Integrated semigroups,C-semigroups and the abstract Cauchy problem

LetA be a linear operator on a Banach space. We consider when the following holds. (*)u′(t,x)=A(u(t,x)) (t≥0),u(0,x)=x, has a unique solution, for allx in the domain ofA ⁿ⁺¹ . We discuss the relationship between (*), integrated semigroups, and C-semigroups. We use this to obtain new results about integrated semigroups and the abstract Cauchy problem. We give several examples where (*) may be easily shown using C-semigroups. Many of these examples may not be done directly using integrated semigroups.     

Equations d’evolution du second ordre associees a des operateurs monotones

This paper extends some recent results of V. Barbu. It is concerned with bounded solutions of the problem:u″∈Au, u′(0)∈ϖj(u(0)−a) whereA is a maximal monotone operator in a Hilbert spaceH, a∈D(A) andj is a strictly convex l.s.c. function fromH to [0,+∞]. An existence and uniqueness theorem for this problem is proved. Takingj to be the indicator function of a pointu ₀∈D(A), one obtains a bounded solutionu(t) of the initial value problem:u″∈Au, u(0)=u ₀. Denotingu(t)=S _1/2(t)u₀ one obtains a semi-group of contractions onD(A). The generator of this semigroup is denoted byA _1/2. Further properties ofS _1/2(t) andA _1/2 are studied.      

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators

In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x₀, where A is an accretive operator in a general Banach space X and f ε L¹(0,T;X). Their method involves proving convergence in the L^∞-norm of a sequence of step function approximations α_n(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions α_n(σ, τ) in the L¹-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

Perturbations of an abstract Euler-Poisson-Darboux equation

The stability of the uniform correctness of the Cauchy problem ,t>0,u(0)=u ₀,u′(0)=0 fork>0 with respect to perturbations of the operator is studied. Translated fromMatematickeskie Zameiki, Vol. 60, No. 3, pp. 363–369, September, 1996.     

On the non-negative non-increasing solutions of non-linear second order differential equations

Summary The sufficient conditions for the existence and uniqueness of solution u(t) of the differential equation u″=f(t, u, u′), are established, satisfying the condition u(t)= =u₀, u(t)≥0 and u′(t)≥0 for t ε (0,+∞). Entrata in Redazione il 26 aprile 1968.     

Heat-kernels and maximalL p -L q -Estimates: The non-autonomous case

In this paper, we establish maximal L^p−L^q estimates for non-autonomous parabolic equations of the type u′(t)+A(t)u(t)=f(t), u(0)=0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t∈[0,T]. We apply this result on semilinear problems of the form u′(t)+A(t)u(t)=f(t, u(t)), u(0)=0.     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

Retarded functional differential equations with nondense domain operators

In this work, we use integrated semigroups to state results on the existence and uniqueness of integral solutions and solutions for the abstract Cauchy problem x^′(t)=Bx(t)+Lx_t, t⩾0, where B is a nondensely defined linear operator on a Banach space X. This revised version was published online in June 2006 with corrections to the Cover Date.     

Operator Matrices as Generators of Cosine Operator Functions

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in L^p(0, 1) and in L²(Ω) equipped with dynamical and acoustic-like boundary conditions, respectively.     

Second order incomplete expiring Cauchy problems

We extend results of Balakrishnan and Dorroh, on 2^nd-order incomplete Cauchy problems, from differentiable to stronglycontinuous semigroups of operators. We show that the Cauchy problem (*) $$\begin{gathered} u''(t) = A(u(t)), t \geqslant 0, u(0) = x, \hfill \\ \mathop {lim}\limits_{t \to \infty } \left\| {u^{(k)} (t)} \right\| = 0, k = 0, 1, 2, \hfill \\ \end{gathered} $$ where A is a linear operator with nonempty resolvent on a Banach space, is well-posed if and only if A has a squares root that generates a C_o semigroup, {T(t)} t>0, that converges to zero, as t goes to infinity, in the strong operator topology. This extension leads to the following application. If A is a linear constant coefficient partial differential operator on L²(?ⁿ), then there exist orthogonal closed subspaces, H₁, H₂, such that H_l⊕H₂=L²(?ⁿ), and (*), on H₁, is well-posed, while the complete Cauchy problem u″(t)=A(u(t)), t??, u(O)=x, u′ (O)=y is well-posed on H₂. We also apply our results to the dying wave equation, on C_o[0, ∞) and L^p(?dv) (1≤p <∞), for a large class of measures v.     

Remarks on bifurcations ofu + μu − u k=0 (4≤kεZ +)

In this paper, we investigate the bifurcations of one class of steady-state reaction-diffusion equations of the formu″ + μu − u ^k=0, subjectu(0)=u(π)=0, where μ is a parameter, 4≤kεZ ⁺. Using the singularity theory based on the Liapunov-Schmidt reduction, some satisfactory results are obtained. This work is supported by the National Natural Science Foundation of China (No.19971057) and the Youth Science Foundation of Shanghai Municipal Commission of Education (No.99QA66).     

Periodic solutions of second order non-Lagrangian systems

Using results in bifurcation theory, we show the existence of periodic solutions of a large class of non-Lagrangian systems of the formu″+A ₁ v′+B ₁u+F₁ (w, w′, w″)=0v″+A ₂ u′+B ₂v+F₂ (w, w′, w″)=0 wherew=(u, v).     

Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.     

On then-dimensional harmonic oscillator

Summary In this paper, various oscillatory properties of solutions of the scalar equation x″+q(t)x=0 are extended to the vector equation u″+Q(t)u=0. Entrata in Redazione il 1^o giugno 1977.     

Singular semi-linear equations inL 1(R)

Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L ¹(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫ _−∞ ^∞ . Some extensions of this result are also given. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS 75-05501.     

Perturbations of existence families for abstract Cauchy problems

In this paper, we establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for arbitrary order abstract Cauchy problems in a Banach space: ; . As a consequence, we obtain such perturbation results for regularized semigroups and regularized cosine operator functions. An example is also given to illustrate possible applications.

     

On multivalued evolution equations in Hilbert spaces

The Cauchy problemdu/dt+Au+B(t,u)∋0,u(0)=u ₀ is studied in a separable Hilbert space setting, whenA is a multivalued maximal monotone operator, andB is a multivalued operator which is measurable with respect to the time variable and upper semi-continuous with respect to the space variable. Under some boundedness conditions onB, an existence theorem is proved, with the extra assumption, in the infinite dimensional case thatA is the subdifferential of a proper lower semi-continuous inf-compact convex function. A theorem of dependence upon the initial condition is also given.     

Impedance Passive and Conservative Boundary Control Systems

The external Cayley transform is used for the conversion between the linear dynamical systems in scattering form and in impedance form. We use this transform to define a class of formal impedance conservative boundary control systems (colligations), without assuming a priori that the associated Cauchy problems are solvable. We give sufficient and necessary conditions when impedance conservative colligations are internally well-posed boundary nodes; i.e., when the associated Cauchy problems are solvable and governed by C ₀ semigroups. We define a “strong” variant of such colligations, and we show that “strong” impedance conservative boundary colligation is a slight generalization of the “abstract boundary space” construction for a symmetric operator in the Russian literature. Many aspects of the theory is illustated by examples involving the transmission line and the wave equations. Received: August 21, 2006. Accepted: October 22, 2006.