Local k-convoluted c-semigroups and complete second order abstract Cauchy problems

Convoluted C-cosine Functions and Convoluted C-semigroups

Convergence Estimates for Abstract Second Order Singularly Perturbed Cauchy Problems with Monotone Nonlinearities∗

We study the behavior of solutions to the problem  ε ( u′′ ε (t) +A1uε(t) ) + uε(t) +A0uε(t) +B ( uε(t) ) = fε(t), t ∈ (0, T ), uε(0) = u0ε, u ′ ε(0) = u1ε, in the Hilbert space H as ε → 0, where A1, A0 are two linear selfadjoint operators and B is a locally Lipschitz and monotone operator. ∗Accepted for publication in revised form on June 15, 2012. †Department of Mathematics and Informatics,...

Almost Periodic Solutions of First- and Second-Order Cauchy Problems

Almost periodicity of solutions of firstand second-order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by Ruess Vu~ and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vector-valued version) of Loomi...

Second order abstract initial - boundary value problems

Introduction Partial differential equations on bounded domains of R n have traditionally been equipped with homogeneous boundary conditions (usually Dirichlet, Neumann, or Robin). However, other kinds of boundary conditions can also be considered, and for a number of concrete application it seems that dynamic (i.e., time-dependent) boundary conditions are the right ones. Motivated by physical p...

Evolution Semigroups for Nonautonomous Cauchy Problems

In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems (NCP ) { u̇(t) = A(t)u(t) u(s) = x ∈ X on a Banach space X by the existence of certain evolution semigroups. Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so called “...