De Bruijn's identity in information theory states that if u is the solution of the heat equation, then the time derivative of the Shannon entropy for this solution is equal to the amount of Fisher information at u. In this article, we show how this identity changes if we replace the heat channel by the Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck channels. Through these passages we investigate the different properties of these solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck semigroup given by the Mehler formula expression.
{"title":"De Bruijn identities in different Markovian channels","authors":"H. Emamirad, A. Rougirel","doi":"10.58997/ejde.2023.12","DOIUrl":"https://doi.org/10.58997/ejde.2023.12","url":null,"abstract":"De Bruijn's identity in information theory states that if u is the solution of the heat equation, then the time derivative of the Shannon entropy for this solution is equal to the amount of Fisher information at u. In this article, we show how this identity changes if we replace the heat channel by the Fokker Planck, or passing from Fokker Planck to Ornstein-Uhlenbeck channels. Through these passages we investigate the different properties of these solutions. We exclusively dissect different properties of Ornstein-Uhlenbeck semigroup given by the Mehler formula expression.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43537271","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 article we study the smoothness properties of solutions to a two-dimensional coupled Zakharov-Kuznetsov system. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u0,v0) possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t),v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution.
{"title":"Smoothing properties for a coupled Zakharov-Kuznetsov system","authors":"Julie L. Levandosky, O. Vera","doi":"10.58997/ejde.2023.11","DOIUrl":"https://doi.org/10.58997/ejde.2023.11","url":null,"abstract":"In this article we study the smoothness properties of solutions to a two-dimensional coupled Zakharov-Kuznetsov system. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u0,v0) possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t),v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49416182","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 article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, [displaylines{(-Delta_p)^su-mu frac{|u|^{p-2}u}{|x|^{sp}}=lambda u^{-alpha}+frac{|u|^{p_s^*(t)-2}u}{|x|^t}, quadhbox{in }Omega, u>0,quadtext{in }Omega, quad u=0, quadtext{in } mathbb{R}^N setminusOmega }] where (Omega subset mathbb{R}^N) is a bounded domain with Lipschitz boundary and( (-Delta_p)^s) is the fractional p-Laplacian operator.We combine some variational techniques with a perturbation method to show the existenceof multiple solutions.
{"title":"Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent","authors":"A. Daoues, A. Hammami, K. Saoudi","doi":"10.58997/ejde.2023.10","DOIUrl":"https://doi.org/10.58997/ejde.2023.10","url":null,"abstract":" In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, [displaylines{(-Delta_p)^su-mu frac{|u|^{p-2}u}{|x|^{sp}}=lambda u^{-alpha}+frac{|u|^{p_s^*(t)-2}u}{|x|^t}, quadhbox{in }Omega, u>0,quadtext{in }Omega, quad u=0, quadtext{in } mathbb{R}^N setminusOmega }] where (Omega subset mathbb{R}^N) is a bounded domain with Lipschitz boundary and( (-Delta_p)^s) is the fractional p-Laplacian operator.We combine some variational techniques with a perturbation method to show the existenceof multiple solutions.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45918360","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 transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.
{"title":"Paradigm for the creation of scales and phases in nonlinear evolution equations","authors":"C. Cheverry, Shahnaz Farhat","doi":"10.58997/ejde.2023.09","DOIUrl":"https://doi.org/10.58997/ejde.2023.09","url":null,"abstract":"The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43936958","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 propose a cholera model with coupled reaction-diffusion equations and ordinary differential equations for discussing the effects of spatial heterogeneity, horizontal transmission, environmental viruses and phages on the spread of vibrio cholerae. We establish the well-posedness of this model which includes the existence of unique global positive solution, asymptotic smoothness of semiflow, and existence of a global attractor. The basic reproduction number R0 is obtained to describe the persistence and extinction of the disease. That is, the disease-free steady state is globally asymptotically stable for R0≤1, while it is unstable for R0>1. And, the disease is persistence and the model has the phage-free and phage-present endemic steady states in this case. Further, the global asymptotic stability of phage-free and phage-present endemic steady states are discussed for spatially homogeneous model. Finally, some numerical examples are displayed in order to illustrate the main theoretical results and our opening questions.
{"title":"Dynamics of a partially degenerate reaction-diffusion cholera model with horizontal transmission and phage-bacteria interaction","authors":"Zhenxiang Hu, Shengfu Wang, L. Nie","doi":"10.58997/ejde.2023.08","DOIUrl":"https://doi.org/10.58997/ejde.2023.08","url":null,"abstract":"We propose a cholera model with coupled reaction-diffusion equations and ordinary differential equations for discussing the effects of spatial heterogeneity, horizontal transmission, environmental viruses and phages on the spread of vibrio cholerae. We establish the well-posedness of this model which includes the existence of unique global positive solution, asymptotic smoothness of semiflow, and existence of a global attractor. The basic reproduction number R0 is obtained to describe the persistence and extinction of the disease. That is, the disease-free steady state is globally asymptotically stable for R0≤1, while it is unstable for R0>1. And, the disease is persistence and the model has the phage-free and phage-present endemic steady states in this case. Further, the global asymptotic stability of phage-free and phage-present endemic steady states are discussed for spatially homogeneous model. Finally, some numerical examples are displayed in order to illustrate the main theoretical results and our opening questions.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41481350","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 article, we study the existence of the integral solution to the neutral functional differential inclusion $${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t), quadtext{for a.e. }t in J:=[0,infty), y_0=phi in C_E=C([-r,0];E),quad r>0,}$$ and the controllability of the corresponding neutral inclusion $${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t)+Bu(t),quad text{for a.e. } t in J, y_0=phi in C_E,}$$ on a half-line via the nonlinear alternative of Leray-Schauder type for contractive multivalued mappings given by Frigon. We illustrate our results with applications to a neutral partial differential inclusion with diffusion, and to a neutral functional partial differential equation with obstacle constrains.
本文利用Frigon给出的压缩多值映射的Leray-Schauder型的非线性替代，研究中立型泛函微分包体$${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t), quadtext{for a.e. }t in J:=[0,infty), y_0=phi in C_E=C([-r,0];E),quad r>0,}$$的积分解的存在性和相应中立型包体$${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t)+Bu(t),quad text{for a.e. } t in J, y_0=phi in C_E,}$$在半线上的可控性。我们将结果应用于具有扩散的中立型偏微分包含，以及具有障碍约束的中立型泛函偏微分方程。
{"title":"Existence and controllability for neutral partial differential inclusions nondenselly defined on a half-line","authors":"Nguyen Thi Van Anh, Bui Thi Hai Yen","doi":"10.58997/ejde.2023.07","DOIUrl":"https://doi.org/10.58997/ejde.2023.07","url":null,"abstract":"In this article, we study the existence of the integral solution to the neutral functional differential inclusion\u0000$${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t), quadtext{for a.e. }t in J:=[0,infty), y_0=phi in C_E=C([-r,0];E),quad r>0,}$$\u0000and the controllability of the corresponding neutral inclusion\u0000$${frac{d}{dt}mathcal{D}y_t-Amathcal{D}y_t-Ly_t in F(t,y_t)+Bu(t),quad text{for a.e. } t in J, y_0=phi in C_E,}$$\u0000on a half-line via the nonlinear alternative of Leray-Schauder type for contractive multivalued mappings given by Frigon. We illustrate our results with applications to a neutral partial differential inclusion with diffusion, and to a neutral functional partial differential equation with obstacle constrains.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47573178","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 article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s
{"title":"Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations","authors":"Pascal Remy","doi":"10.58997/ejde.2023.06","DOIUrl":"https://doi.org/10.58997/ejde.2023.06","url":null,"abstract":"In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47417720","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 the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.
{"title":"Linear higher-order fractional differential and integral equations","authors":"K. Lan","doi":"10.58997/ejde.2023.01","DOIUrl":"https://doi.org/10.58997/ejde.2023.01","url":null,"abstract":"We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L1(a,b) and C[a,b] when n≥ 2. We establish the equivalence in C[a,b] of the IVP of the nth order FDE subject to the initial condition u(i)(a)=ui for i in {0,1,...,n-1} when n≥2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n≥2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n≥2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47753244","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 article concerns the internal stabilization problem of 1-D interconnected heat-wave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribution and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable.
{"title":"Internal stabilization of interconnected heat-wave equations","authors":"Xiulan Yu, Jun‐min Wang, Han-Wen Zhang","doi":"10.58997/ejde.2023.03","DOIUrl":"https://doi.org/10.58997/ejde.2023.03","url":null,"abstract":"This article concerns the internal stabilization problem of 1-D interconnected heat-wave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribution and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71218463","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 reaction-diffusion problem formulated with a higher-order operator, a non-linear advection, and a Fisher-KPP reaction term depending on the spatial variable. The higher-order operator induces solutions to oscillate in the proximity of an equilibrium condition. Given this oscillatory character, solutions are studied in a set of bounded domains. We introduce a new extension operator, that allows us to study the solutions in the open domain RN, but departing from a sequence of bounded domains. The analysis about regularity of solutions is built based on semigroup theory. In this approach, the solutions are interpreted as an abstract evolution given by a bounded continuous operator. Afterward, asymptotic profiles of solutions are studied based on a Hamilton-Jacobi equation that is obtained with a single point exponential scaling. Finally, a numerical assessment, with the function bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis.
{"title":"Semigroup theory and asymptotic profiles of solutions for a higher-order Fisher-KPP problem in R^N","authors":"José Luis Díaz Palencia","doi":"10.58997/ejde.2023.04","DOIUrl":"https://doi.org/10.58997/ejde.2023.04","url":null,"abstract":"We study a reaction-diffusion problem formulated with a higher-order operator, a non-linear advection, and a Fisher-KPP reaction term depending on the spatial variable. The higher-order operator induces solutions to oscillate in the proximity of an equilibrium condition. Given this oscillatory character, solutions are studied in a set of bounded domains. We introduce a new extension operator, that allows us to study the solutions in the open domain RN, but departing from a sequence of bounded domains. The analysis about regularity of solutions is built based on semigroup theory. In this approach, the solutions are interpreted as an abstract evolution given by a bounded continuous operator. Afterward, asymptotic profiles of solutions are studied based on a Hamilton-Jacobi equation that is obtained with a single point exponential scaling. Finally, a numerical assessment, with the function bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71218513","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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