This work is concerned with the proof of decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation . The coefficient consists of an increasing smooth function and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).
In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.
In this paper we prove the -well posedness of the Cauchy problem for quasi-linear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0,T] and smooth in x ∈ ℝⁿ.
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