This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.
{"title":"Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems","authors":"Yijun Chen, Xiaohu Wang, Kenan Wu","doi":"10.3934/cpaa.2022059","DOIUrl":"https://doi.org/10.3934/cpaa.2022059","url":null,"abstract":"This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131240532","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}
First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining begin{document}$ 2n+3[frac{n+1}{2}] $end{document} limit cycles near the double generalized homoclinic loop.
First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining begin{document}$ 2n+3[frac{n+1}{2}] $end{document} limit cycles near the double generalized homoclinic loop.
{"title":"The number of limit cycles by perturbing a piecewise linear system with three zones","authors":"Xiaolei Zhang, Yanqin Xiong, Yi Zhang","doi":"10.3934/cpaa.2022049","DOIUrl":"https://doi.org/10.3934/cpaa.2022049","url":null,"abstract":"<p style='text-indent:20px;'>First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining <inline-formula><tex-math id=\"M1\">begin{document}$ 2n+3[frac{n+1}{2}] $end{document}</tex-math></inline-formula> limit cycles near the double generalized homoclinic loop.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124791395","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 long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.
{"title":"Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay","authors":"Lin Yang, Yejuan Wang, P. Kloeden","doi":"10.3934/cpaa.2022048","DOIUrl":"https://doi.org/10.3934/cpaa.2022048","url":null,"abstract":"In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128425571","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}
with the homogeneous Dirichlet boundary conditions. Here, begin{document}$ mathcal{L} = (-Delta)^m $end{document} is a harmonic operator of order begin{document}$ m $end{document}, begin{document}$ v = v(t, x):[0, tau]timesOmegarightarrow mathbb{R}^n $end{document} is the unknown, begin{document}$ t $end{document} is a parameter, begin{document}$ F_{S, T} $end{document} is a function related to given functions begin{document}$ S $end{document} and begin{document}$ T $end{document}, and begin{document}$ G(v)(t, x) $end{document} is defined by the solution begin{document}$ y^v(s;t, x) $end{document} of an ODE-IVP begin{document}$ {rm d}y/mathrm{d}s = v(s, y), quad y(t) = x $end{document}. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.
We study the existence of the solution to a semilinear higher-order elliptic system begin{document}$ mathcal{L}v(t, cdot) = F_{S, T}circ G(v)(t, cdot), quad forall tin [0, tau], $end{document} with the homogeneous Dirichlet boundary conditions. Here, begin{document}$ mathcal{L} = (-Delta)^m $end{document} is a harmonic operator of order begin{document}$ m $end{document}, begin{document}$ v = v(t, x):[0, tau]timesOmegarightarrow mathbb{R}^n $end{document} is the unknown, begin{document}$ t $end{document} is a parameter, begin{document}$ F_{S, T} $end{document} is a function related to given functions begin{document}$ S $end{document} and begin{document}$ T $end{document}, and begin{document}$ G(v)(t, x) $end{document} is defined by the solution begin{document}$ y^v(s;t, x) $end{document} of an ODE-IVP begin{document}$ {rm d}y/mathrm{d}s = v(s, y), quad y(t) = x $end{document}. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.
{"title":"On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration","authors":"Xiaojun Zheng, Zhongdan Huan, Jun Liu","doi":"10.3934/cpaa.2022068","DOIUrl":"https://doi.org/10.3934/cpaa.2022068","url":null,"abstract":"<p style='text-indent:20px;'>We study the existence of the solution to a semilinear higher-order elliptic system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ mathcal{L}v(t, cdot) = F_{S, T}circ G(v)(t, cdot), quad forall tin [0, tau], $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with the homogeneous Dirichlet boundary conditions. Here, <inline-formula><tex-math id=\"M1\">begin{document}$ mathcal{L} = (-Delta)^m $end{document}</tex-math></inline-formula> is a harmonic operator of order <inline-formula><tex-math id=\"M2\">begin{document}$ m $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M3\">begin{document}$ v = v(t, x):[0, tau]timesOmegarightarrow mathbb{R}^n $end{document}</tex-math></inline-formula> is the unknown, <inline-formula><tex-math id=\"M4\">begin{document}$ t $end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id=\"M5\">begin{document}$ F_{S, T} $end{document}</tex-math></inline-formula> is a function related to given functions <inline-formula><tex-math id=\"M6\">begin{document}$ S $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M7\">begin{document}$ T $end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M8\">begin{document}$ G(v)(t, x) $end{document}</tex-math></inline-formula> is defined by the solution <inline-formula><tex-math id=\"M9\">begin{document}$ y^v(s;t, x) $end{document}</tex-math></inline-formula> of an ODE-IVP <inline-formula><tex-math id=\"M10\">begin{document}$ {rm d}y/mathrm{d}s = v(s, y), quad y(t) = x $end{document}</tex-math></inline-formula>. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129191679","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 investigate transients in the oceanic Ekman layer in the presence of time-varying winds and a constant-in-depth but time dependent eddy viscosity, where the initial state is taken to be the steady state corresponding to the initial wind and the initial eddy viscosity. For this specific situation, a formula for the evolution of the surface current can be derived explicitly. We show that, if the wind and the eddy viscosity converge toward constant values for large times, under mild assumptions on their convergence rate the solution converges toward the corresponding steady state. The time evolution of the surface current and the surface deflection angle is visualized with the aid of simple numerical plots for some specific examples.
{"title":"The surface current of Ekman flows with time-dependent eddy viscosity","authors":"L. Roberti","doi":"10.3934/cpaa.2022064","DOIUrl":"https://doi.org/10.3934/cpaa.2022064","url":null,"abstract":"In this paper we investigate transients in the oceanic Ekman layer in the presence of time-varying winds and a constant-in-depth but time dependent eddy viscosity, where the initial state is taken to be the steady state corresponding to the initial wind and the initial eddy viscosity. For this specific situation, a formula for the evolution of the surface current can be derived explicitly. We show that, if the wind and the eddy viscosity converge toward constant values for large times, under mild assumptions on their convergence rate the solution converges toward the corresponding steady state. The time evolution of the surface current and the surface deflection angle is visualized with the aid of simple numerical plots for some specific examples.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114952412","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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