The Cauchy problem for quasilinear SG‐hyperbolic systems

Abstract

Key words Cauchy problem, quasilinear hyperbolic systems, SG-pseudo-differential operators MSC (2000) 35L45, 35L60, 35S10We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × R n and presenting a linear growth for |x| → ∞. We prove well-posedness in the Schwartz space S(R n ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument.

Log-Lipschitz regularity for SG hyperbolic systems

“…where, for every fixed w ∈ R m , the symbol k(t, x, w, ξ ) satisfies (1.2) for some C αβ smoothly depending on w. Combining the techniques used in [1,2], we prove that (1.4) is well posed in S(R n ). In this way, we provide also an extension of the results in [1] to systems of SG type.…”

“…The results about Sobolev continuity and composition of operators given in Section 2 can be extended to operators from LG m M in the following way, cf. [2].…”

“…The results in [9] have been extended by S. Coriasco [10], S. Coriasco and L. Rodino [11] to weakly hyperbolic systems with constant multiplicities and, more recently, to quasilinear symmetrizable systems by the second author and L. Zanghirati [2]. In this paper we examine the role played by Log-Lipschitz regularity in SG systems of the type considered in [2,9].…”

“…Following [2,20], we replace now v σ q in (5.9) by P σ q V , V in (5.7), where P σ q is the operator of order (−s 1 + |σ | + n/2, −s 2 + q) defined by…”

“…Enlargement of the system. Following [2,20], we enlarge the size of the system (5.5) by introducing new equations for some weighted derivatives of v, in order to obtain (see Proposition 5.3) continuous operators on H s spaces. We define…”