The reduced (in the angular coordinate φ) wave equation and Klein-Gordon equation are considered on a Kerr background and in the framework of C-semigroup theory. Each equation is shown to have a well-posed initial value problem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroup’s generator coincides with the spectrum of a...

In this paper, we study the logarithmic stability for the hyperbolic equations by arbitrary boundary observation. Based on Carleman estimate, we first prove an estimate of the resolvent operator of such equation. Then we prove the logarithmic stability estimate for the hyperbolic equations without any assumption on an observation sub-boundary.

Adomian decomposition method has been applied to solve many functional equations so far. In this article, we have used this method to solve the fuzzy differential equation under generalized differentiability. We interpret a fuzzy differential equation by using the strongly generalized differentiability. Also one concrete application for ordinary fuzzy differential equation with fuzzy input data...

In order to propose a deeper analysis of the general quartic equation with real coefficients, analytical solutions for all cubic and equations were reviewed here; then, it was found that there can only be one form resolvent satisfies following two conditions at same time: (1) Its discriminant is identical equation. (2) It has least positive root whenever non-biquadratic. This unique special def...

We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the non...

The Webster-Lokshin equation is a partial differential equation considered in this paper. It models the sound velocity in an acoustic domain. The dynamics contains linear fractional derivatives which can admit an infinite dimensional representation of diffusive type. The boundary conditions are described by impedance condition, which can be represented by two finite dimensional systems. Under t...

We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension.

In this expository paper, we discuss the use of the Evans function in finding resonances, which are poles of the analytic continuation of the resolvent. We illustrate the utility of the general theory developed in [13, 14] by applying it to two physically interesting test cases: the linear Schrödinger operator and the linearization associated with the integrable nonlinear Schrödinger equation.