Matthias Eller

ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.

Acoustic source identification using multiple frequency information

We consider the inverse problem of identifying the location and shape of a finitely supported acoustic source function, separable with respect to space and frequency, from measurements of the acoustic field on a closed surface for many frequencies. A simple uniqueness proof and an error estimate for the unknown source function are presented. From the uniqueness proof an efficient numerical algorithm for the solution is developed. The algorithm is tested using numerically generated data in dimensions 2 and 3.

Finite Dimensionality of the Attractor for a Semilinear Wave Equation with Nonlinear Boundary Dissipation

Abstract Long-time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary point at an exponential rate. This result makes it possible to prove that the global attractor is exponential, i.e., it attracts every bounded set with exponential speed.

Carleman estimates for some elliptic systems

A Carleman estimate for a certain first order elliptic system is proved. The proof is elementary and does not rely on pseudo-differential calculus. This estimate is used to prove Carleman estimates for the isotropic Lame system as well as for the isotropic Maxwell system with C1 coefficients.

Unique continuation for solutions to Maxwell's system with non-analytic anisotropic coefficients

We consider Maxwells system with anisotropic coefficients, i.e., the electric permittivity and the magnetic permeability are assumed to be matrices with non-analytic entries. Under the additional assumption that one matrix is a scalar multiple of the other we prove unique continuation of the system across non-characteristic surfaces. The proof relies on differential geometry as well as on Carleman estimates.

A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Holmgrens theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwells equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgrens Theorem.

SemiGlobal Exact Controllability of Nonlinear Plates

The exact controllability problem for several semilinear thin plate models is considered. A distributed control affects a collar of the plates boundary. The result is semiglobal in the sense that there is no restriction on the size of the initial and the target states, but the controllability time is uniform only with respect to a given bounded set containing these states. Both (i) closed-loop-based and (ii) “pure” open-loop constructions are discussed. Strategy (i) describes exact controls for an abstract class of second-order evolution equations. It applies to the Berger plate with small in-plane stresses (and, depending on some open questions, possibly to the von Karman model). This method partly relies on uniform stabilization and offers no apparent leeway to improve the controllability time. Strategy (ii) is demonstrated on the example of a Kirchhoff model with a dissipative polynomial source term. Such a source serves as a prototype for ultimately considering the same control construction for the B...