{"title":"A convergence result for a Stefan problem with phase relaxation","authors":"V. Recupero","doi":"10.3934/dcdss.2023119","DOIUrl":"https://doi.org/10.3934/dcdss.2023119","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80724236","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":"Dafermos regularization and viscous wave fan profiles for Riemann solutions of Burger's equation","authors":"Weishi Liu","doi":"10.3934/dcdss.2023047","DOIUrl":"https://doi.org/10.3934/dcdss.2023047","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74778813","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 study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term $q$. For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term $q$. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term $q$. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as $tto +infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.
{"title":"Well-posedness and long-time behavior of a bulk-surface coupled Cahn-Hilliard-diffusion system with singular potential for lipid raft formation","authors":"Hao Wu, Shengqin Xu","doi":"10.3934/dcdss.2023169","DOIUrl":"https://doi.org/10.3934/dcdss.2023169","url":null,"abstract":"We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term $q$. For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term $q$. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term $q$. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as $tto +infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135400449","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}
The 1D Burger's equation with Dirichlet boundary conditions exhibits a first transition from the trivial steady state to a sinusoidal patterned steady state as the parameter $ lambda $ which controls the linear term exceeds 1. The main goal of this paper is to present two different approaches regarding the transition of this patterned steady state. We believe that these approaches can be extended to study the dynamics of more interesting models. As a first approach, we consider an external forcing on the equation which supports a sinusoidal solution as a stable steady state which loses its stability at a critical threshold. We use the method of continued fractions to rigorously analyze the associated linear problem. In particular, we find that the system exhibits a mixed type transition with two distinct basins for initial conditions one of which leads to a local steady state and the other leaves a small neighborhood of the origin. As a second approach, we consider the dynamics on the center-unstable manifold of the first two modes of the unforced system. In this approach, the secondary transition produces two branches of steady state solutions. On one of these branches there is another transition which indicates a symmetry breaking phenomena.
{"title":"Two approaches to instability analysis of the viscous Burgers' equation","authors":"Messoud Efendiev, Taylan Sengul, Burhan Tiryakioglu","doi":"10.3934/dcdss.2023202","DOIUrl":"https://doi.org/10.3934/dcdss.2023202","url":null,"abstract":"The 1D Burger's equation with Dirichlet boundary conditions exhibits a first transition from the trivial steady state to a sinusoidal patterned steady state as the parameter $ lambda $ which controls the linear term exceeds 1. The main goal of this paper is to present two different approaches regarding the transition of this patterned steady state. We believe that these approaches can be extended to study the dynamics of more interesting models. As a first approach, we consider an external forcing on the equation which supports a sinusoidal solution as a stable steady state which loses its stability at a critical threshold. We use the method of continued fractions to rigorously analyze the associated linear problem. In particular, we find that the system exhibits a mixed type transition with two distinct basins for initial conditions one of which leads to a local steady state and the other leaves a small neighborhood of the origin. As a second approach, we consider the dynamics on the center-unstable manifold of the first two modes of the unforced system. In this approach, the secondary transition produces two branches of steady state solutions. On one of these branches there is another transition which indicates a symmetry breaking phenomena.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135506329","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 discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab $ Omegatimes (0, h) $ with $ Omegasubset mathbb{R}^2 $ and $ h>0 $ we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder $ Omegatimes (0, h) $. The Dirichlet datum has the form $ (g, 0) $, where $ gcolonpartialOmegato mathbb{S}^1 $ has non-zero winding number. Under appropriate conditions on the scaling, in the limit as $ hto 0 $ we obtain a behavior that is similar to the one observed in the asymptotic analysis (see [7]) of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in $ Omega $ having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and Hélein [7].
{"title":"Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals","authors":"Giacomo Canevari, Antonio Segatti","doi":"10.3934/dcdss.2023174","DOIUrl":"https://doi.org/10.3934/dcdss.2023174","url":null,"abstract":"In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab $ Omegatimes (0, h) $ with $ Omegasubset mathbb{R}^2 $ and $ h>0 $ we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder $ Omegatimes (0, h) $. The Dirichlet datum has the form $ (g, 0) $, where $ gcolonpartialOmegato mathbb{S}^1 $ has non-zero winding number. Under appropriate conditions on the scaling, in the limit as $ hto 0 $ we obtain a behavior that is similar to the one observed in the asymptotic analysis (see [7]) of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in $ Omega $ having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and Hélein [7].","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135556893","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}
Moritz Ebeling-Rump, Dietmar Hömberg, Robert Lasarzik
In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg–Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system.
{"title":"On a two-scale phasefield model for topology optimization","authors":"Moritz Ebeling-Rump, Dietmar Hömberg, Robert Lasarzik","doi":"10.3934/dcdss.2023206","DOIUrl":"https://doi.org/10.3934/dcdss.2023206","url":null,"abstract":"In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg–Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135609327","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}
This paper is concerned with the existence and limiting behavior of invariant probability measures or periodic probability measures for a type of widely used Hopfield-type lattice models with two nonlinear terms of arbitrary polynomial growth on the entire integer set $ mathbb{Z}^d $ driven by nonlinear white noise and Lévy noise. First, when the noise intensity is within a controllable range, we prove that the family probability distribution laws solutions and use the weak convergence method to prove the existence of invariant probability measures. Then, when the terms that change over time are periodic we also discussed the periodic probability measures existence in a weighted $ ell_rho^2 $ space. Finally, the limiting behavior of the collection of all invariant or periodic probability measures weakly compact are studied for Hopfield models driven by nonlinear white noise and Lévy noise about with noise intensity.
{"title":"Limiting behavior of invariant or periodic measure of Hopfield neural models driven by locally Lipschitz Lévy noise","authors":"Hailang Bai, Yan Wang, Yu Wang","doi":"10.3934/dcdss.2023195","DOIUrl":"https://doi.org/10.3934/dcdss.2023195","url":null,"abstract":"This paper is concerned with the existence and limiting behavior of invariant probability measures or periodic probability measures for a type of widely used Hopfield-type lattice models with two nonlinear terms of arbitrary polynomial growth on the entire integer set $ mathbb{Z}^d $ driven by nonlinear white noise and Lévy noise. First, when the noise intensity is within a controllable range, we prove that the family probability distribution laws solutions and use the weak convergence method to prove the existence of invariant probability measures. Then, when the terms that change over time are periodic we also discussed the periodic probability measures existence in a weighted $ ell_rho^2 $ space. Finally, the limiting behavior of the collection of all invariant or periodic probability measures weakly compact are studied for Hopfield models driven by nonlinear white noise and Lévy noise about with noise intensity.","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135212429","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":"Fractional weighted $ p $-Kirchhoff equations with general nonlinearity","authors":"Mingqi Xiang, Chaoqun Song","doi":"10.3934/dcdss.2023128","DOIUrl":"https://doi.org/10.3934/dcdss.2023128","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86764453","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":"Instantaneous blow-up for evolution inequalities of Sobolev type with nonlinear convolution terms","authors":"Ibtehal Alazman, M. Jleli","doi":"10.3934/dcdss.2023026","DOIUrl":"https://doi.org/10.3934/dcdss.2023026","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90424491","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":"Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents","authors":"E. Azroul, M. Bouziani, H. Hjiaj","doi":"10.3934/dcdss.2023034","DOIUrl":"https://doi.org/10.3934/dcdss.2023034","url":null,"abstract":"","PeriodicalId":48838,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series S","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81148832","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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