Pub Date : 2022-02-22DOI: 10.57262/ade027-0910-647
Raffaele Folino, Luis Fernando Lopez Rios, M. Strani
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.
{"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}
Pub Date : 2022-02-08DOI: 10.57262/ade028-0708-537
T. Anoop, K. Kumar
For $dgeq 2$ and $frac{2d+2}{d+2}
对于$dgeq2$和$frac｛2d+2｝｛d+2｝＜p
{"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}
Pub Date : 2022-01-07DOI: 10.57262/ade028-0910-779
D. Ganguly, D. Karmakar, Saikat Mazumdar
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.
我们研究了双曲空间上半线性热方程的柯西问题: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}对于属于$C(mathbb{H}^{n}) cap L^{infty}(mathbb{H}^{n})$的非负初始数据$u_0$和形式$f(u,t) = h(t)g(u).$的$f$的不同选择，我们研究了Fujita现象。众所周知，对于$u,$中的幂非线性，幂权$h(t) = t^q$是次临界的，因为对于小初始数据存在非负全局解。另一方面，对于指数权$h(t) = e^{mu t},$，它表现出Fujita现象，即存在一个临界指数$mu^*$，如果$mu>mu^*$则所有非负解在有限时间内爆炸，如果$mu leq mu^*$存在小初始数据的非负全局解。本文的主要目标之一是在$u$中找到一个适当的非线性，以便上面提到的具有功率权重$h(t) = t^q$的柯西问题确实表现出藤田现象。在本文的剩余部分，我们研究了$u.$中指数非线性的Fujita现象，并进一步将这些结果推广到Cartan-Hadamard流形。
{"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}
{"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}
Pub Date : 2021-12-06DOI: 10.57262/ade027-1112-735
Maria L. de Borb'on, Leandro Martin Del Pezzo, Pablo Ochoa
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.
{"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}
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.
{"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}
Pub Date : 2021-11-15DOI: 10.57262/ade028-0910-807
K. Kinra, M. T. Mohan
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).
{"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}
Pub Date : 2021-11-09DOI: 10.57262/ade028-0102-113
E. Colorado, A. Ortega
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).
{"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}
Pub Date : 2021-11-01DOI: 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}
Pub Date : 2021-09-23DOI: 10.57262/ade028-0708-637
M. Bardi, Alessandro Goffi
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.
{"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}
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