Pub Date : 2022-05-01DOI: 10.57262/die035-0506-359
Shuaishuai Liang, Yueqiang Song
{"title":"Nontrivial solutions of quasilinear Choquard equation involving the $p$-Laplacian operator and critical nonlinearities","authors":"Shuaishuai Liang, Yueqiang Song","doi":"10.57262/die035-0506-359","DOIUrl":"https://doi.org/10.57262/die035-0506-359","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43809323","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 : 2022-04-19DOI: 10.57262/die036-0304-229
Stefano Biagi, F. Esposito, E. Vecchi
where Ω ⊂ R , 1 < p < q < N and a(·) ≥ 0. This class of functionals naturally appear in homogenization theory and in the study of strongly anisotropic materials (see, e.g., [39]), and falls into the framework of the so called functionals with non-standard growth introduced by Marcellini [27, 28]. The literature concerning functionals like (1.1) is pretty vast and concerns as a main topic the regularity of minimizers, see e.g. [2, 11, 12, 23] and the references therein.
其中Ω⊂R、1
{"title":"Symmetry of intrinsically singular solutions of double phase problems","authors":"Stefano Biagi, F. Esposito, E. Vecchi","doi":"10.57262/die036-0304-229","DOIUrl":"https://doi.org/10.57262/die036-0304-229","url":null,"abstract":"where Ω ⊂ R , 1 < p < q < N and a(·) ≥ 0. This class of functionals naturally appear in homogenization theory and in the study of strongly anisotropic materials (see, e.g., [39]), and falls into the framework of the so called functionals with non-standard growth introduced by Marcellini [27, 28]. The literature concerning functionals like (1.1) is pretty vast and concerns as a main topic the regularity of minimizers, see e.g. [2, 11, 12, 23] and the references therein.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41410774","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 : 2022-03-01DOI: 10.57262/die035-0304-151
F. Paronetto
{"title":"Local boundedness for forward-backward parabolic De Giorgi classes without assuming higher regularity","authors":"F. Paronetto","doi":"10.57262/die035-0304-151","DOIUrl":"https://doi.org/10.57262/die035-0304-151","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41614001","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 : 2022-03-01DOI: 10.57262/die035-0304-173
H. Hajaiej, K. Perera
{"title":"Ground state and least positive energy solutions of elliptic problems involving mixed fractional $p$-Laplacians","authors":"H. Hajaiej, K. Perera","doi":"10.57262/die035-0304-173","DOIUrl":"https://doi.org/10.57262/die035-0304-173","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47068861","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 : 2022-01-01DOI: 10.57262/die035-0102-123
Guangze Gu, Zhipeng Yang
{"title":"Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well","authors":"Guangze Gu, Zhipeng Yang","doi":"10.57262/die035-0102-123","DOIUrl":"https://doi.org/10.57262/die035-0102-123","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49041241","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}
{"title":"Positive solutions of fractional Schrödinger-Poisson systems involving critical nonlinearities with potential","authors":"H. Fan, Zhaosheng Feng, Xingjie Yan","doi":"10.57262/die035-0102-1","DOIUrl":"https://doi.org/10.57262/die035-0102-1","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47283086","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 : 2021-12-28DOI: 10.57262/die035-0910-511
A. Bachir, J. Giacomoni, G. Warnault
In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: { ∆pu = v |∇u| in Ω ∆pv = v β |∇u| in Ω, where Ω ⊂ R (N ≥ 2) is either equal to R or equal to a ball BR centered at the origin and having radius R > 0, 1 < p < ∞, m, q > 0, α ≥ 0, 0 ≤ β ≤ m and δ := (p− 1− α)(p− 1− β)− qm 6= 0. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that,we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in R.
在本文中,我们处理以下拟线性椭圆系统涉及梯度项的形式:{∆pu v = | |∇u在Ω∆p - v = vβ|∇u |Ω,哪里Ω⊂R (N≥2)等于R或等于一个球BR为中心在原点,半径R > 0, 1 < p <∞,m q > 0,α≥0,0≤β≤m和δ:= (p−−1α)(p−−1)β−qm 6 = 0。我们的目的是建立上述系统的爆破径向解的渐近性。准确地说，我们给出了这类爆破径向解在边界处的精确渐近性质。为此，我们证明了独立兴趣问题的一个强极大原理，并研究了R中的一个辅助渐近自治系统。
{"title":"Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids","authors":"A. Bachir, J. Giacomoni, G. Warnault","doi":"10.57262/die035-0910-511","DOIUrl":"https://doi.org/10.57262/die035-0910-511","url":null,"abstract":"In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: { ∆pu = v |∇u| in Ω ∆pv = v β |∇u| in Ω, where Ω ⊂ R (N ≥ 2) is either equal to R or equal to a ball BR centered at the origin and having radius R > 0, 1 < p < ∞, m, q > 0, α ≥ 0, 0 ≤ β ≤ m and δ := (p− 1− α)(p− 1− β)− qm 6= 0. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that,we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in R.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43644838","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 : 2021-12-23DOI: 10.57262/die035-0304-191
J. I. D'iaz, A. V. Podolskiy, T. Shaposhnikova
We consider the homogenization of an optimal control problem in which the control v is placed on a part Γ0 of the boundary and the spatial domain contains a thin layer of “small particles”, very close to the controlling boundary, and a Robin boundary condition is assumed on the boundary of those “small particles”. This problem can be associated with the climatization modeling of Bioclimatic Double Skin Façades which was developed in modern architecture as a tool for energy optimization. We assume that the size of the particles and the parameters involved in the Robin boundary condition are critical (and so they justify the occurrence of some “strange terms” in the homogenized problem). The cost functional is given by a weighted balance of the distance (in a H-type metric) to a prescribed target internal temperature uT and the proper cost of the control v (given by its L(Γ0) norm). We prove the (weak) convergence of states uε and of the controls vε to some functions, u0 and v0, respectively, which are completely identified: u0 satisfies an artificial boundary condition on Γ0 and v0 is the optimal control associated to a limit cost functional J0 in which the “boundary strange term” on Γ0 arises. This information on the limit problem makes much more manageable the study of the optimal climatization of such double skin structures.
{"title":"Boundary control and homogenization: Optimal climatization through smart double skin boundaries","authors":"J. I. D'iaz, A. V. Podolskiy, T. Shaposhnikova","doi":"10.57262/die035-0304-191","DOIUrl":"https://doi.org/10.57262/die035-0304-191","url":null,"abstract":"We consider the homogenization of an optimal control problem in which the control v is placed on a part Γ0 of the boundary and the spatial domain contains a thin layer of “small particles”, very close to the controlling boundary, and a Robin boundary condition is assumed on the boundary of those “small particles”. This problem can be associated with the climatization modeling of Bioclimatic Double Skin Façades which was developed in modern architecture as a tool for energy optimization. We assume that the size of the particles and the parameters involved in the Robin boundary condition are critical (and so they justify the occurrence of some “strange terms” in the homogenized problem). The cost functional is given by a weighted balance of the distance (in a H-type metric) to a prescribed target internal temperature uT and the proper cost of the control v (given by its L(Γ0) norm). We prove the (weak) convergence of states uε and of the controls vε to some functions, u0 and v0, respectively, which are completely identified: u0 satisfies an artificial boundary condition on Γ0 and v0 is the optimal control associated to a limit cost functional J0 in which the “boundary strange term” on Γ0 arises. This information on the limit problem makes much more manageable the study of the optimal climatization of such double skin structures.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44368013","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 : 2021-11-15DOI: 10.57262/die035-1112-729
M. Fila, Petra Mackov'a, J. Takahashi, E. Yanagida
Abstract. The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.
摘要研究一类具有特定持久奇异行为的快速扩散方程的正解。首先，构造了具有各向异性奇异点的新型解。根据参数的不同，这些解要么在分布意义上解原方程，要么在时空中不局部可积。我们证明后者也适用于具有蛇形奇点的解，蛇形奇点的存在性最近已被M. Fila, J.R. King, J. Takahashi和E. Yanagida证明。此外，我们建立了在分布意义上，M. Fila, J. Takahashi和E. Yanagida在2019年证明的各向同性解的存在性实际上解决了带有移动Dirac源项的相应问题。最后，讨论了一类临界情况下各向异性奇异解的存在性。
{"title":"Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation","authors":"M. Fila, Petra Mackov'a, J. Takahashi, E. Yanagida","doi":"10.57262/die035-1112-729","DOIUrl":"https://doi.org/10.57262/die035-1112-729","url":null,"abstract":"Abstract. The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47733555","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 : 2021-11-04DOI: 10.57262/die036-0910-727
D. Daners, Jochen Gluck, J. Mui
We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behaviour and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalisations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.
{"title":"Local uniform convergence and eventual positivity of solutions to biharmonic heat equations","authors":"D. Daners, Jochen Gluck, J. Mui","doi":"10.57262/die036-0910-727","DOIUrl":"https://doi.org/10.57262/die036-0910-727","url":null,"abstract":"We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behaviour and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalisations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43501794","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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