Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

Abstract

In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

“…By putting ω −1 (ε) ∼ (log ε) r for an appropriate r , and repeating as in the proof of Theorem 2.3 (b), from (7.6) we conclude (a similar argument is also in [25]) that there exists η > 0 and, for p = 0, 1 there exist c p > 0 and N p > 0 such that…”

“…For such equations, the authors of [25] introduced the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. We now apply a modification of this notion to the Cauchy problem (4.3).…”

“…The mathematical analysis of hyperbolic equations with discontinuous coefficients goes back to Hurd and Sattinger [31]. We refer to [25] for the historical review of this topic.…”

On a class of hyperbolic equations in Weyl–Hörmander calculus

“…The L p boundedness in the setting of S(m, g) calculus for fractional powers of subelliptic operators has been studied in [Del06] and [Del]. Recent developments on hyperbolic equations with rough time dependent coefficients can be found in [GR15] and the references therein.…”

On hyperbolic systems with time-dependent Hölder characteristics

“…Further extreme cases: analytic coefficients and distributional coefficients have been also investigated (see, e.g. authors' papers [6,9,10], respectively, and references therein). Hyperbolic systems of the form (1) have been also investigated (see, e.g.…”