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

Kunihiko Kajitani, Yasuo Yuzawa (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We discuss the local existence 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 and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.

The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

Neil Watson (2000)

Annales Polonici Mathematici

Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip X = n × ] 0 , a [ . We prove a representation theorem for an arbitrary C 2 , 1 function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with C coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...

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