The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to time

Abstract

We discuss the local existance and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones, then we shall prove them by use of Tanabe-Sobolevski's method.

On hyperbolic systems with time-dependent Hölder characteristics

“…This allows us to implement the ideas developed in [7] for scalar equations, where the reduction in a scalar equation to a Sylvester form system was performed. To summarise our result, here we first note that combining the results in [14,15] we already know that the Cauchy problem (1) is well-posed in the Gevrey class γ s , with…”

“…For example, this is an improvement of Yuzawa's and Kajitani's order (5) from [14,15]. See Remark 2.3 for more details.…”

“…By choosing κ small enough we can conclude that |V (t, This is an improvement in terms of Gevrey order of Yuzawa's and Kajitani's result in [14,15]. Indeed, Yuzawa first for t-dependent systems (without lower-order terms) in [15] …”

“…Hyperbolic systems of the form (1) have been also investigated (see, e.g. Garetto [6], Kajitani and Yuzawa [14] and Yuzawa [15]). The main new idea behind this paper enabling us to obtain an improvement in the wellposedness results for the system in (1) is the transformation of the system (1) to a larger system which, however, enjoys the property of being in block Sylvester form.…”

On $$C^\infty $$ C ∞ well-posedness of hyperbolic systems with multiplicities

“…Ultradistributional well-posedness. For convenience of the reader we recall Yuzawa's well-posedness result proven in [18]. We begin by introducing for ρ > 0 and s > 1, the space H l Λ(ρ,s) of all f ∈ L 2 (R n ) such that…”