. In this article we analyze the boundary value problem governing stagnation-point fl ow of a fl uid with a power law outer fl ow over a surface moving with a speed proportional to the outer fl ow. The fl ow is characterized by two physical parameters; ε , which measures the stretching ( ε > 0) or shrinking ( ε < 0) of the sheet relative to the outer fl ow, and n > 0, the power law exponent. In the case of aiding fl ow ( ε > 0), where the (stretching) surface and the outer fl ow move in the same direction, we prove existence of a solution for all values of n . For opposing fl ow ( ε < 0), where the (shrinking) surface and the outer fl ow move in opposite directions, the situation is much more complicated. For − 1 < ε < 0 and all n we prove a solution exists. However, for ε (cid:2) − 1, we prove there exists a value, ε crit ( n ) (cid:2) − 1, such that no solutions exist for ε (cid:2) ε crit . For n = 1 / 7 and n = 1 / 3 we prove that ε crit = − 1. For other values of n , we derive bounds which illustrate the complicated nature of the existence/nonexistence boundary for opposing ( ε < 0) fl ows.
{"title":"Analysis of stagnation point flow over a stretching/shrinking surface","authors":"M'bagne F. 'bengue, J. Paullet","doi":"10.7153/dea-2021-13-23","DOIUrl":"https://doi.org/10.7153/dea-2021-13-23","url":null,"abstract":". In this article we analyze the boundary value problem governing stagnation-point fl ow of a fl uid with a power law outer fl ow over a surface moving with a speed proportional to the outer fl ow. The fl ow is characterized by two physical parameters; ε , which measures the stretching ( ε > 0) or shrinking ( ε < 0) of the sheet relative to the outer fl ow, and n > 0, the power law exponent. In the case of aiding fl ow ( ε > 0), where the (stretching) surface and the outer fl ow move in the same direction, we prove existence of a solution for all values of n . For opposing fl ow ( ε < 0), where the (shrinking) surface and the outer fl ow move in opposite directions, the situation is much more complicated. For − 1 < ε < 0 and all n we prove a solution exists. However, for ε (cid:2) − 1, we prove there exists a value, ε crit ( n ) (cid:2) − 1, such that no solutions exist for ε (cid:2) ε crit . For n = 1 / 7 and n = 1 / 3 we prove that ε crit = − 1. For other values of n , we derive bounds which illustrate the complicated nature of the existence/nonexistence boundary for opposing ( ε < 0) fl ows.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130782","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 time-independent fractional equation: where Ω is a smooth bounded domain, s ∈ ( 0 , 1 ) , N > 2 s 0 < q < 1, the coef fi cient functions f and g may change sign. We fi rst obtain the existence of ground state solution by the Nehari method under the combined effect of coef fi cient functions. Then we fi nd the multiplicity of positive solutions by Mountain pass theorem under some stronger conditions, and one of them is a ground state solution.
． 本文考虑一个与时间无关的分数阶方程，其中Ω是光滑有界域，s∈(0,1)，N bbb20 s 0 < q < 1，系数函数f和g可以改变符号。首先用Nehari方法得到了在系数函数联合作用下基态解的存在性。然后利用山口定理，在一些较强的条件下求出正解的多重性，其中一个正解是基态解。
{"title":"Multiplicity results for critical fractional equations with sign-changing weight functions","authors":"Yang Pu, Jia‐Feng Liao","doi":"10.7153/DEA-2021-13-09","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-09","url":null,"abstract":". In this paper, we consider a time-independent fractional equation: where Ω is a smooth bounded domain, s ∈ ( 0 , 1 ) , N > 2 s 0 < q < 1, the coef fi cient functions f and g may change sign. We fi rst obtain the existence of ground state solution by the Nehari method under the combined effect of coef fi cient functions. Then we fi nd the multiplicity of positive solutions by Mountain pass theorem under some stronger conditions, and one of them is a ground state solution.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"129 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130595","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}
We are interested in the existence result for a class of nonlinear third-order three-point boundary value problem with integral condition at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The result are illustrated with an example. Mathematics subject classification (2010): 34B15, 34B18.
{"title":"A class of nonlinear third-order boundary value problem with integral condition at resonance","authors":"H. Djourdem","doi":"10.7153/DEA-2021-13-04","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-04","url":null,"abstract":"We are interested in the existence result for a class of nonlinear third-order three-point boundary value problem with integral condition at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The result are illustrated with an example. Mathematics subject classification (2010): 34B15, 34B18.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71131019","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}
We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-Delta u=|x|^alpha |u|^{p}+zeta(t),{mathbf w}(x)$ in $(0,infty)times{mathbb{R}}^N$, where $Ngeq 3$, $p>1$, $alpha>-2$, $zeta, {mathbf w}$ are continuous functions such that $zeta(t)sim t^sigma$ as $tto 0$, $zeta(t)sim t^m$ as $ttoinfty$ . We obtain local existence for $sigma>-1$. We also show the following: $-$ If $mleq 0$, $p 0$, then all solutions blow up in finite time; $-$ If $m> 0$, $p>1$ and $int_{mathbb{R}^N}{mathbf w}(x)dx>0$, then all solutions blow up in finite time. The main novelty in this paper is that blow up depends on the behavior of $zeta$ at infinity.
{"title":"Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation","authors":"M. Majdoub","doi":"10.7153/DEA-2021-13-06","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-06","url":null,"abstract":"We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-Delta u=|x|^alpha |u|^{p}+zeta(t),{mathbf w}(x)$ in $(0,infty)times{mathbb{R}}^N$, where $Ngeq 3$, $p>1$, $alpha>-2$, $zeta, {mathbf w}$ are continuous functions such that $zeta(t)sim t^sigma$ as $tto 0$, $zeta(t)sim t^m$ as $ttoinfty$ . We obtain local existence for $sigma>-1$. We also show the following: \u0000$-$ If $mleq 0$, $p 0$, then all solutions blow up in finite time; \u0000$-$ If $m> 0$, $p>1$ and $int_{mathbb{R}^N}{mathbf w}(x)dx>0$, then all solutions blow up in finite time. \u0000The main novelty in this paper is that blow up depends on the behavior of $zeta$ at infinity.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49114865","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 weakly nonlinear boundary value problems on infinite intervals. For such problems, we provide criteria for the existence of solutions as well as a qualitative description of the behavior of solutions depending on a parameter. We investigate the relationship between solutions to these weakly nonlinear problems and the solutions to a set of corresponding linear problems.
{"title":"On weakly nonlinear boundary value problems on infinite intervals","authors":"B. Freedman, Jesús F. Rodríguez","doi":"10.7153/dea-2020-12-12","DOIUrl":"https://doi.org/10.7153/dea-2020-12-12","url":null,"abstract":"In this paper, we study weakly nonlinear boundary value problems on infinite intervals. For such problems, we provide criteria for the existence of solutions as well as a qualitative description of the behavior of solutions depending on a parameter. We investigate the relationship between solutions to these weakly nonlinear problems and the solutions to a set of corresponding linear problems.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46676655","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}
. We consider a class of nonlocal problem with strong singularity and linear term. Com- bining with the variational method and Nehari manifold, a necessary and suf fi cient condition that shows the existence of positive solution is obtained.
{"title":"Existence of positive solution for a class of nonlocal problem with strong singularity and linear term","authors":"A. Hou, Jia‐Feng Liao","doi":"10.7153/dea-2020-12-18","DOIUrl":"https://doi.org/10.7153/dea-2020-12-18","url":null,"abstract":". We consider a class of nonlocal problem with strong singularity and linear term. Com- bining with the variational method and Nehari manifold, a necessary and suf fi cient condition that shows the existence of positive solution is obtained.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130288","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 are concerned with the study of a class of second-order quasilinear parabolic equations involving Leray-Lions type operators with anisotropic growth conditions. By an approximation argument, we estabilsh the existence of entropy solutions in the framework of anisotropic parabolic Sobolev spaces when the initial condition and the data are assumed to be merely integrable. In addition, we prove that entropy solutions coincide with the renormalized solutions.
{"title":"Parabolic anisotropic problems with lower order terms and integrable data","authors":"M. Chrif, S. E. Manouni, H. Hjiaj","doi":"10.7153/dea-2020-12-26","DOIUrl":"https://doi.org/10.7153/dea-2020-12-26","url":null,"abstract":"In this paper we are concerned with the study of a class of second-order quasilinear parabolic equations involving Leray-Lions type operators with anisotropic growth conditions. By an approximation argument, we estabilsh the existence of entropy solutions in the framework of anisotropic parabolic Sobolev spaces when the initial condition and the data are assumed to be merely integrable. In addition, we prove that entropy solutions coincide with the renormalized solutions.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130585","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}
Mohammed El Mokhtar Ould El Mokhtar, Zeid I. Almuhiameed
. This article is devoted to the existence and multiplicity to the following singular ellip- tic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights: where Ω is a smooth bounded domain in R N ( N (cid:2) 3 ) , 0 ∈ Ω , λ > 0, 0 (cid:3) μ < μ N : = ( N − 2 ) 2 / 4, p = 2 ∗ ( s ) = 2 ( N − s ) / ( N − 2 ) with 0 < s < 2 is the critical Hardy-Sobolev critical exponent, 0 (cid:3) α < N ( p − 1 + β ) / p , 0 < β < 1 and 2 < p (cid:3) 2 ∗ : = 2 N / ( N − 2 ) is the critical Sobolev exponent. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained.
． 这文章devoted to the》的存在和multiplicity跟踪单数和单数ellip - tic equation nonlinearities, Hardy-Sobolev连接exponent和举重:Ω在哪里a smooth bounded域名在R N (N (cid: 2.0) 3),∈Ω,λ> 0,0 (cid): 3)μ<μN = (N−2)2 / 4,p =∗(s) = 2 (N−s) / s (N−2)和0 < < 2就是连接Hardy-Sobolev连接exponent, 0 (cid) 3:α< N (p−1 +β)- p, 0 <β< 1和2 < p (cid): 3)∗:= N / (N−2)是《连接Sobolev exponent。通过使用Nehari manifold和mountain pass theorem，最不重要的四种关键解决方案是保密的。
{"title":"On singular elliptic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights","authors":"Mohammed El Mokhtar Ould El Mokhtar, Zeid I. Almuhiameed","doi":"10.7153/dea-2020-12-25","DOIUrl":"https://doi.org/10.7153/dea-2020-12-25","url":null,"abstract":". This article is devoted to the existence and multiplicity to the following singular ellip- tic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights: where Ω is a smooth bounded domain in R N ( N (cid:2) 3 ) , 0 ∈ Ω , λ > 0, 0 (cid:3) μ < μ N : = ( N − 2 ) 2 / 4, p = 2 ∗ ( s ) = 2 ( N − s ) / ( N − 2 ) with 0 < s < 2 is the critical Hardy-Sobolev critical exponent, 0 (cid:3) α < N ( p − 1 + β ) / p , 0 < β < 1 and 2 < p (cid:3) 2 ∗ : = 2 N / ( N − 2 ) is the critical Sobolev exponent. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130572","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}
. We consider the large time asymptotic behavior of solutions to the initial-boundary value problem where n ∈ N . We fi nd large time asymptotic formulas of solutions for three different cases 1 ) a =
{"title":"Asymptotics for the Sobolev type equations with pumping","authors":"Jhon J. Pérez","doi":"10.7153/dea-2020-12-08","DOIUrl":"https://doi.org/10.7153/dea-2020-12-08","url":null,"abstract":". We consider the large time asymptotic behavior of solutions to the initial-boundary value problem where n ∈ N . We fi nd large time asymptotic formulas of solutions for three different cases 1 ) a =","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"48 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130132","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}
Summary: This work is concerned with the existence of a solution for non-autonomous measure driven semilinear equation in Banach spaces. The Schauder fixed point theorem is utilized to explore the existence of a solution. Finally, we construct an example to demonstrate the acquired outcomes.
{"title":"Existence of solution for non-autonomous semilinear measure driven equations","authors":"Surendra Kumar, R. Agarwal","doi":"10.7153/dea-2020-12-20","DOIUrl":"https://doi.org/10.7153/dea-2020-12-20","url":null,"abstract":"Summary: This work is concerned with the existence of a solution for non-autonomous measure driven semilinear equation in Banach spaces. The Schauder fixed point theorem is utilized to explore the existence of a solution. Finally, we construct an example to demonstrate the acquired outcomes.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"11 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130903","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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