Strictly hyperbolic equations with coefficients low-regular in time and smooth in space

Abstract

We consider the Cauchy problem for strictly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is L 2 well-posed in the case of Lipschitz continuous coefficients in time, H s well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of Hölder continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and Hölder coefficients are recovered.Mathematics Subject Classification (2010). 35S05, 35L30, 47G30.Keywords. higher order strictly hyperbolic Cauchy problem, modulus of continuity, loss of derivatives, pseudodifferential operators.As to be expected from the know results of the above-mentioned authors, the modulus of continuity µ is linked to the weight function η. In this paper, we describe how µ and η are related to each other and give sufficient conditions for the well-posedness of problem (1.1) which link µ to η and the sequence K p .

Extended Gevrey Regularity via the Short-Time Fourier Transform

“…. When m is a polynomial weight of the form m(x, ω) = x t ω s we will use the notation M p,q s,t (R d ) for the modulation spaces which consists of all f ∈ (S (1)…”

“…This turned out to be important e.g. in the study of strictly hyperbolic equations, see [1]. Paley-Wiener type theorem for compactly supported extended Gevrey regular functions is given in [24,25], and it turns out that the Fourier-Laplace transform of such functions have certain logarithmic decay at infinity which can be expressed in terms of Lambert W function.…”

A generalized Levi condition for weakly hyperbolic Cauchy problems with coefficients low regular in time and smooth in space

“…With this weight sequence, it is clear that assumption (A7) is satisfied. For more detailed considerations about finding and choosing a weight sequence in this setting, we refer the reader to the example with Log-Log [m] -Lip continuous coefficients in [4]. In these cases, we have well-posedness in spaces that are very close but a little bit smaller than the classical Gevrey space G s .…”

“…More recently, in [4] the authors established a general condition linking the regularity of the coefficients in time to possible solution spaces and the required regularity of the coefficients in space. They assumed that the coefficients satisfy the relation D β x a m−j, γ (t, x) − D β x a m−j, γ (s, x) ≤ CK |β| µ(|t − s|), 0 ≤ |t − s| ≤ 1, x ∈ R n , (1.2) where µ is a modulus of continuity describing their regularity in time and {K p } p is a weight sequence describing their regularity in space.…”

“…Some examples of moduli of continuity and how they are commonly referred to. Following [4], we also use weight sequences {K p } p to describe the regularity of the coefficients in space. Definition 1.2.…”