Discrete & Continuous Dynamical Systems - S最新文献

{"title":"Quantitative destruction of invariant circles","authors":"Lin Wang","doi":"10.3934/dcds.2021164","DOIUrl":"https://doi.org/10.3934/dcds.2021164","url":null,"abstract":"<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id=\"M1\">begin{document}$ omega $end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id=\"M2\">begin{document}$ N $end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id=\"M3\">begin{document}$ R_N $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M4\">begin{document}$ |R_N|_{C^r}<epsilon $end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id=\"M5\">begin{document}$ N $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M6\">begin{document}$ r $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M7\">begin{document}$ epsilon $end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id=\"M8\">begin{document}$ omega $end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id=\"M9\">begin{document}$ omega $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82054257","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}

{"title":"Global well-posedness for fractional Sobolev-Galpern type equations","authors":"Huy Tuan Nguyen, N. Tuan, Chaoxia Yang","doi":"10.3934/dcds.2021206","DOIUrl":"https://doi.org/10.3934/dcds.2021206","url":null,"abstract":"This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74921836","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}

{"title":"Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term","authors":"P. Colli, G. Gilardi, E. Rocca, J. Sprekels","doi":"10.3934/dcdss.2022001","DOIUrl":"https://doi.org/10.3934/dcdss.2022001","url":null,"abstract":"<p style='text-indent:20px;'>The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in <inline-formula><tex-math id=\"M1\">begin{document}$ {{mathbb{R}}}^d $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ 1leq dleq 3 $end{document}</tex-math></inline-formula>, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case <inline-formula><tex-math id=\"M3\">begin{document}$ d = 2 $end{document}</tex-math></inline-formula>. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79690962","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}

{"title":"On $ n $-tuplewise IP-sensitivity and thick sensitivity","authors":"Jian Li, Yini Yang","doi":"10.3934/dcds.2021211","DOIUrl":"https://doi.org/10.3934/dcds.2021211","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M2\">begin{document}$ (X,T) $end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id=\"M3\">begin{document}$ ngeq 2 $end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id=\"M4\">begin{document}$ (X,T) $end{document}</tex-math></inline-formula> is <inline-formula><tex-math id=\"M5\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id=\"M6\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id=\"M7\">begin{document}$ delta>0 $end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id=\"M8\">begin{document}$ U $end{document}</tex-math></inline-formula> of <inline-formula><tex-math id=\"M9\">begin{document}$ X $end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id=\"M10\">begin{document}$ x_1,x_2,dotsc,x_nin U $end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ Bigl{kin mathbb{N}colon minlimits_{1le i<jle n}d(T^k x_i,T^k x_j)>deltaBigr} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id=\"M11\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id=\"M12\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id=\"M13\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id=\"M14\">begin{document}$ ngeq 2 $end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id=\"M15\">begin{document}$ n $end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id=\"M16\">begin{document}$ n $end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id=\"M17\">begin{document}$ ^* $end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id=\"M18\">begin{document}$ ^* $end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system adm","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82998315","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}

{"title":"Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity","authors":"Marcelo Bongarti, I. Lasiecka, J. H. Rodrigues","doi":"10.3934/dcdss.2022020","DOIUrl":"https://doi.org/10.3934/dcdss.2022020","url":null,"abstract":"The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called \"critical\" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not \"controlled\", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73457258","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}

{"title":"Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources","authors":"Enzo Vitillaro","doi":"10.3934/dcdss.2021130","DOIUrl":"https://doi.org/10.3934/dcdss.2021130","url":null,"abstract":"<p style='text-indent:20px;'>The aim of this paper is to give global nonexistence and blow–up results for the problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{cases} u_{tt}-Delta u+P(x,u_t) = f(x,u) qquad &text{in $(0,infty)timesOmega$,} u = 0 &text{on $(0,infty)times Gamma_0$,} u_{tt}+partial_nu u-Delta_Gamma u+Q(x,u_t) = g(x,u)qquad &text{on $(0,infty)times Gamma_1$,} u(0,x) = u_0(x),quad u_t(0,x) = u_1(x) & text{in $overline{Omega}$,} end{cases} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ Omega $end{document}</tex-math></inline-formula> is a bounded open <inline-formula><tex-math id=\"M2\">begin{document}$ C^1 $end{document}</tex-math></inline-formula> subset of <inline-formula><tex-math id=\"M3\">begin{document}$ {mathbb R}^N $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M4\">begin{document}$ Nge 2 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M5\">begin{document}$ Gamma = partialOmega $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M6\">begin{document}$ (Gamma_0,Gamma_1) $end{document}</tex-math></inline-formula> is a partition of <inline-formula><tex-math id=\"M7\">begin{document}$ Gamma $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M8\">begin{document}$ Gamma_1not = emptyset $end{document}</tex-math></inline-formula> being relatively open in <inline-formula><tex-math id=\"M9\">begin{document}$ Gamma $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M10\">begin{document}$ Delta_Gamma $end{document}</tex-math></inline-formula> denotes the Laplace–Beltrami operator on <inline-formula><tex-math id=\"M11\">begin{document}$ Gamma $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M12\">begin{document}$ nu $end{document}</tex-math></inline-formula> is the outward normal to <inline-formula><tex-math id=\"M13\">begin{document}$ Omega $end{document}</tex-math></inline-formula>, and the terms <inline-formula><tex-math id=\"M14\">begin{document}$ P $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M15\">begin{document}$ Q $end{document}</tex-math></inline-formula> represent nonlinear damping terms, while <inline-formula><tex-math id=\"M16\">begin{document}$ f $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M17\">begin{document}$ g $end{document}</tex-math></inline-formula> are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77961785","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}

{"title":"Aubry-Mather theory for contact Hamiltonian systems II","authors":"Kaizhi Wang, Lin Wang, Jun Yan","doi":"10.3934/dcds.2021128","DOIUrl":"https://doi.org/10.3934/dcds.2021128","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems <inline-formula><tex-math id=\"M1\">begin{document}$ H(x,u,p) $end{document}</tex-math></inline-formula> with certain dependence on the contact variable <inline-formula><tex-math id=\"M2\">begin{document}$ u $end{document}</tex-math></inline-formula>. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set <inline-formula><tex-math id=\"M3\">begin{document}$ tilde{mathcal{S}}_s $end{document}</tex-math></inline-formula> consists of <i>strongly</i> static orbits, which coincides with the Aubry set <inline-formula><tex-math id=\"M4\">begin{document}$ tilde{mathcal{A}} $end{document}</tex-math></inline-formula> in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show <inline-formula><tex-math id=\"M5\">begin{document}$ tilde{mathcal{S}}_ssubsetneqqtilde{mathcal{A}} $end{document}</tex-math></inline-formula> in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of <inline-formula><tex-math id=\"M6\">begin{document}$ H $end{document}</tex-math></inline-formula> on <inline-formula><tex-math id=\"M7\">begin{document}$ u $end{document}</tex-math></inline-formula> fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the <i>minimal</i> viscosity solution and <i>non-minimal</i> ones.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85043416","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}

{"title":"Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation","authors":"Sangdon Jin, Younghun Hong","doi":"10.3934/dcds.2022010","DOIUrl":"https://doi.org/10.3934/dcds.2022010","url":null,"abstract":"For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [2]. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84594968","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}

{"title":"On stochastic porous-medium equations with critical-growth conservative multiplicative noise","authors":"N. Dirr, Hubertus Grillmeier, Guenther Grün","doi":"10.3934/dcds.2020388","DOIUrl":"https://doi.org/10.3934/dcds.2020388","url":null,"abstract":"First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74378463","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}

{"title":"Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives","authors":"M. Ammi, M. Tahiri, Delfim F. M. Torres","doi":"10.3934/dcdss.2021155","DOIUrl":"https://doi.org/10.3934/dcdss.2021155","url":null,"abstract":"The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89890846","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}