The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.
{"title":"The anisotropic Cahn–Hilliard equation: Regularity theory and strict separation properties","authors":"H. Garcke, P. Knopf, J. Wittmann","doi":"10.3934/dcdss.2023146","DOIUrl":"https://doi.org/10.3934/dcdss.2023146","url":null,"abstract":"The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"31 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89481144","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}
Boštjan Gabrovšek, Giovanni Molica Bisci, Dušan D. Repovš
In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term begin{document}$ f $end{document} has a suitable oscillating behaviour either at the origin or at infinity.
In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term begin{document}$ f $end{document} has a suitable oscillating behaviour either at the origin or at infinity.
{"title":"On nonlocal Dirichlet problems with oscillating term","authors":"Boštjan Gabrovšek, Giovanni Molica Bisci, Dušan D. Repovš","doi":"10.3934/dcdss.2022130","DOIUrl":"https://doi.org/10.3934/dcdss.2022130","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term <inline-formula><tex-math id=\"M1\">begin{document}$ f $end{document}</tex-math></inline-formula> has a suitable oscillating behaviour either at the origin or at infinity.</p>","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"121 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78428342","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}
We consider a phase-field model which describes the interactions between the blood flow and the thrombus. The latter is supposed to be a viscoelastic material. The potential describing the cohesive energy of the mixture is assumed to be of Flory-Huggins type (i.e. logarithmic). This ensures the boundedness from below of the dissipation energy. In the two dimensional case, we prove the local (in time) existence and uniqueness of a strong solution, provided that the two viscosities of the pure fluid phases are close enough. We also show that the order parameter remains strictly separated from the pure phases if it is so at the initial time.
{"title":"A phase-field system arising from multiscale modeling of thrombus biomechanics in blood vessels: Local well-posedness in dimension two","authors":"M. Grasselli, A. Poiatti","doi":"10.3934/dcdss.2023105","DOIUrl":"https://doi.org/10.3934/dcdss.2023105","url":null,"abstract":"We consider a phase-field model which describes the interactions between the blood flow and the thrombus. The latter is supposed to be a viscoelastic material. The potential describing the cohesive energy of the mixture is assumed to be of Flory-Huggins type (i.e. logarithmic). This ensures the boundedness from below of the dissipation energy. In the two dimensional case, we prove the local (in time) existence and uniqueness of a strong solution, provided that the two viscosities of the pure fluid phases are close enough. We also show that the order parameter remains strictly separated from the pure phases if it is so at the initial time.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"33 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91284359","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}
Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.
{"title":"Degenerate diffusion with Preisach hysteresis","authors":"Chiara Gavioli, Pavel Krejvc'i","doi":"10.3934/dcdss.2023154","DOIUrl":"https://doi.org/10.3934/dcdss.2023154","url":null,"abstract":"Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82380023","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}
In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $mathcal{R}_0$.
{"title":"The linear stability and basic reproduction numbers for autonomous FDEs","authors":"Xiao-Qiang Zhao","doi":"10.3934/dcdss.2023082","DOIUrl":"https://doi.org/10.3934/dcdss.2023082","url":null,"abstract":"In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $mathcal{R}_0$ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $mathcal{R}_0$.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"10 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063197","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}
Xinghui Zhang, Enyong Liu, Jianlong Qiu, A. Zhang, Zhi Liu
{"title":"Output feedback finite-time stabilization of a class of large-scale high-order nonlinear stochastic feedforward systems","authors":"Xinghui Zhang, Enyong Liu, Jianlong Qiu, A. Zhang, Zhi Liu","doi":"10.3934/dcdss.2023008","DOIUrl":"https://doi.org/10.3934/dcdss.2023008","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90214021","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}
Bui Duc Nam, Le Nhat Huynh, Le Dinh Long, Yusuf Gurefe
In this paper, we consider the terminal value problem for fractional super diffusive equation in case linear source function. This equation has many applications in physical phenomena. The results in this study are mainly provide the existence and regularity of the mild solution under the various assumptions of the input data.
{"title":"On terminal value problem for fractional superdiffusive of Sobolev equation type","authors":"Bui Duc Nam, Le Nhat Huynh, Le Dinh Long, Yusuf Gurefe","doi":"10.3934/dcdss.2023172","DOIUrl":"https://doi.org/10.3934/dcdss.2023172","url":null,"abstract":"In this paper, we consider the terminal value problem for fractional super diffusive equation in case linear source function. This equation has many applications in physical phenomena. The results in this study are mainly provide the existence and regularity of the mild solution under the various assumptions of the input data.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549391","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}
In [2] we established an existence theorem on bounded/periodic solutions for a class of scalar delay differential equations of the form$ begin{equation} frac{du}{dt} = f(t,u(t), u(t-r_1), cdots, u(t-r_n)), qquad t in mathbb R, ;;;;;(1)end{equation} $under the assumptions that the constant delays $ r_k>0 $, $ k = 1, cdots, n $, are 'small' and $ f $ satisfies a one-sided Lipschitz condition on the variables $ u(t), u(t-r_1), cdots, u(t-r_n) $. In this paper, we improve this result in the case that $ f $ is strictly increasing in some variables $ u(t-r_k) $ and obtain a new result that allows larger values of $ r_k $ with which the equation (1) still has a bounded/periodic solution. We illustrate this result via some population models.
在[2]中，我们建立了一类形式为$ begin{equation} frac{du}{dt} = f(t,u(t), u(t-r_1), cdots, u(t-r_n)), qquad t in mathbb R, ;;;;;(1)end{equation} $的标量延迟微分方程的有界/周期解的存在性定理，假设常数延迟$ r_k>0 $, $ k = 1, cdots, n $是“小”的，并且$ f $满足变量$ u(t), u(t-r_1), cdots, u(t-r_n) $上的单侧Lipschitz条件。在本文中，我们改进了$ f $在某些变量$ u(t-r_k) $中是严格递增的情况下的结果，得到了一个新的结果，允许更大的$ r_k $值，使得方程(1)仍然有有界/周期解。我们通过一些人口模型来说明这一结果。
{"title":"An improved result on bounded/periodic solutions for some scalar delay differential equations","authors":"Shangbing Ai","doi":"10.3934/dcdss.2023205","DOIUrl":"https://doi.org/10.3934/dcdss.2023205","url":null,"abstract":"In [2] we established an existence theorem on bounded/periodic solutions for a class of scalar delay differential equations of the form$ begin{equation} frac{du}{dt} = f(t,u(t), u(t-r_1), cdots, u(t-r_n)), qquad t in mathbb R, ;;;;;(1)end{equation} $under the assumptions that the constant delays $ r_k>0 $, $ k = 1, cdots, n $, are 'small' and $ f $ satisfies a one-sided Lipschitz condition on the variables $ u(t), u(t-r_1), cdots, u(t-r_n) $. In this paper, we improve this result in the case that $ f $ is strictly increasing in some variables $ u(t-r_k) $ and obtain a new result that allows larger values of $ r_k $ with which the equation (1) still has a bounded/periodic solution. We illustrate this result via some population models.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135559412","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":"On the Cahn–Hilliard–Oono equation with singular potential and volume constraint","authors":"Takeshi Fukao, Giulio Schimperna","doi":"10.3934/dcdss.2023198","DOIUrl":"https://doi.org/10.3934/dcdss.2023198","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135561006","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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