Global solvability for Kirchhoff equation in special classes of non-analytic functions

Abstract

In this paper we derive the following two properties: the first one is a precise representation of WKB solution to the Cauchy problem of a linear wave equation with a variable coefficient with respect to time, and the second one is the global solvability for Kirchhoff equation in some special classes of nonrealanalytic functions, which is proved by grace of the first property.

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

“…We have introduced an example for a(t) by (17), which ensures the optimality of the estimate (7). Actually, one can prove the following statement: Proposition 2 Let m be a positive integer, p and q be real numbers satisfying 0 ≤ q < p + (1 − p)/m.…”

“…On the other hand, the results from C 2 property of a(t) are the conclusions from one further step of diagonalization procedure; indeed, we can expect more precise representation of WKB solutions from the C 2 property of a(t) which implies an improvement of the result from C 1 property. The diagonalization procedure taking into account the C m property of a(t) for m ≥ 3 was introduced by Hirosawa [7] in order to prove the global solvability for Kirchhoff equations in non-realanalytic classes. In [1], Cicognani and Hirosawa proved the local well-posedness for non-Lipschitz continuous coefficients a(t) under suitable conditions generalizing (11) to C m property and a stabilization property corresponding to (6) (see Remark 6).…”

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

“…Some positive effects from further steps of diagonalization procedure with C 2 property were introduced in [17] for the proof of L 2 and C ∞ well-posedness of (1.3) with some singular coefficients; this argument is developed in [6], [8] and [16] for instance. A certain effect of the C m property with m ≥ 3 without stabilization was studied in [12] for m = 3 as an application to the global solvability for a quasi-linear wave equation, and it was developed to general m in [11] and [13]. We have observed from our theorems and examples that the C m property of the coefficient on the Cauchy problem (1.3) is meaningful only if we suppose the stabilization property (1.7) simultaneously.…”

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

“…The crucial points of this procedure are the regularity of the coefficient and the symmetricity of the characteristic roots. (Originally, this method was introduced in [6]. )…”