaka Tamada","doi":"10.57262/die034-1112-675","DOIUrl":"https://doi.org/10.57262/die034-1112-675","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41827770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in Cb([0,∞); L c(R)). As a direct consequence, global existence for data in strong critical Lebesgue Lc (R) follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution u ≡ 0 is shown to be asymptotically stable in Lc (R) – it is the only self-similar solution which is initially small in Lc (R). Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.
本文讨论了hardy - hsamnon方程的柯西问题(及其分数阶类比)。研究了一类在无穷远处缓慢衰减的连续函数的初始数据的局部适定性。在Cb([0，∞)上得到了小数据(临界弱- lebesgue空间)的全局适定性;L c (R))。其直接结果是，强临界Lebesgue Lc (R)中的数据在一个小条件下具有全局存在性，而唯一性是无条件的。此外，我们证明了自相似解的存在性，并检验了全局定义解的长时间行为。证明了零解u≡0在Lc (R)中是渐近稳定的——它是Lc (R)中唯一初始较小的自相似解。此外，在初始数据的温和假设下得到了爆破结果，并找到了相应的Fujita临界指数。
{"title":"On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior","authors":"Gael Diebou Yomgne","doi":"10.57262/die035-0102-57","DOIUrl":"https://doi.org/10.57262/die035-0102-57","url":null,"abstract":"This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in Cb([0,∞); L c(R)). As a direct consequence, global existence for data in strong critical Lebesgue Lc (R) follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution u ≡ 0 is shown to be asymptotically stable in Lc (R) – it is the only self-similar solution which is initially small in Lc (R). Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48199158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-18DOI: 10.57262/die035-0304-211
Philippe Laurencçot, Christoph Walker
describes the dynamics of the size distribution function φ = φ(t, x) ≥ 0 of particles of size x ∈ (0,∞) at time t > 0. Particles modify their sizes according to three different mechanisms: random fluctuations, here accounted for by size diffusion at a constant diffusion rate D > 0 (hereafter normalized toD = 1), spontaneous fragmentation with overall fragmentation rate a ≥ 0 and daughter distribution function b ≥ 0, and binary coalescence with coagulation kernel k ≥ 0. Nucleation is not taken into account in this model, an assumption which leads to the homogeneous Dirichlet boundary condition (1.1b) at x = 0. Let us recall that the coagulation-fragmentation equation without size diffusion, corresponding to setting D = 0 in (1.1a), arises in several fields of physics (grain growth, aerosol and raindrops formation, polymer and colloidal chemistry) and biology (hematology, animal grouping) and has been studied extensively in the mathematical literature since the pioneering works
{"title":"Well-posedness of the coagulation-fragmentation equation with size diffusion","authors":"Philippe Laurencçot, Christoph Walker","doi":"10.57262/die035-0304-211","DOIUrl":"https://doi.org/10.57262/die035-0304-211","url":null,"abstract":"describes the dynamics of the size distribution function φ = φ(t, x) ≥ 0 of particles of size x ∈ (0,∞) at time t > 0. Particles modify their sizes according to three different mechanisms: random fluctuations, here accounted for by size diffusion at a constant diffusion rate D > 0 (hereafter normalized toD = 1), spontaneous fragmentation with overall fragmentation rate a ≥ 0 and daughter distribution function b ≥ 0, and binary coalescence with coagulation kernel k ≥ 0. Nucleation is not taken into account in this model, an assumption which leads to the homogeneous Dirichlet boundary condition (1.1b) at x = 0. Let us recall that the coagulation-fragmentation equation without size diffusion, corresponding to setting D = 0 in (1.1a), arises in several fields of physics (grain growth, aerosol and raindrops formation, polymer and colloidal chemistry) and biology (hematology, animal grouping) and has been studied extensively in the mathematical literature since the pioneering works","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46636783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-02DOI: 10.57262/die036-0506-413
H. R. Quoirin, Jefferson S. Silva, K. Silva
We investigate zero energy critical points for a class of functionals $Phi_mu$ defined on a uniformly convex Banach space, and depending on a real parameter $mu$. More precisely, we show the existence of a sequence $(mu_n)$ such that $Phi_{mu_n}$ has a pair of critical points $pm u_n$ satisfying $Phi_{mu_n}(pm u_n)=0$, for every $n$. In addition, we provide some properties of $mu_n$ and $u_n$. This result, which is proved via a fibering map approach (based on the {it nonlinear generalized Rayleigh quotient} method cite{I1}) combined with the Ljusternik-Schnirelman theory, is then applied to several classes of elliptic pdes.
{"title":"Zero energy critical points of functionals depending on a parameter","authors":"H. R. Quoirin, Jefferson S. Silva, K. Silva","doi":"10.57262/die036-0506-413","DOIUrl":"https://doi.org/10.57262/die036-0506-413","url":null,"abstract":"We investigate zero energy critical points for a class of functionals $Phi_mu$ defined on a uniformly convex Banach space, and depending on a real parameter $mu$. More precisely, we show the existence of a sequence $(mu_n)$ such that $Phi_{mu_n}$ has a pair of critical points $pm u_n$ satisfying $Phi_{mu_n}(pm u_n)=0$, for every $n$. In addition, we provide some properties of $mu_n$ and $u_n$. This result, which is proved via a fibering map approach (based on the {it nonlinear generalized Rayleigh quotient} method cite{I1}) combined with the Ljusternik-Schnirelman theory, is then applied to several classes of elliptic pdes.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41961289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-01DOI: 10.57262/die034-0910-491
P. Pedregal
{"title":"Existence under lack of convexity","authors":"P. Pedregal","doi":"10.57262/die034-0910-491","DOIUrl":"https://doi.org/10.57262/die034-0910-491","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46282328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-01DOI: 10.57262/die034-0910-467
Lü-Yi Ma, Zhi-Cheng Wang
{"title":"On the existence of cylindrically symmetric traveling fronts of fractional Allen-Cahn equation in $mathbb{R}^{3}$","authors":"Lü-Yi Ma, Zhi-Cheng Wang","doi":"10.57262/die034-0910-467","DOIUrl":"https://doi.org/10.57262/die034-0910-467","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41954518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-01DOI: 10.57262/die034-0910-519
Zhiguo Xu
{"title":"Infinitely many solutions for the fractional $p$&$q$ problem with critical Sobolev-Hardy exponents and sign-changing weight functions","authors":"Zhiguo Xu","doi":"10.57262/die034-0910-519","DOIUrl":"https://doi.org/10.57262/die034-0910-519","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48302070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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