Frederick Bloom

Asymptotic Bounds for Solutions to a System of Damped Integrodifferential Equations of Electromagnetic Theory.

Abstract : For the system of damped integrodifferential equations which govern the evolution of the electric induction field in a class of rigid holohedral isotropic dielectrics of the type introduced by Toupin and Rivlin, conditions on the memory functions are deduced which imply that the L2 norms of such induction fields are bounded away from zero even as the damping grows in an unbounded manner; explicit lower bounds for the L2 norms of the induction fields in such dielectrics are derived as t increases without limit. (Author)

On the damped nonlinear evolution equation wtt = σ(w)xx − ywt

Abstract Initial boundary value problems for the damped nonlinear wave equation wtt = σ(w)xx − ywt arise in several areas of applied mathematics and, in particular, in studies of shearing flow in a nonlinear viscoelastic fluid; the problems of global existence and nonexistence of smooth solutions have been extensively studied in the strictly hyperbolic case σ′(δ) ⩾ e > 0, ∀δ ϵ R1 as well as in the case where σ′(0) > 0 and the initial data are chosen so small that σ′(w) > 0 for as long as a smooth solution w(x, t) exists. In this paper the global nonexistence problem is studied for the cases σ′(0) = 0 and σ′(0) > 0 but σ′(δ)

Growth estimates for electric displacement fields and bounds for constitutive constants in the Maxwell-Hopkinson theory of dielectrics

Abstract Employing a modified version of a concavity argument for abstract differential equations, we obtain growth estimates for solutions to a class of initial-value problems associated with an undamped linear integrodifferential equation in Hilbert space. Our results are applied to the derivation of growth estimates for the gradients of electric displacement fields occurring in rigid nonconducting material dielectrics of Maxwell-Hopkinson type and these, in turn, are used to bound constitutive constants appearing in theories associated with such material response.

Stability and growth estimates for electric fields in non-conducting material dielectrics

Abstract Employing results derived by the author for solutions of an abstract integrodifferential equation in Hilbert space, we obtain stability and growth estimates for electric fields in nonconducting material dielectrics. It is assumed that a linear constitutive equation of Maxwell-Hopkinson type relates the electric field and the electric displacement field in the dielectric; specific results for a simple memory function of exponential type are given.

Concavity arguments and growth estimates for damped linear integrodifferential equations with applications to a class of holohedral isotropic dielectrics

Concavity arguments are employed so as to obtain growth estimates for solutions to two initial-value problems associated with a class of damped integrodifferential equations in Hilbert space. By applying the results obtained in this abstract setting we obtain growth estimates for the gradients of electric displacement fields which occur in a class of holohedral isotropic nonconducting rigid dielectrics; such estimates may be used to bound constitutive constants appearing in the equations which define such material response.ZusammenfassungWir betrachten eine Klasse gedämpfter Integrodifferentialgleichungen in einem Hilbertraum und erhalten mit Hilfe von Konkavitätsargumenten Abschätzugen für das Wachstum Lösungen. Durch Anwendung dieser abstrakten Resultate auf elektrische Felder in einem holohedrischen, isotropen, nichtleitenden Medium, wie es 1960 von Toupin und Rivlin betrachtet wurde, gewinnen wir Abschätzungen für das Wachstum der elektrischen Verschiebungsvektoren. Mit Hilfe solcher Abschätzungen können Schranken für die verallgemeinerten Dielektrizitätskonstanten konstruiert werden, die in den Materialgleichungen von Toupin und Rivlin auftreten.

Some stability theorems for an abstract equation in Hilbert space with applications to linear elastodynamics

Abstract Previous results of Knops and Payne concerning continuous data dependence in linear elastodynamics are extended to include the case where the elasticities may be time dependent. Our results, which include stability under perturbations of both the elasticities and the initial geometry, are obtained by applying a logarithmic convexity argument to the Cauchy problem associated with an abstract differential equation in Hilbert space.

On continuous dependence for a class of dynamical problems in linear thermoelasticity

Given that the temperature dependent elasticities of a linear anisotropic elastic material satisfy certain Lipschitz type conditions, it is shown that the displacement vector depends continuously, in an appropriate norm, on temperature deviations from a fixed but arbitrary steady state temperature distribution.ZusammenfassungUnter der Voraussetzung, dass die temperaturabhängigen Elastizitätskonstanten eines anisotropen, linearelastischen Materials gewissen Lipschitzartigen Bedingungen genügen, wird hier bewiesen, dass der Verschiebungsvektor von den Abweichungen aus einer festgegebenen sonst aber beliebigen, stationären Temperaturverteilung nach einer geeigneten Norm stetig abhängt.

Asymptotic behavior of solutions to the damped quasilinear equation ∂2∂t2 u(x, t) + γ∂u∂t (x, t) − ∂∂xσ ∂u∂x (x, t)) = 0☆

Abstract Asymptotic lower bounds for the L 2 norms of solutions of initial-boundary value problems associated with the equation of the title are derived for a simple case in which the equation fails to exhibit strict hyperbolicity. It is shown that in such cases it can be expected that the norm of a solution will be bounded away from zero as t → +∞ even as the damping factor γ becomes infinitely large.

On the Nonexistence of Globally Bounded Solutions to Initial-History Value Problems for Integrodifferential Equations.

Abstract : Conditions are given under which it is impossible that there exist a solution of the integrodifferential initial-history value problem which lies in a class of bounded perturbations. An application is given to the nonexistence of globally bounded solutions, to initial-history boundary value problems for Maxwell-Hopkinson dielectrics, which lie in classes of bounded functions.

Continuous dependence on initial geometry for a class of abstract equations in Hilbert space

Abstract Conditions are given under which members of a class of uniformly bounded solutions to the Cauchy problem associated with equations of the form Mutt − Nu = F, in Hilbert space, depend continuously on perturbations of the initial geometry; our results generalize a similar theorem of Knops and Payne for classical solutions to initial-boundary value problems in linear elastodynamics.