On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Abstract

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: "smoothness", "oscillations", "degeneration" and "stabilization". Gevrey and C ∞ well-posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples.

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…”

On second order weakly hyperbolic equations and the ultradifferentiable classes

“…[4] in case of a 0 > 0, and [5,8,9,10,11] in case of a 0 = 0 for instance. In particular, it is examined in [2,5,6,7,13,19] that a(t) is singular only at t = T , and our main theorem is based on their researches. Here we note that the linear wave equations with singular coefficients are studied by motivated to apply the time global solvability of Kirchhoff equation, which is a sort of non-linear wave equations with non-local nonlinearity; for the details refer to [12,15,17,18].…”