Meryem El Attaouy, K. Ezzinbi, Gaston Mandata ˜N'Guerekata
This article establishes a reduction principle for partial functional differential equation without compactness of the semigroup generated by the linear part. Under conditions more general than the compactness of the C0-semigroup generated by the linear part, we establish the quasi-compactness of the C0-semigroup associated to the linear part of the partial functional differential equation. This result allows as to construct a reduced system that is posed by an ordinary differential equation posed in a finite dimensional space. Through this result we study the existence of almost automorphic and almost periodic solutions for partial functional differential equations. For illustration, we study a transport model.
{"title":"Reduction principle for partial functional differential equation without compactness","authors":"Meryem El Attaouy, K. Ezzinbi, Gaston Mandata ˜N'Guerekata","doi":"10.58997/ejde.2023.39","DOIUrl":"https://doi.org/10.58997/ejde.2023.39","url":null,"abstract":"This article establishes a reduction principle for partial functional differential equation without compactness of the semigroup generated by the linear part. Under conditions more general than the compactness of the C0-semigroup generated by the linear part, we establish the quasi-compactness of the C0-semigroup associated to the linear part of the partial functional differential equation. This result allows as to construct a reduced system that is posed by an ordinary differential equation posed in a finite dimensional space. Through this result we study the existence of almost automorphic and almost periodic solutions for partial functional differential equations. For illustration, we study a transport model.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45321327","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}
Let (Omega) be a bounded regular domain of ( mathbb{R}^N), (Ngeqslant 1), (pin (1,+infty)), and ( sin (0,1) ). We consider the eigenvalue problem $$displaylines{ (-Delta_p)^s u + V|u|^{p-2}u= lambda m(x)|u|^{p-2}u quadhbox{in } Omega cr u=0 quad hbox{in } mathbb{R}^N setminus Omega, }$$ where the potential V and the weight m are possibly unbounded and are sign-changing. After establishing the boundedness and regularity of weak solutions, we prove that this problem admits principal eigenvalues under certain conditions. We also show that when such eigenvalues exist, they are simple and isolated in the spectrum of the operator.
{"title":"Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights","authors":"Oumarou Asso, M. Cuesta, J. Doumatè, L. Leadi","doi":"10.58997/ejde.2023.38","DOIUrl":"https://doi.org/10.58997/ejde.2023.38","url":null,"abstract":"Let (Omega) be a bounded regular domain of ( mathbb{R}^N), (Ngeqslant 1), (pin (1,+infty)), and ( sin (0,1) ). We consider the eigenvalue problem $$displaylines{ (-Delta_p)^s u + V|u|^{p-2}u= lambda m(x)|u|^{p-2}u quadhbox{in } Omega cr u=0 quad hbox{in } mathbb{R}^N setminus Omega, }$$ where the potential V and the weight m are possibly unbounded and are sign-changing. After establishing the boundedness and regularity of weak solutions, we prove that this problem admits principal eigenvalues under certain conditions. We also show that when such eigenvalues exist, they are simple and isolated in the spectrum of the operator.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47017322","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 prove the existence of solutions to the integrodifferential equation of mixed type begin{gather*}x^Delta (t)=f Big( t,x(t), int_0^t k_1 (t,s)g(s,x(s)) Delta s, int_0^a k_2(t,s)h(s,x(s)) Delta s Big),cr x(0)=x_0, quad x_0 in E,; t in I_a=[0,a] cap mathbb{T},; a>0, end{gather*} where (mathbb{T}) denotes a time scale (nonempty closed subset of real numbers (mathbb{R})), (I_a) is a time scale interval. In the first part of this paper functions (f,g,h) are Caratheodory functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil delta integrals, which generalizes the Henstock-Kurzweil integrals. In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis delta integrals. Additionally, functions f, g, h satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness.
本文证明了混合型begin｛collecte*｝x^Delta（t）=fBig（t，x（t），int_0^tk_1（t，s）g（s，x（s））Delta s，int:0^a_2（t、s）h（s，x（s），Delta sBig），cr x（0）=x_0，quad x_0在E，中解的存在性；t in I_a=[0，a]capmathbb｛t｝，；a> 0，end｛collecte*｝，其中（mathbb｛T｝）表示时间尺度（实数的非空闭子集（math bb｛R｝）），（I_a）是时间尺度间隔。在本文的第一部分中，函数（f，g，h）是Banach空间E中具有值的Caratheodory函数，并且积分是在Henstock-Korzweil-delta积分的意义上取的，它推广了Henstock-Kurzweil积分。在第二部分中，f，g，h，x是弱弱序连续函数，积分取Henstock-Kurzweil-Pettis-delta积分的意义。此外，函数f，g，h满足一些边界条件和用非紧测度表示的条件。
{"title":"Integrodifferential equations of mixed type on time scales with Delta-HK and Delta-HKP integrals","authors":"A. Sikorska-Nowak","doi":"10.58997/ejde.2023.29","DOIUrl":"https://doi.org/10.58997/ejde.2023.29","url":null,"abstract":"In this article we prove the existence of solutions to the integrodifferential equation of mixed type begin{gather*}x^Delta (t)=f Big( t,x(t), int_0^t k_1 (t,s)g(s,x(s)) Delta s, int_0^a k_2(t,s)h(s,x(s)) Delta s Big),cr x(0)=x_0, quad x_0 in E,; t in I_a=[0,a] cap mathbb{T},; a>0, end{gather*} where (mathbb{T}) denotes a time scale (nonempty closed subset of real numbers (mathbb{R})), (I_a) is a time scale interval. In the first part of this paper functions (f,g,h) are Caratheodory functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil delta integrals, which generalizes the Henstock-Kurzweil integrals. In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis delta integrals. Additionally, functions f, g, h satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46715677","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 structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis.
{"title":"Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis","authors":"Zhihong Zhao, Huanqin Hu","doi":"10.58997/ejde.2023.37","DOIUrl":"https://doi.org/10.58997/ejde.2023.37","url":null,"abstract":"This article concerns the structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47270899","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}
Jason Clark, Oleksandr Misiats, V. Mogylova, Oleksandr Stanzhytskyi
In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.
{"title":"Asymptotic behavior of stochastic functional differential evolution equation","authors":"Jason Clark, Oleksandr Misiats, V. Mogylova, Oleksandr Stanzhytskyi","doi":"10.58997/ejde.2023.35","DOIUrl":"https://doi.org/10.58997/ejde.2023.35","url":null,"abstract":"In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42465534","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 introduce the concept of p-mean θ-pseudo almost periodic stochastic processes, which is slightly weaker than p-mean pseudo almost periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of square-mean θ-pseudo almost periodic mild solutions for a semilinear stochastic differential equation in infinite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in square-mean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic differential equations (SDEs) in finite dimensions.
{"title":"Pseudo almost periodicity for stochastic differential equations in infinite dimensions","authors":"Ye-Jun Chen, H. Ding","doi":"10.58997/ejde.2023.34","DOIUrl":"https://doi.org/10.58997/ejde.2023.34","url":null,"abstract":"In this article, we introduce the concept of p-mean θ-pseudo almost periodic stochastic processes, which is slightly weaker than p-mean pseudo almost periodic stochastic processes. Using the operator semigroup theory and stochastic analysis theory, we obtain the existence and uniqueness of square-mean θ-pseudo almost periodic mild solutions for a semilinear stochastic differential equation in infinite dimensions. Moreover, we prove that the obtained solution is also pseudo almost periodic in path distribution. It is noteworthy that the ergodic part of the obtained solution is not only ergodic in square-mean but also ergodic in path distribution. Our main results are even new for the corresponding stochastic differential equations (SDEs) in finite dimensions.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44699148","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 investigate the asymptotic behavior at infinity of viscosity solutions to degenerate elliptic equations. We obtain Holder estimates, up to the flat boundary, by using the rescaling method. Also as a byproduct we obtain a Liouville type result on Baouendi-Grushin type operators.
{"title":"Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space","authors":"Xiaobiao Jia, Shan Ma","doi":"10.58997/ejde.2023.33","DOIUrl":"https://doi.org/10.58997/ejde.2023.33","url":null,"abstract":"In this article we investigate the asymptotic behavior at infinity of viscosity solutions to degenerate elliptic equations. We obtain Holder estimates, up to the flat boundary, by using the rescaling method. Also as a byproduct we obtain a Liouville type result on Baouendi-Grushin type operators.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45497757","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 focuses on the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equations in Rn with n≥3. By constructing a couple of proper supersolution and subsolution, we prove the existence of periodic pyramidal traveling front in R3 and then in Rn with n>3.
{"title":"Pyramidal traveling fronts of a time periodic diffusion equation with degenerate monostable nonlinearity","authors":"Z. Bu, Chen-Lu Wang, Xin-Tian Zhang","doi":"10.58997/ejde.2023.31","DOIUrl":"https://doi.org/10.58997/ejde.2023.31","url":null,"abstract":"This article focuses on the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equations in Rn with n≥3. By constructing a couple of proper supersolution and subsolution, we prove the existence of periodic pyramidal traveling front in R3 and then in Rn with n>3.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43121625","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 consider a Dirichlet problem driven by a weighted (p,2)-Laplacian with a reaction which is resonant both at (pminfty) and at zero (double resonance). We prove a multiplicity theorem producing three nontrivial smooth solutions with sign information and ordered. In the appendix we develop the spectral properties of the weighted r-Laplace differential operator.
{"title":"A weighted (p,2)-equation with double resonance","authors":"Zhenhai Liu, Nikolaos S. Papageorgiou","doi":"10.58997/ejde.2023.30","DOIUrl":"https://doi.org/10.58997/ejde.2023.30","url":null,"abstract":"We consider a Dirichlet problem driven by a weighted (p,2)-Laplacian with a reaction which is resonant both at (pminfty) and at zero (double resonance). We prove a multiplicity theorem producing three nontrivial smooth solutions with sign information and ordered. In the appendix we develop the spectral properties of the weighted r-Laplace differential operator.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42095355","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 analyze a partial differential equation that models the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a diffusion term corresponding to the random movement of particles, and a nonlocal advection term corresponding to the movement of particles in the transverse directionperpendicular to their forward movement. The model is compared to the Schrodinger equation model of the experiment. The model supports the ensemble interpretation of quantum mechanics.
See also https://ejde.math.txstate.edu/special/02/w1/abstr.html
{"title":"Nonlocal advection diffusion equations and the two-slit experiment in quantum mechanics","authors":"Glenn Webb","doi":"10.58997/ejde.sp.02.w1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.w1","url":null,"abstract":"We analyze a partial differential equation that models the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a diffusion term corresponding to the random movement of particles, and a nonlocal advection term corresponding to the movement of particles in the transverse directionperpendicular to their forward movement. The model is compared to the Schrodinger equation model of the experiment. The model supports the ensemble interpretation of quantum mechanics.
 See also https://ejde.math.txstate.edu/special/02/w1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135891565","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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