Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n,$$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.Comment: 22 page

Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems

This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.Comment: 20 page

Preduals of Campanato spaces and Sobolev-Campanato spaces: A general construction

In this paper we describe two limiting processes for families of Banach spaces closely related to the standard definition of projective and inductive limits. These processes lead again to Banach spaces. Information about linear operators and duality between basic families of spaces is carried over to the corresponding limit spaces. The abstract results are shown to be applicable to Campanato spaces and Sobolev-Campanato spaces. In particular, we obtain the existence and a characterization of predual spaces. Some imbedding relations are investigated in more detail

Abstract Forced Symmetry Breaking and Forced Frequency Locking of Modulated Waves

AbstractWe consider abstract forced symmetry breaking problems of the typeF(x, λ)=y. It is supposed that for allλthe mapsF(·, λ) are equivariant with respect to the action of a compact Lie group, thatF(x0, λ0)=0 and, hence, thatF(x, λ0)=0 for all elementsxof the group orbit O(x0) ofx0. We look for solutionsxwhich bifurcate from the solution family O(x0) asλandymove away fromλ0and zero, respectively. Especially, we describe the number of different solutionsx(for fixed control parametersλandy), their dynamic stability and their asymptotic behavior forytending to zero. Further, generalizations are given to problems of the typeF(x, λ)=y(x, λ). Finally, our results are applied to a forced frequency locking problem of the typex(t)=f(x(t), λ)−y(t). Here it is supposed that the vector fieldsf(·, λ) areS1-equivariant, that the unperturbed equationx=f(x, λ0) has an orbitally stable modulated wave solution and that the forcingy(t) is a modulated wave

Existence and uniqueness of weak solutions of an initial boundary value problem arising in laser dynamics

In this paper a mathematical model for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers in the case of arbitrarily space depending carrier densities is investigated. We introduce a suitable weak formulation of the initial boundary value problem and prove existence, uniqueness and some regularity properties of the solution. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous coefficients is included

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type ∂2tu(t,x)−a(x,λ)2∂2xu(t,x)=b(x,λ,u(t,x),u(t−τ,x),∂tu(t,x),∂xu(t,x)),x∈(0,1) with smooth coefficient functions a and b such that a(x,λ)>0 and b(x,λ,0,0,0,0)=0 for all x and λ. We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on τ and λ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution u=0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the “loss of derivatives” property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays τ.Peer Reviewe

Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems

This paper concerns general singularly perturbed second order semilinear elliptic equations on bounded domains $Omega subset R^n$ with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, we prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and we estimate the distance between the approximate and the exact solutions. Here ''spike solution'' means that there exists a point in $Omega$ such that the solution has a spike-like shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign

Linear Elliptic Boundary Value Problems with Non-smooth Data: Normal Solvability on Sobolev-Campanato Spaces

In this paper linear elliptic boundary value problems of second order with non-smooth data (L∞-coefficients, Lipschitz domains, regular sets, non-homogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fredholm operators between appropriate Sobolev-Campanato spaces, that the weak solutions are Hölder continuous up to the boundary and that they depend smoothly (in the sense of a Hölder norm) on the coefficients and on the right hand sides of the equations and boundary conditions