On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients

Abstract

We consider the loss of regularity of the solution to the backward Cauchy problem for a second order strictly hyperbolic equation on the time interval [0, T ] with time depending coefficients which have a singularity only at the end point t = 0. The main purpose of this paper is to show that the loss of regularity of the solution on the Gevrey scale can be described by the order of differentiability of the coefficients on (0, T ], the order of singularities of each derivatives as t → 0 and a stabilization condition of the amplitude of oscillations described by an integral on (0, T ). Moreover, we prove the optimality of the conditions for C ∞ coefficients on (0, T ] by constructing a counterexample.

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

“…well-posed for s < β/(β − 1) and L 2 well-posed for β = 1 by [1]. We observe that both additional conditions of (I) and (IV) continuously contribute to be permitted a worse oscillation for the well-posedness in the same Gevrey class without assuming them.…”

“…Thus, a more precise analysis will be possible if we additionally introduce the stabilization property. Remark 1.2 The stabilization property (1.7) with λ(t) ≡ 1 was introduced in [1] for strictly hyperbolic case, thus (1.7) for degenerate λ(t) should be a reasonable generalization for weakly hyperbolic case. The degeneration of λ(t) contributes to stabilize the amplitude of the coefficient, but such λ(t) may bring a bad effect from a point of view of a perturbation of constant coefficient.…”

“…We can estimate the deviation of variable from constant coefficients by another scale, which is determined by some integrals of the amplitude of the coefficients. Such an idea was introduce in [2], [3] and [7] associated with the Hölder continuity of the coefficients, but we adopt the following formulation, which is introduced in [1] as the stabilization property of a(t) as t → t 0 with a positive parameter δ:…”

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].…”

“…since κ(α, β, m) ≥ 0. Here the condition (1.7) was introduced in [2] as the stabilization property, and the constant a T is uniquely determined if a constant α ∈ (1, β) exists. We observe that κ(1, β, m) = 1 − 1/β, and κ(α, β, m) is strictly decreasing with respect to m only if α > 1.…”