共查询到20条相似文献,搜索用时 484 毫秒
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Consider in a real Hilbert space H the Cauchy problem (P0): u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, where −A is the infinitesimal generator of a C0-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (P0) the following regularization (Pε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, u′(T)=uT, where ε>0 is a small parameter. We investigate existence, uniqueness and higher regularity for problem (Pε). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (Pε). Problem (Pε) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H). However, the boundary layer of order one is not visible through the norm of L2(0,T;H). 相似文献
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We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form ut+Au(t)=f(u(t),t), u(1)=φ, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0) is well-posed and that its solution Uβ(t) converges on [0,1] to the exact solution u(t) as β→0+. These results extend some earlier works on the nonlinear backward problem. 相似文献
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Guang-Xin Huang Feng Yin Ke Guo 《Journal of Computational and Applied Mathematics》2008,217(1):259-267
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Let u(t)=−Fx(t) be the optimal control of the open-loop system x′(t)=Ax(t)+Bu(t) in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system x′(t)=(A−BF)x(t). Main attention is given to the case of a skew-Hermitian matrix A. Given an operator A, for a class of cases, we find a matrix B that provides an almost optimal decay rate. 相似文献
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Mohamed Ali Toumi 《Expositiones Mathematicae》2010,28(3):269-275
Let A be an Archimedean f -algebra and let N(A) be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A is a derivation if and only if d(A)⊂N(A) and d(A2)={0}, where A2 is the set of all products ab in A. 相似文献
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Let P(D) be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on Rn and V be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e−tH generated by H=P(D)+V with Kato type perturbing potential V , which naturally generalizes the classical result for Schrödinger semigroup e−t(Δ+V) as V∈K2(Rn), the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e−tP(D) on L1(Rn). As a consequence of the Gaussian upper bound, the Lp-spectral independence of H is concluded. 相似文献
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In the present work, we introduce the concept of almost automorphic functions on time scales and present the first results about their basic properties. Then, we study the nonautonomous dynamic equations on time scales given by xΔ(t)=A(t)x(t)+f(t) and xΔ(t)=A(t)x(t)+g(t,x(t)), t∈T where T is a special case of time scales that we define in this article. We prove a result ensuring the existence of an almost automorphic solution for both equations, assuming that the associated homogeneous equation of this system admits an exponential dichotomy. Also, assuming that the function g satisfies the global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear dynamic equation on time scales. Further, we present some applications of our results for some new almost automorphic time scales. Finally, we present some interesting models in which our main results can be applied. 相似文献
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We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ) on a given domain Ω=(0,T)×Rn. It is known that, if the Hamiltonian H=H(t,p) is not a convex (or concave) function in p , or H(⋅,p) may change its sign on (0,T), then the Hopf-type formula does not define a viscosity solution on Ω . Under some assumptions for H(t,p) on the subdomains (ti,ti+1)×Rn⊂Ω, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics. 相似文献
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We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) ε0, the same happens for the solution u(t,⋅) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows. 相似文献
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For a tridiagonal, singular matrix A we present a method for the computation of the polynomial p(λ) such that AD=p(A) holds, where AD is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A. 相似文献