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Advances in Differential Equations最新文献

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On a generalized Cahn--Hilliard model with $p$-Laplacian 关于具有$p$-拉普拉斯算子的广义Cahn—Hilliard模型
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2022-02-22 DOI: 10.57262/ade027-0910-647
Raffaele Folino, Luis Fernando Lopez Rios, M. Strani
{"title":"On a generalized Cahn--Hilliard model with $p$-Laplacian","authors":"Raffaele Folino, Luis Fernando Lopez Rios, M. Strani","doi":"10.57262/ade027-0910-647","DOIUrl":"https://doi.org/10.57262/ade027-0910-647","url":null,"abstract":"A generalized Cahn-Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label\"generalized\"refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p>1$ and a double well potential of the form $F(u)=frac{1}{2theta}|1-u^2|^theta$, with $theta>1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F\"(pm1)>0$, which are typical of the standard Cahn-Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $thetageq p>1$. In the $critical$ $theta=p>1$, we prove $exponentially$ $slow$ $motion$ of profiles with a transition layer structure, thus extending the well know results of the standard model, where $theta=p=2$; conversely, in the $supercritical$ case $theta>p>1$, we prove $algebraic$ $slow$ $motion$ of layered profiles.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45610819","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}
引用次数: 1
Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality 通过严格Faber-Krahn型不等式的第一特征值的定义域变分
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2022-02-08 DOI: 10.57262/ade028-0708-537
T. Anoop, K. Kumar
{"title":"Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality","authors":"T. Anoop, K. Kumar","doi":"10.57262/ade028-0708-537","DOIUrl":"https://doi.org/10.57262/ade028-0708-537","url":null,"abstract":"For $dgeq 2$ and $frac{2d+2}{d+2}<p<infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $lambda _1(Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $Omega subset mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $Omega setminus mathscr{O}$, where $mathscr{O}subset subset Omega $ is an obstacle. Under some geometric assumptions on $Omega $ and $mathscr{O}$, we prove the strict monotonicity of $lambda _1 (Omega setminus mathscr{O})$ with respect to certain translations and rotations of $mathscr{O}$ in $Omega $.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48668976","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}
引用次数: 5
Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up 双曲空间上的非线性热方程:整体存在性和有限时间爆破
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2022-01-07 DOI: 10.57262/ade028-0910-779
D. Ganguly, D. Karmakar, Saikat Mazumdar
{"title":"Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up","authors":"D. Ganguly, D. Karmakar, Saikat Mazumdar","doi":"10.57262/ade028-0910-779","DOIUrl":"https://doi.org/10.57262/ade028-0910-779","url":null,"abstract":"We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space: begin{align}label{abs:eqn} left{begin{array}{ll} partial_{t}u=Delta_{mathbb{H}^{n}} u+ f(u, t)&hbox{ in }~ mathbb{H}^{n}times (0, T), quad u =u_{0}&hbox{ in }~ mathbb{H}^{n}times {0}. end{array}right. end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(mathbb{H}^{n}) cap L^{infty}(mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, it exhibits Fujita phenomena for the exponential weight $h(t) = e^{mu t},$ i.e. there exists a critical exponent $mu^*$ such that if $mu>mu^*$ then all non-negative solutions blow-up in finite time and if $mu leq mu^*$ there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that the above mentioned Cauchy problem with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47392905","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}
引用次数: 0
Quasilinear double phase problems in the whole space via perturbation methods 用摄动方法求解全空间拟线性双相位问题
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2022-01-01 DOI: 10.57262/ade027-0102-1
B. Ge, P. Pucci
{"title":"Quasilinear double phase problems in the whole space via perturbation methods","authors":"B. Ge, P. Pucci","doi":"10.57262/ade027-0102-1","DOIUrl":"https://doi.org/10.57262/ade027-0102-1","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41426437","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}
引用次数: 7
Weak and viscosity solutions for non-homogeneous fractional equations in Orlicz spaces Orlicz空间中非齐次分式方程的弱解和粘性解
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-12-06 DOI: 10.57262/ade027-1112-735
Maria L. de Borb'on, Leandro Martin Del Pezzo, Pablo Ochoa
{"title":"Weak and viscosity solutions for non-homogeneous fractional equations in Orlicz spaces","authors":"Maria L. de Borb'on, Leandro Martin Del Pezzo, Pablo Ochoa","doi":"10.57262/ade027-1112-735","DOIUrl":"https://doi.org/10.57262/ade027-1112-735","url":null,"abstract":"In this paper, we consider non-homogeneous fractional equations in Orlicz spaces, with a source depending on the spatial variable, the unknown function and its fractional gradient. The latter is adapted to the Orlicz framework. The main contribution of the article is to establish the equivalence between weak and viscosity solutions for such equations.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49651942","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}
引用次数: 4
Gradient estimate for solutions of second-order elliptic equations 二阶椭圆方程解的梯度估计
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-11-22 DOI: 10.57262/ade027-0102-77
V. Maz'ya, R. McOwen
{"title":"Gradient estimate for solutions of second-order elliptic equations","authors":"V. Maz'ya, R. McOwen","doi":"10.57262/ade027-0102-77","DOIUrl":"https://doi.org/10.57262/ade027-0102-77","url":null,"abstract":"We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x = 0. In particular, we treat the case of solutions that are not Lipschitz continuous at x = 0. We show that our estimate is sharp.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44870120","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}
引用次数: 0
$mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains 无界区域中二维随机对流Brinkman-Forchheimer方程的H^1 -随机吸引子
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-11-15 DOI: 10.57262/ade028-0910-807
K. Kinra, M. T. Mohan
{"title":"$mathbb H^1$-random attractors for 2d stochastic convective Brinkman-Forchheimer equations in unbounded domains","authors":"K. Kinra, M. T. Mohan","doi":"10.57262/ade028-0910-807","DOIUrl":"https://doi.org/10.57262/ade028-0910-807","url":null,"abstract":"The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincar'e domains). We first prove the existence of $mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $rin[1,3]$) perturbed by an additive noise on Poincar'e domains. Furthermore, we deduce the existence of a unique invariant measure in $mathbb{H}^1$ for the 2D SCBF equations defined on Poincar'e domains. In addition, a remark on the extension of these results to general unbounded domains is also discussed. Finally, for 2D SCBF equations forced by additive one-dimensional Wiener noise, we prove the upper semicontinuity of the random attractors, when the domain changes from bounded to unbounded (Poincar'e).","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48208442","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}
引用次数: 1
Nonlinear Fractional Schrödinger Equations coupled by power--type nonlinearities 幂型非线性耦合的非线性分数阶薛定谔方程
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-11-09 DOI: 10.57262/ade028-0102-113
E. Colorado, A. Ortega
{"title":"Nonlinear Fractional Schrödinger Equations coupled by power--type nonlinearities","authors":"E. Colorado, A. Ortega","doi":"10.57262/ade028-0102-113","DOIUrl":"https://doi.org/10.57262/ade028-0102-113","url":null,"abstract":"In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schrödinger equations, { (−∆)u1 + λ1u1 = μ1|u1|u1 + β|u2||u1|u1 in R , (−∆)u2 + λ2u2 = μ2|u2|u2 + β|u1||u2|u2 in R , where u1, u2 ∈ W (R ), with N = 1, 2, 3; λj , μj > 0, j = 1, 2, β ∈ R, p ≥ 2 and p− 1 2p N < s < 1. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters β, p, λj , μj , (j = 1, 2) satisfy appropriate conditions. We also study the previous system with m-equations, (−∆)uj + λjuj = μj |uj |uj + m ∑ k=1 k 6=j βjk|uk||uj |uj , uj ∈W (R ); j = 1, . . . ,m where λj , μj > 0 for j = 1, . . . ,m ≥ 3, the coupling parameters βjk = βkj ∈ R for j, k = 1, . . . ,m, j 6= k. For this system we prove similar results as for m = 2, depending on the values of the parameters βjk, p, λj , μj , (for j, k = 1, . . . ,m, j 6= k).","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47500355","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}
引用次数: 1
The Paneitz curvature problem on $S^n$ S^n$上的Paneitz曲率问题
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-11-01 DOI: 10.57262/ade026-1112-585
A. Alghanemi, Aymen Bensouf, H. Chtioui
{"title":"The Paneitz curvature problem on $S^n$","authors":"A. Alghanemi, Aymen Bensouf, H. Chtioui","doi":"10.57262/ade026-1112-585","DOIUrl":"https://doi.org/10.57262/ade026-1112-585","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47149874","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}
引用次数: 1
Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries 基于Hörmander向量场模型的完全非线性方程的Liouville结果：Ⅱ。卡诺群与格鲁申几何
IF 1.4 3区 数学
Advances in Differential Equations Pub Date : 2021-09-23 DOI: 10.57262/ade028-0708-637
M. Bardi, Alessandro Goffi
{"title":"Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries","authors":"M. Bardi, Alessandro Goffi","doi":"10.57262/ade028-0708-637","DOIUrl":"https://doi.org/10.57262/ade028-0708-637","url":null,"abstract":"The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, that under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper we proved an abstract result and discussed operators on the Heisenberg group. Here we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and zero-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47458145","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}
引用次数: 1
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