Hyperbolic Equations with Non-analytic Coefficients

Abstract

We prove that the Cauchy problem for a hyperbolic, homogeneous equation with C ∞ coefficients depending on time, is well posed in every Gevrey class, although in general it is not well-posed in C ∞ , provided the characteristic roots satisfy the condition

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

“…Note that this is a straightforward extension of Lemma 2 in [23] valid for two parameter (that is δ, t) dependent matrices. …”

“…We start by recalling the known results for coefficients which are regular: in [17], extending the one-dimensional result of Kinoshita and Spagnolo in [23], we have obtained the following well-posedness result (for the special case of b j = 0, see also [9]): For the sake of the reader we briefly recall the definitions of the spaces γ s (R n ) and γ (s) (R n ) of (Roumieu) Gevrey functions and (Beurling) Gevrey functions, respectively. These are intermediate classes between analytic functions (s = 1) and smooth functions.…”

“…employed by Kinoshita and Spagnolo in [23] to obtain Gevrey well-posedness for the corresponding Cauchy problem. It is clear that the regularised Equation (1) and the corresponding first order system have solutions (u ε ) ε and (U ε ) ε , respectively, depending on the parameter ε ∈ (0, 1].…”

“…so the condition on the roots used in [23] and in [17] is trivially fulfilled with constant M = 2 (see (6) in [17]). By analysing Lemma 3.2 and Proposition 3.1 in this particular case we get the following results on the quasi-symmetriser Q (2) δ (λ ε ).…”

“…A careful analysis of the proof of Lemma 2 in [23] allows us to extend Lemma 3.4 to the family of quasi-symmetrisers (Q (2) δ (λ ε )) ε . The constant…”