Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Abstract

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under C k -regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the C ∞ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

“…We conclude this paper by showing that when the coefficients are regular enough and the initial data are Gevrey then the very weak solution coincides with the classical and ultradistributional ones obtained in [17,23]. Since the initial data do not need to be regularised because they are already Gevrey, there exists a representative (u ε ) ε of u such that …”

“…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.…”

“…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 ).…”

“…We know from [17] that the Cauchy problem We also know that the Cauchy problem (55) has a unique solution u in G([0, T ]; G s (R n )) after suitable embedding of coefficients and initial data. This means that there exists a representative (u ε ) ε of classical smooth solutions such that…”

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

“…Analogues of Parts (a) of the above theorems for the wave equation on R n go back to Colombini, de Giorgi, and Spagnolo [11]. For higher order hyperbolic equations in R the Gevrey well-posedness was considered in [14] and [36] under assumptions corresponding to Cases I.2 (a) and I.3 (a), which were extended to R n in [22] and [23], respectively. Other low regularity or multiple characteristics situations were considered in e.g.…”