In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the bou...In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the boundary value is . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation in. We propose two different conditions for the right hand side and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.展开更多
We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all sma...We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.展开更多
Mimicking vascular systems in living beings,designers have realized microvascular composites to achieve thermal regulation and other functionalities,such as electromagnetic modulation,sensing,and healing.Such material...Mimicking vascular systems in living beings,designers have realized microvascular composites to achieve thermal regulation and other functionalities,such as electromagnetic modulation,sensing,and healing.Such material systems avail circulating fluids through embedded vasculatures to accomplish the mentioned functionalities that benefit various aerospace,military,and civilian applications.Although heat transfer is a mature field,control of thermal characteristics in synthetic microvascular systems via circulating fluids is new,and a theoretical underpinning is lacking.What will benefit designers are predictive mathematical models and an in-depth qualitative understanding of vascular-based active cooling/heating.So,the central focus of this paper is to address the remarked knowledge gap.First,we present a reduced-order model with broad applicability,allowing the inlet temperature to differ from the ambient temperature.Second,we apply mathematical analysis tools to this reduced-order model and reveal many heat transfer properties of fluid-sequestered vascular systems.We derive point-wise properties(minimum,maximum,and comparison principles)and global properties(e.g.,bounds on performance metrics such as the mean surface temperature and thermal efficiency).These newfound results deepen our understanding of active cooling/heating and propel the perfecting of thermal regulation systems.展开更多
This paper is devoted to investigating the selection mechanism of the minimal wave speed for traveling waves to an epidemic model.The determinacy of linear and nonlinear selections is further discussed by the upper-lo...This paper is devoted to investigating the selection mechanism of the minimal wave speed for traveling waves to an epidemic model.The determinacy of linear and nonlinear selections is further discussed by the upper-lower solutions and comparison principle.A threshold is defined by the eigenvalue problem of the linearized system.We show that the nonlinear determinacy is obtained as long as there exists a lower solution with a faster decay and a speed parameter that is larger than the threshold.When the speed parameter equals to the threshold,if there exists an upper solution satisfying proper limit behavior,then the linear selection is realized.For a special function of infection rate,we obtain a threshold parameter that determines the linear and nonlinear selections.展开更多
This paper concerns a class of impulsive discrete systems and offers sufficient conditions of boundedness in terms of two measures for such systems, by employing vector Lyapunov functions and a new comparison principl...This paper concerns a class of impulsive discrete systems and offers sufficient conditions of boundedness in terms of two measures for such systems, by employing vector Lyapunov functions and a new comparison principle. The result obtained generalizes those in known literatures.展开更多
This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks.As a continuity of the past work(Wu and Niu,2016;Yu,et al.,2011)on the existence and uniquen...This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks.As a continuity of the past work(Wu and Niu,2016;Yu,et al.,2011)on the existence and uniqueness of the traveling wave solutions,it is very reasonable and interesting to consider the exponential stability of the traveling wave solutions.By the weighted energy method,comparison principle and the first integral mean value theorem,this paper proves that,for all monotone traveling waves with the wave speed c<c1*<0 or c>c2*>0,the solutions converge time-exponentially to the corresponding traveling waves,when the initial perturbations decay at some fields.展开更多
This paper is concerned with stability of traveling wave fronts for nonlocal diffusive system.We adopt L^(1),-weighted,L^(1)-and L^(2)-energy estimates for the perturbation systems,and show that all solutions of...This paper is concerned with stability of traveling wave fronts for nonlocal diffusive system.We adopt L^(1),-weighted,L^(1)-and L^(2)-energy estimates for the perturbation systems,and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev spaces.展开更多
In this short note we consider reflected backward stochastic differential equations(RBSDEs)with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous.In this case,the barrier is rep...In this short note we consider reflected backward stochastic differential equations(RBSDEs)with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous.In this case,the barrier is represented as a nondecreasing limit of right continuous with left limit(RCLL)barriers.We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions.Finally,we highlight the connection of these RBSDEs with standard RCLL BSDEs.展开更多
We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear cas...We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.展开更多
In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy conditi...In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy condition needed in our earlier works(Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:1212–1253,2016a;Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:2507–2553,2016b)to establish the uniqueness result.We establish the comparison principle under natural and mild conditions.Moreover,we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations,induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem,respectively.展开更多
文摘In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the boundary value is . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation in. We propose two different conditions for the right hand side and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.
基金NNSF of China(Grant No.12071413)NSF of Guangxi(Grant No.2023GXNSFAA026085)the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No.823731 CONMECH。
文摘We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.
文摘Mimicking vascular systems in living beings,designers have realized microvascular composites to achieve thermal regulation and other functionalities,such as electromagnetic modulation,sensing,and healing.Such material systems avail circulating fluids through embedded vasculatures to accomplish the mentioned functionalities that benefit various aerospace,military,and civilian applications.Although heat transfer is a mature field,control of thermal characteristics in synthetic microvascular systems via circulating fluids is new,and a theoretical underpinning is lacking.What will benefit designers are predictive mathematical models and an in-depth qualitative understanding of vascular-based active cooling/heating.So,the central focus of this paper is to address the remarked knowledge gap.First,we present a reduced-order model with broad applicability,allowing the inlet temperature to differ from the ambient temperature.Second,we apply mathematical analysis tools to this reduced-order model and reveal many heat transfer properties of fluid-sequestered vascular systems.We derive point-wise properties(minimum,maximum,and comparison principles)and global properties(e.g.,bounds on performance metrics such as the mean surface temperature and thermal efficiency).These newfound results deepen our understanding of active cooling/heating and propel the perfecting of thermal regulation systems.
基金supported by NSF of China (No.11971213)Natural Science Foundation of Gansu Province of China (No.21JR7RA535).
文摘This paper is devoted to investigating the selection mechanism of the minimal wave speed for traveling waves to an epidemic model.The determinacy of linear and nonlinear selections is further discussed by the upper-lower solutions and comparison principle.A threshold is defined by the eigenvalue problem of the linearized system.We show that the nonlinear determinacy is obtained as long as there exists a lower solution with a faster decay and a speed parameter that is larger than the threshold.When the speed parameter equals to the threshold,if there exists an upper solution satisfying proper limit behavior,then the linear selection is realized.For a special function of infection rate,we obtain a threshold parameter that determines the linear and nonlinear selections.
基金Supported by the Project-sponsored by SRF for ROCS, SEM of China(48371109) the Natural Science Foundation of Hebei Province (A2006000941).
文摘This paper concerns a class of impulsive discrete systems and offers sufficient conditions of boundedness in terms of two measures for such systems, by employing vector Lyapunov functions and a new comparison principle. The result obtained generalizes those in known literatures.
基金supported by the Natural Science Foundation of Shandong Province under Grant No.ZR2017MA045the National Natural Science Foundation of China under Grant No.61873144。
文摘This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks.As a continuity of the past work(Wu and Niu,2016;Yu,et al.,2011)on the existence and uniqueness of the traveling wave solutions,it is very reasonable and interesting to consider the exponential stability of the traveling wave solutions.By the weighted energy method,comparison principle and the first integral mean value theorem,this paper proves that,for all monotone traveling waves with the wave speed c<c1*<0 or c>c2*>0,the solutions converge time-exponentially to the corresponding traveling waves,when the initial perturbations decay at some fields.
基金supported by the China Postdoctoral Science Foundation(No.2020M670963)supported by the Natural Science Foundation of China(No.12071297)the Natural Science Foundation of Shanghai(No.18ZR1426500).
文摘This paper is concerned with stability of traveling wave fronts for nonlocal diffusive system.We adopt L^(1),-weighted,L^(1)-and L^(2)-energy estimates for the perturbation systems,and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev spaces.
文摘In this short note we consider reflected backward stochastic differential equations(RBSDEs)with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous.In this case,the barrier is represented as a nondecreasing limit of right continuous with left limit(RCLL)barriers.We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions.Finally,we highlight the connection of these RBSDEs with standard RCLL BSDEs.
文摘We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.
基金Research supported in part by NSF grant DMS 1413717。
文摘In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy condition needed in our earlier works(Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:1212–1253,2016a;Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:2507–2553,2016b)to establish the uniqueness result.We establish the comparison principle under natural and mild conditions.Moreover,we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations,induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem,respectively.
