Search

We propose and analyze a Strang splitting method for a cubic semilinear Schrodinger equation with forcing and damping terms and subject to periodic boundary conditions. The nonlinear part is solved analytically, whereas the linear part – space derivatives, damping and forcing – is approximated by the exponential trapezoidal rule. The necessary operator exponentials and ϕ -functions can be computed efficiently by fast Fourier transforms if space is discretized by spectral collocation. Under natural regularity assumptions, we first show global existence of the problem in H 4 ( T ) and establish global bounds depending on properties of the forcing. The main result of our error analysis is first-order convergence in H 1 ( T ) and second-order convergence in L 2 ( T ) on bounded time-intervals.

We investigate the Lie and the Strang splitting for the cubic nonlinear Schrodinger equation on the full space and on the torus in up to three spatial dimensions. We prove that the Strang splitting converges in L 2 with order 1 + θ for initial values in H 2 + 2 θ with θ ∈ ( 0 , 1 ) and that the Lie splitting converges with order one for initial values in H 2 .

We investigate sectoriality and maximal regularity in L p – L q -Sobolev spaces for the structurally damped plate equation with Dirichlet–Neumann (clamped) boundary conditions. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the half-space, and in bounded domains of class C 4 . It turns out that the first-order system related to the scalar equation on R n is sectorial only after a shift in the operator. On the half-space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions. We further show that the semigroup solving the problem on bounded domains is exponentially stable.

In this paper, we introduce locally Lipschitz observation systems for nonlinear semigroups and show that they can be represented by an ‘admissible’ nonlinear output operator defined on a suitable subspace. In the semilinear case, this concept fits well to the Lebesgue extension known from linear system theory. For semilinear systems, we show robustness of exact observability near equilibria under locally small Lipschitz perturbations. Finally, we apply our results to a damped nonlinear plate equation and a semilinear thermo-elastic system.

We comprehensively study a one dimensional elliptic operator degenerating of first order at the boundary in an L p setting. Here the coefficient of the drift term determines the regularity and the possible boundary conditions of the problem.

We show that the realization Ap of the elliptic operator Au=div(Q∇u)+F⋅∇u+Vu in Lp(RN,RN), p∈[1,+∞[, generates a strongly continuous semigroup, and we determine its domain D(Ap)={u∈W2,p(RN,RN):F⋅∇u+Vu∈Lp(RN,RN)} if 1

For finite-dimensional systems the Hautus test is a well known and easily checkable condition for observability. Russell and Weiss [A general necessary condition for exact observability, SIAM J. Control Optim. 32 (1994) 1–23] suggested an infinite-dimensional version of the Hautus test, which is necessary for exact infinite-time observability and sufficient for approximate infinite-time observability of exponentially stable systems. In this paper the notion of observability is studied for polynomially stable systems. Several known results for exponentially stable systems are extended to the setting of polynomially stable systems. By means of an example the obtained results are illustrated.

We establish maximal regularity of type Lp for a parabolic evolution equation u′(t) = A(t)u(t) + f(t) with A( · ) ∈ C([0, T], L (D(A(0)), X)) and construct the corresponding evolution family on the underlying Banach space X. Our proofs are based on the operator sum method and the use of evolution semigroups. The results are applied to parabolic partial differential equations with continuous coefficients.

Evans and Howland used an abstract characterization of evolution semigroups and a perturbation result for C 0 -semigroups to treat non-autonomous Schrodinger equations and problems in scattering theory. Here we proceed in a similar way. At first we characterize evolution semigroups on spaces of vector-valued functions E ( X ) which are induced by a strongly continuous evolution family ( U ( t , s)) t ≥ s defined on the Banach space X . From this and a perturbation result of Voigt we obtain a sufficient condition for an operator on L p ( R , X ) to be the generator of an evolution semigroup. This will be applied to non-autonomous heat equations with absorption.