Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for...Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for corresponding semigroup. Moreover, a Dresher＇s type inequality for two-parameter family of means, is also proved.展开更多
The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capa...The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in W pl,γ（; E0, E） is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.展开更多
For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigro...For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.展开更多
This paper is concerned first with the behaviour of differences T（t） - T（s） near the origin, where （T（t）） is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in ...This paper is concerned first with the behaviour of differences T（t） - T（s） near the origin, where （T（t）） is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane （in which case it is assumed analytic）. For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T（t） -T（2t） have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.展开更多
In this paper,the existence,uniqueness and Strichartz type estimates to solutions of multipoint problem for abstract linear and nonlinear wave equations are obtained.The equation includes a linear operator A defined i...In this paper,the existence,uniqueness and Strichartz type estimates to solutions of multipoint problem for abstract linear and nonlinear wave equations are obtained.The equation includes a linear operator A defined in a Hilbert space H.We obtain the existence,uniqueness regularity properties,and Strichartz type estimates to solutions of a wide class of wave equations which appear in the fields of elastic rod,hydro-dynamical process,plasma,materials science and physics,by choosing the space H and the operator A.展开更多
The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challen...The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales,particularly those described by evolution PDEs with time-derivative terms.This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups.The proposed approach is an iterative algorithm where each computational unit,termed the single-step neural operator solver(SSNOS for short),approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step.The SSNOS consists of both linear and nonlinear components:The linear part approximates the linear operator in the solution map;in contrast,the nonlinear part captures deviations in the solution function caused by the equations nonlinearities.To evaluate the performance of the algorithm,the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation.The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator.Moreover,the model can achieve a relatively high approximation accuracy with simple pre-training.展开更多
A model which describes the dynamics of an SIS epidemic in an age-structured and time dependent population is considered. The global behavior of the model with a special form for the infective force is obtained. The e...A model which describes the dynamics of an SIS epidemic in an age-structured and time dependent population is considered. The global behavior of the model with a special form for the infective force is obtained. The existence and uniqueness of the solution and nontrivial endemic equilibrium state are proved.展开更多
文摘Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for corresponding semigroup. Moreover, a Dresher＇s type inequality for two-parameter family of means, is also proved.
文摘The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in W pl,γ（; E0, E） is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.
文摘For injective, bounded operator C on a Banach space X , the author defines the C -dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive C -semigroups, where a C -semigroup T(·) is quasi-contractive if ‖T(t)x‖‖Cx‖ for all t0 and x∈X . This kind of generators guarantee that the associate abstract Cauchy problem u′(t,x)=Au(t,x) has a unique nonincreasing solution when the initial data is in C(D(A)) (here D(A) is the domain of A ). Also, the generators of quasi isometric C -semigroups are characterized.
基金Supported by EPSRC (EP/F020341/1)partially supported by the research project AHPIfunded by ANR
文摘This paper is concerned first with the behaviour of differences T（t） - T（s） near the origin, where （T（t）） is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane （in which case it is assumed analytic）. For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T（t） -T（2t） have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.
文摘In this paper,the existence,uniqueness and Strichartz type estimates to solutions of multipoint problem for abstract linear and nonlinear wave equations are obtained.The equation includes a linear operator A defined in a Hilbert space H.We obtain the existence,uniqueness regularity properties,and Strichartz type estimates to solutions of a wide class of wave equations which appear in the fields of elastic rod,hydro-dynamical process,plasma,materials science and physics,by choosing the space H and the operator A.
基金supported by the National Natural Science Foundation Major Program of China(No.12494544)the National Natural Science Foundation General Program of China(No.12171039)+1 种基金the New Cornerstone Science Foundation through the XPLORER PRIZE and Sino-German Center Mobility Programme(No.M-0548)the Shanghai Science and Technology Program(No.21JC1400600)。
文摘The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales,particularly those described by evolution PDEs with time-derivative terms.This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups.The proposed approach is an iterative algorithm where each computational unit,termed the single-step neural operator solver(SSNOS for short),approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step.The SSNOS consists of both linear and nonlinear components:The linear part approximates the linear operator in the solution map;in contrast,the nonlinear part captures deviations in the solution function caused by the equations nonlinearities.To evaluate the performance of the algorithm,the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation.The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator.Moreover,the model can achieve a relatively high approximation accuracy with simple pre-training.
基金This work is supported by the National Natural Sciences Foundation of China.
文摘A model which describes the dynamics of an SIS epidemic in an age-structured and time dependent population is considered. The global behavior of the model with a special form for the infective force is obtained. The existence and uniqueness of the solution and nontrivial endemic equilibrium state are proved.
