{"title":"Normalized solutions for Kirchhoff type equations with combined nonlinearities: The Sobolev critical case","authors":"Xiaojing Feng, Haidong Liu, Zhitao Zhang","doi":"10.3934/dcds.2023035","DOIUrl":"https://doi.org/10.3934/dcds.2023035","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84313208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, $ begin{equation} begin{cases} u_t = Delta u-chi_1 nablacdot (frac{u}{w} nabla w)+u(a_1-b_1u-c_1v) , quad &xin Omegacr v_t = Delta v-chi_2 nablacdot (frac{v}{w} nabla w)+v(a_2-b_2v-c_2u), quad &xin Omegacr 0 = Delta w-mu w +nu u+ lambda v, quad &xin Omega cr frac{partial u}{partial n} = frac{partial v}{partial n} = frac{partial w}{partial n} = 0, quad &xinpartialOmega, end{cases} end{equation}~~~~(1) $ where $ Omega subset mathbb{R}^N $ is a bounded smooth domain, and $ chi_i $, $ a_i $, $ b_i $, $ c_i $ ($ i = 1, 2 $) and $ mu, , nu, , lambda $ are positive constants. In [25], among others, we proved that for any given nonnegative initial data $ u_0, v_0in C^0(barOmega) $ with $ u_0+v_0not equiv 0 $, (1) has a unique globally defined classical solution $ (u(t, x;u_0, v_0), v(t, x;u_0, v_0), w(t, x;u_0, v_0)) $ with $ u(0, x;u_0, v_0) = u_0(x) $ and $ v(0, x;u_0, v_0) = v_0(x) $ in any space dimensional setting with any positive constants $ chi_i, a_i, b_i, c_i $ ($ i = 1, 2 $) and $ mu, nu, lambda $. In this paper, we assume that the competition in (1) is weak in the sense that $ frac{c_1}{b_2}
{"title":"Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics","authors":"Halil ibrahim Kurt, Wenxian Shen","doi":"10.3934/dcds.2023130","DOIUrl":"https://doi.org/10.3934/dcds.2023130","url":null,"abstract":"The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, $ begin{equation} begin{cases} u_t = Delta u-chi_1 nablacdot (frac{u}{w} nabla w)+u(a_1-b_1u-c_1v) , quad &xin Omegacr v_t = Delta v-chi_2 nablacdot (frac{v}{w} nabla w)+v(a_2-b_2v-c_2u), quad &xin Omegacr 0 = Delta w-mu w +nu u+ lambda v, quad &xin Omega cr frac{partial u}{partial n} = frac{partial v}{partial n} = frac{partial w}{partial n} = 0, quad &xinpartialOmega, end{cases} end{equation}~~~~(1) $ where $ Omega subset mathbb{R}^N $ is a bounded smooth domain, and $ chi_i $, $ a_i $, $ b_i $, $ c_i $ ($ i = 1, 2 $) and $ mu, , nu, , lambda $ are positive constants. In [25], among others, we proved that for any given nonnegative initial data $ u_0, v_0in C^0(barOmega) $ with $ u_0+v_0not equiv 0 $, (1) has a unique globally defined classical solution $ (u(t, x;u_0, v_0), v(t, x;u_0, v_0), w(t, x;u_0, v_0)) $ with $ u(0, x;u_0, v_0) = u_0(x) $ and $ v(0, x;u_0, v_0) = v_0(x) $ in any space dimensional setting with any positive constants $ chi_i, a_i, b_i, c_i $ ($ i = 1, 2 $) and $ mu, nu, lambda $. In this paper, we assume that the competition in (1) is weak in the sense that $ frac{c_1}{b_2}<frac{a_1}{a_2}, quad frac{c_2}{b_1}<frac{a_2}{a_1}. $ Then (1) has a unique positive constant solution $ (u^*, v^*, w^*) $, where $ u^* = frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2}, quad v^* = frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, quad w^* = frac{nu}{mu}u^*+frac{lambda}{mu} v^*. $ We obtain some explicit conditions on $ chi_1, chi_2 $ which ensure that the positive constant solution $ (u^*, v^*, w^*) $ is globally stable, that is, for any given nonnegative initial data $ u_0, v_0in C^0(barOmega) $ with $ u_0not equiv 0 $ and $ v_0not equiv 0 $, $ limlimits_{ttoinfty}Big(|u(t, cdot;u_0, v_0)-u^*|_infty +|v(t, cdot;u_0, v_0)-v^*|_infty+|w(t, cdot;u_0, v_0)-w^*|_inftyBig) = 0. $","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"135319721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $ F: mathbb{R}^nrightarrow [0, , infty) $ be a convex function of class $ C^2(mathbb{R}^nbackslash{0}) $, which is even and positively homogeneous of degree $ 1 $. In this paper, we prove that$ suplimits_{uin W^{1, n}(mathbb{R}^n), , displaystyle{int}_{mathbb{R}^n}(F^n(nabla u)+|u|^n)dxleq1}displaystyle{int}_{mathbb{R}^n}frac{Phi(lambda_{n}(1-frac{beta}{n})(1+alpha|u|^{n}_n)^{frac{1}{n-1}}|u|^{frac{n}{n-1}})}{F^o(x)^beta}dx $is finite for $ 0leqalpha<1 $, and the supremum is infinity for $ alphageq1 $, where $ F^o(x) $ is the polar function of $ F $, $ Phi(t) = e^t-sum_{j = 0}^{n-2}frac{t^j}{j!} $, $ betain[0, n) $, $ lambda_n = n^{frac{n}{n-1}}kappa_n^{frac{1}{n-1}} $ and $ kappa_n $ is the volume of the unit Wulff ball. Moreover, by using the method of blow-up analysis, we also obtain the existence of extremal functions for the supremum when $ 0leqalpha<1 $.
{"title":"Anisotropic singular Trudinger-Moser inequalities on the whole Euclidean space","authors":"Xiaomeng Li","doi":"10.3934/dcds.2023111","DOIUrl":"https://doi.org/10.3934/dcds.2023111","url":null,"abstract":"Let $ F: mathbb{R}^nrightarrow [0, , infty) $ be a convex function of class $ C^2(mathbb{R}^nbackslash{0}) $, which is even and positively homogeneous of degree $ 1 $. In this paper, we prove that$ suplimits_{uin W^{1, n}(mathbb{R}^n), , displaystyle{int}_{mathbb{R}^n}(F^n(nabla u)+|u|^n)dxleq1}displaystyle{int}_{mathbb{R}^n}frac{Phi(lambda_{n}(1-frac{beta}{n})(1+alpha|u|^{n}_n)^{frac{1}{n-1}}|u|^{frac{n}{n-1}})}{F^o(x)^beta}dx $is finite for $ 0leqalpha<1 $, and the supremum is infinity for $ alphageq1 $, where $ F^o(x) $ is the polar function of $ F $, $ Phi(t) = e^t-sum_{j = 0}^{n-2}frac{t^j}{j!} $, $ betain[0, n) $, $ lambda_n = n^{frac{n}{n-1}}kappa_n^{frac{1}{n-1}} $ and $ kappa_n $ is the volume of the unit Wulff ball. Moreover, by using the method of blow-up analysis, we also obtain the existence of extremal functions for the supremum when $ 0leqalpha<1 $.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"136257651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers a special class of quasi-periodic systems with a small parameter, whose unperturbed part has a degenerate equilibrium point. We prove the existence of response solutions for many sufficiently small parameters. The proof is based on some formal KAM techniques and the Leray-Schauder Continuation Theorem.
{"title":"Response solutions of a class of degenerate quasi-periodic systems with a small parameter","authors":"Xiaomei Yang, Junxiang Xu","doi":"10.3934/dcds.2023114","DOIUrl":"https://doi.org/10.3934/dcds.2023114","url":null,"abstract":"This paper considers a special class of quasi-periodic systems with a small parameter, whose unperturbed part has a degenerate equilibrium point. We prove the existence of response solutions for many sufficiently small parameters. The proof is based on some formal KAM techniques and the Leray-Schauder Continuation Theorem.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"136306278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper extends the generalized Wiener–Wintner Theorem built by Host and Kra to the multilinear case under the hypothesis of pointwise convergence of multilinear ergodic averages. In particular, we have the following result:Let $ (X, {mathcal B}, mu, T) $ be a measure preserving system. Let $ a $ and $ b $ be two distinct non-zero integers. Then for any $ f_{1}, f_{2}in L^{infty}(mu) $, there exists a full measure subset $ X(f_{1}, f_{2}) $ of $ X $ such that for any $ xin X(f_{1}, f_{2}) $, and any nilsequence $ {textbf b} = {b_n}_{nin {mathbb Z}} $,$ limlimits_{Nrightarrow infty}frac{1}{N}sumlimits_{n = 0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x) $exists.
{"title":"Multilinear Wiener-Wintner type ergodic averages and its application","authors":"Rongzhong Xiao","doi":"10.3934/dcds.2023109","DOIUrl":"https://doi.org/10.3934/dcds.2023109","url":null,"abstract":"This paper extends the generalized Wiener–Wintner Theorem built by Host and Kra to the multilinear case under the hypothesis of pointwise convergence of multilinear ergodic averages. In particular, we have the following result:Let $ (X, {mathcal B}, mu, T) $ be a measure preserving system. Let $ a $ and $ b $ be two distinct non-zero integers. Then for any $ f_{1}, f_{2}in L^{infty}(mu) $, there exists a full measure subset $ X(f_{1}, f_{2}) $ of $ X $ such that for any $ xin X(f_{1}, f_{2}) $, and any nilsequence $ {textbf b} = {b_n}_{nin {mathbb Z}} $,$ limlimits_{Nrightarrow infty}frac{1}{N}sumlimits_{n = 0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x) $exists.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"135700495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 consider the long time well-posedness of solutions to two-dimensional compressible magnetohydrodynamic (MHD) boundary layer equations. When the initial data is a small perturbation of a steady solution with size of $ varepsilon $, then the lifespan of solutions in Sobolev spaces is proved to be greater than $ varepsilon^{-frac43} $. And such a result can be extended to the case that both initial data and far-field state are small perturbations around the steady states. Moreover, it holds true for both isentropic and non-isentropic magnetohydrodynamic boundary layer equations.
{"title":"Long time well-posedness of compressible magnetohydrodynamic boundary layer equations in Sobolev spaces","authors":"Shengxin Li, Feng Xie","doi":"10.3934/dcds.2023133","DOIUrl":"https://doi.org/10.3934/dcds.2023133","url":null,"abstract":"In this paper we consider the long time well-posedness of solutions to two-dimensional compressible magnetohydrodynamic (MHD) boundary layer equations. When the initial data is a small perturbation of a steady solution with size of $ varepsilon $, then the lifespan of solutions in Sobolev spaces is proved to be greater than $ varepsilon^{-frac43} $. And such a result can be extended to the case that both initial data and far-field state are small perturbations around the steady states. Moreover, it holds true for both isentropic and non-isentropic magnetohydrodynamic boundary layer equations.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"135560497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space $ mathbb{R}^{3}. $ We prove that the corresponding weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a smallness condition in the critical case.
{"title":"Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin condition","authors":"Debayan Maity, Takéo Takahashi","doi":"10.3934/dcds.2023123","DOIUrl":"https://doi.org/10.3934/dcds.2023123","url":null,"abstract":"In this article, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space $ mathbb{R}^{3}. $ We prove that the corresponding weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a smallness condition in the critical case.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","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":"135213132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Evolutionary bifurcation diagrams of A $ P $-Laplacian generalized logistic problem with nonnegative constant yield harvesting","authors":"Kuo-Chih Hung, Shin-Hwa Wang, Jhih-Jyun Zeng","doi":"10.3934/dcds.2023019","DOIUrl":"https://doi.org/10.3934/dcds.2023019","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73526469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bistable pulsating wave of a competition model in rapidly varying media and its homogenization limit","authors":"Weiwei Ding, Rui Huang, Xiao Yu","doi":"10.3934/dcds.2023012","DOIUrl":"https://doi.org/10.3934/dcds.2023012","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77959575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multi-peak solutions for logarithmic Schrödinger equations with potentials unbounded below","authors":"Xiaoming An, Xian-lin Yang","doi":"10.3934/dcds.2023073","DOIUrl":"https://doi.org/10.3934/dcds.2023073","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77522009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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