In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space whic...In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.展开更多
This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are a...This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L_p norm.展开更多
The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, wher...The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.展开更多
This paper deals with a class of new abstract kinetic operators and studies essential spectrum and distribution of isolated eigenvalues of operators. As an application, the spectral analysis of a kind of transport ope...This paper deals with a class of new abstract kinetic operators and studies essential spectrum and distribution of isolated eigenvalues of operators. As an application, the spectral analysis of a kind of transport operators with periodic boundary condition is discussed.展开更多
Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,q s(Ω;E) is derived.The embedding theorems in Besov-Lions type spaces B l,s p,q(Ω;E0,E) are studied,where E0,E are tw...Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,q s(Ω;E) is derived.The embedding theorems in Besov-Lions type spaces B l,s p,q(Ω;E0,E) are studied,where E0,E are two Banach spaces and E 0 E.The most regular class of interpolation space E α,between E 0 and E are found such that the mixed differential operator D α is bounded and compact from B p,q l,s (Ω;E 0,E) to B p,q s (Ω;E α) and Ehrling-Nirenberg-Gagliardo type sharp estimates established.By using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are obtained.In applications,the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.展开更多
文摘In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.
文摘This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L_p norm.
基金The Natural Science Foundation of Department ofEducation of Jiangsu Province (No06KJD110087)
文摘The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.
基金Project supported by the National Natural Science Foundation of China.
文摘This paper deals with a class of new abstract kinetic operators and studies essential spectrum and distribution of isolated eigenvalues of operators. As an application, the spectral analysis of a kind of transport operators with periodic boundary condition is discussed.
文摘Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,q s(Ω;E) is derived.The embedding theorems in Besov-Lions type spaces B l,s p,q(Ω;E0,E) are studied,where E0,E are two Banach spaces and E 0 E.The most regular class of interpolation space E α,between E 0 and E are found such that the mixed differential operator D α is bounded and compact from B p,q l,s (Ω;E 0,E) to B p,q s (Ω;E α) and Ehrling-Nirenberg-Gagliardo type sharp estimates established.By using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are obtained.In applications,the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.
