HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. STOKES AND NAVIER-STOKES PROBLEMS WITH NAVIER-TYPE BOUNDARY CONDITION IN L P -SPACES Chérif Al Baba, Chérif Amrouche
{"title":"Stokes and Navier-Stokes problems with Navier-type boundary condition in L^p-spaces","authors":"Chérif Al Baba, C. Amrouche","doi":"10.7153/DEA-2019-11-08","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-08","url":null,"abstract":"HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. STOKES AND NAVIER-STOKES PROBLEMS WITH NAVIER-TYPE BOUNDARY CONDITION IN L P -SPACES Chérif Al Baba, Chérif Amrouche","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42791719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation % begin{equation*} left{begin{array}{l} partial_tu+partial_x^3u+L_pu+upartial_xu=0, u(0,,x)=u_0(x). end{array} right. end{equation*} % where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I!!R)$ and sharp ill-posedness in $H^s(I!!R)$ when $s<-p/2$ with $p geq 2$.
本文研究了以下广义KdV- Burgers方程的柯西问题 % begin{equation*} left{begin{array}{l} partial_tu+partial_x^3u+L_pu+upartial_xu=0, u(0,,x)=u_0(x). end{array} right. end{equation*} % where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I!!R)$ and sharp ill-posedness in $H^s(I!!R)$ when $s<-p/2$ with $p geq 2$.
{"title":"Sharp well-posedness and ill-posedness results for dissipative KdV equations on the real line","authors":"X. Carvajal, P. Gamboa, Raphael Santos","doi":"10.7153/dea-2021-13-24","DOIUrl":"https://doi.org/10.7153/dea-2021-13-24","url":null,"abstract":"This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \u0000% \u0000begin{equation*} \u0000left{begin{array}{l} \u0000partial_tu+partial_x^3u+L_pu+upartial_xu=0, \u0000u(0,,x)=u_0(x). \u0000end{array} \u0000right. \u0000end{equation*} \u0000% \u0000where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I!!R)$ and sharp ill-posedness in $H^s(I!!R)$ when $s<-p/2$ with $p geq 2$.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48161559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.
{"title":"Applications of generalized trigonometric functions with two parameters II","authors":"S. Takeuchi","doi":"10.7153/dea-2019-11-28","DOIUrl":"https://doi.org/10.7153/dea-2019-11-28","url":null,"abstract":"Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator. Compared to GTFs with one parameter, there are few applications of GTFs with two parameters to differential equations. We will apply GTFs with two parameters to studies on the inviscid primitive equations of oceanic and atmospheric dynamics, new formulas of Gaussian hypergeometric functions, and the $L^q$-Lyapunov inequality for the one-dimensional $p$-Laplacian.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42254778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider nonlinearly perturbed Legendre differential equations subject to the usual boundary conditions. For such problems we establish sufficient conditions for the existence of solutions and in some cases we provide a qualitative description of solutions depending on a parameter. The results presented depend on the size and limiting behavior of the nonlinearities.
{"title":"Existence of solutions to nonlinear Legendre boundary value problems","authors":"B. Freedman, Jesús F. Rodríguez","doi":"10.7153/dea-2019-11-24","DOIUrl":"https://doi.org/10.7153/dea-2019-11-24","url":null,"abstract":"In this paper, we consider nonlinearly perturbed Legendre differential equations subject to the usual boundary conditions. For such problems we establish sufficient conditions for the existence of solutions and in some cases we provide a qualitative description of solutions depending on a parameter. The results presented depend on the size and limiting behavior of the nonlinearities.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45223855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. C. Pereira, Hoang-Ngan Nguyen, C. Raposo, C. Maranhão
In this manuscript, we consider the nonlinear beam equation with internal damping and source term utt +Δu+M(|∇u|)(−Δu)+ut = |u|r−1u where r > 1 is a constant, M(s) is a continuous function on [0,+∞) . The global solutions are constructed by using the Faedo-Galerkin approximations, taking into account that the initial data is in appropriate set of stability created from the Nehari manifold. The asymptotic behavior is obtained by the Nakao method.
{"title":"On the solutions for an extensible beam equation with internal damping and source terms","authors":"D. C. Pereira, Hoang-Ngan Nguyen, C. Raposo, C. Maranhão","doi":"10.7153/DEA-2019-11-17","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-17","url":null,"abstract":"In this manuscript, we consider the nonlinear beam equation with internal damping and source term utt +Δu+M(|∇u|)(−Δu)+ut = |u|r−1u where r > 1 is a constant, M(s) is a continuous function on [0,+∞) . The global solutions are constructed by using the Faedo-Galerkin approximations, taking into account that the initial data is in appropriate set of stability created from the Nehari manifold. The asymptotic behavior is obtained by the Nakao method.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"32 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This article concerns nonexistence and existence of positive solutions to the fractional differential equation where α ∈ ( 2 , 3 ) , D α is the standard Riemann-Liouville derivative and f : R + × R + → R + is a continuous function. The main results obtained here, are under eigenvalue criteria.
本文研究分数阶微分方程{Dα u(t)+ f (t,u(t)) = 0,0 t <∞，u(0) = Dα−2u(0) =极限→∞Dα−1u(t) = 0，其中α∈(2,3)，Dα是标准Riemann-Liouville导数，f: R+ ×R+→R+是连续函数的不存在性和存在性。这里得到的主要结果是在特征值准则下得到的。数学学科分类(2010):26A33, 34B16, 34B18, 34B27。
{"title":"Eigenvalue criteria for existence and nonexistence of positive solutions for α-order fractional differential equations,(2 < α; < 3), on the half-line","authors":"Abdelhamid Benmezaï, S. Chentout","doi":"10.7153/dea-2019-11-22","DOIUrl":"https://doi.org/10.7153/dea-2019-11-22","url":null,"abstract":". This article concerns nonexistence and existence of positive solutions to the fractional differential equation where α ∈ ( 2 , 3 ) , D α is the standard Riemann-Liouville derivative and f : R + × R + → R + is a continuous function. The main results obtained here, are under eigenvalue criteria.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider a first order dynamic equation on time scales in which the right hand side is a Δ -Carathéodory function, which is not necessarily continuous. We generalize this discontinuous dynamic equation using Henstock–Kurzweil Δ -integral and establish results concerning existence of solutions using simple analysis. Uniqueness of solutions is obtained using an Osgood type condition. Moreover we introduce the concept of Henstock–Kurzweil Δ -equi-integrability and study continuous dependence and convergence of solutions.
{"title":"Generalized first order dynamic equations on time scales with Δ-Carathéodory functions","authors":"Sanket Tikare","doi":"10.7153/dea-2019-11-06","DOIUrl":"https://doi.org/10.7153/dea-2019-11-06","url":null,"abstract":"In this paper we consider a first order dynamic equation on time scales in which the right hand side is a Δ -Carathéodory function, which is not necessarily continuous. We generalize this discontinuous dynamic equation using Henstock–Kurzweil Δ -integral and establish results concerning existence of solutions using simple analysis. Uniqueness of solutions is obtained using an Osgood type condition. Moreover we introduce the concept of Henstock–Kurzweil Δ -equi-integrability and study continuous dependence and convergence of solutions.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper we study the nonlinear parabolic system ∂ t u = Δ u + a | x | − γ | v | p − 1 v , ∂ t v = Δ v + b | x | − ρ | u | q − 1 u , t > 0, x ∈ R N { 0 } , N (cid:2) 1, a , b ∈ R , 0 (cid:3) γ < min ( N , 2 ) , 0 < ρ < min ( N , 2 ) , p , q > 1. Under conditions on the parameters p , q , γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In partic- ular, we show the existence of self-similar solutions with initial value Φ = ( ϕ 1 , ϕ 2 ) , where ϕ 1 , ϕ 2 are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions.
． 在本文中,我们研究了非线性抛物系统∂x t u =Δu + | |−γ| | p−1 v,∂x t v =Δv + b | |−ρ| | q−1 u, t > 0, x∈R N {0}, N (cid: 2) 1, a, b∈R, 0 (cid: 3)γ< min (N, 2), 0 <ρ< min (N, 2), p, q > 1。在参数p, q， γ和ρ的条件下，我们证明了初值小的全局解对某些范数的存在唯一性。特别地，我们证明了初始值Φ = (Φ 1， Φ 2)的自相似解的存在性，其中Φ 1， Φ 2是齐次初始数据。我们还证明了一些全局解在大时间内对自相似解是渐近的。
{"title":"Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems","authors":"B. Slimene","doi":"10.7153/dea-2019-11-21","DOIUrl":"https://doi.org/10.7153/dea-2019-11-21","url":null,"abstract":". In this paper we study the nonlinear parabolic system ∂ t u = Δ u + a | x | − γ | v | p − 1 v , ∂ t v = Δ v + b | x | − ρ | u | q − 1 u , t > 0, x ∈ R N { 0 } , N (cid:2) 1, a , b ∈ R , 0 (cid:3) γ < min ( N , 2 ) , 0 < ρ < min ( N , 2 ) , p , q > 1. Under conditions on the parameters p , q , γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In partic- ular, we show the existence of self-similar solutions with initial value Φ = ( ϕ 1 , ϕ 2 ) , where ϕ 1 , ϕ 2 are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we consider a reaction-diffusion competition model describing population dynamics of two competing species and the interactions between them in a heterogeneous environment. The main goal of this paper is to study the impact of different diffusion strategies on the outcome of competition between two populations while the first species is distributed according to the resource function and the second population is following the regular dispersion. We focus on how directed diffusion in the habitat influences selection. The two populations differ in the diffusion strategies they employ as well as in their environmental intensities. We establish the main results which determine that the competing species may either coexist, or one of them may bring the other to extinction. If higher carrying capacity is incorporated for the directed dispersal population then competitive exclusion of a regularly diffusing population is inevitable. We consider the case when both populations manage to coexist and there is an ideal free pair with identical carrying capacity, and the relevant coexistence equilibrium is a global attractor. The coexistence solution is also presented by showing the influence of diffusion coefficients. In a series of examples, the results have been justified and illustrated numerically. Mathematics subject classification (2010): 92D25, 35K57, 35K50, 37N25.
{"title":"Directed vs regular diffusion strategy: evolutionary stability analysis of a competition model and an ideal free pair","authors":"D. Kamrujjaman","doi":"10.7153/DEA-2019-11-11","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-11","url":null,"abstract":"In this study, we consider a reaction-diffusion competition model describing population dynamics of two competing species and the interactions between them in a heterogeneous environment. The main goal of this paper is to study the impact of different diffusion strategies on the outcome of competition between two populations while the first species is distributed according to the resource function and the second population is following the regular dispersion. We focus on how directed diffusion in the habitat influences selection. The two populations differ in the diffusion strategies they employ as well as in their environmental intensities. We establish the main results which determine that the competing species may either coexist, or one of them may bring the other to extinction. If higher carrying capacity is incorporated for the directed dispersal population then competitive exclusion of a regularly diffusing population is inevitable. We consider the case when both populations manage to coexist and there is an ideal free pair with identical carrying capacity, and the relevant coexistence equilibrium is a global attractor. The coexistence solution is also presented by showing the influence of diffusion coefficients. In a series of examples, the results have been justified and illustrated numerically. Mathematics subject classification (2010): 92D25, 35K57, 35K50, 37N25.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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