On second order weakly hyperbolic equations and the ultradifferentiable classes

Abstract

We consider second order weakly hyperbolic equations with time dependent coefficients in the ultradifferentiable classes. Our main purpose of the present paper is an investigation the relation between the classes of the functions to be well-posed and the following properties of the coefficients: the order of degeneration, stabilization to a monotonic function and their smoothness in the ultradifferentiable classes.

On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

“…In [4], which is a pioneer work for this kind of problem, it is proved that if a(t) > 0 and a(t) ∈ C σ (R + ) with σ ∈ (0, 1), where C σ (R + ) denotes the class of Hölder continuous functions on R + , then (1.1) is well-posed in the Gevrey class of order ν with ν < 1/(1 − σ). After that, the relations between various types of singularities of a(t) and the Gevrey order ν for the well-posedness of (1.1) was studied in many papers, for instance [1,3,5,8,11]. In particular, a sort of stabilization properties corresponding to (1.3) and (2.10) are introduced in [1,8,11].…”

“…After that, the relations between various types of singularities of a(t) and the Gevrey order ν for the well-posedness of (1.1) was studied in many papers, for instance [1,3,5,8,11]. In particular, a sort of stabilization properties corresponding to (1.3) and (2.10) are introduced in [1,8,11]. As a consequence of these results, for any fixed T > 0 there exists a constant C T such that the energy estimate…”

A class of non-analytic functions for the global solvability of Kirchhoff equation

“…The proof of Proposition 3.1 consists of three parts; reduction to first order system and diagonalization, symbol calculus, and these applications for the estimate (3.6). The original idea of the proof was introduced in [8], and the following proof is a modification of it.…”