In this thesis we consider the Cauchy problem for general higher order constant coefficient strictly hyperbolic PDEs with lower order terms and show how the behaviour of the characteristic roots determine the rate of decay in the associated Lp-Lq estimates. In particular, we show under what conditions the solution behaves like that of the standard wave equation, the wave equation with dissipation or the Klein-Gordon equation. We explain the various factors involved, such as the presence of multiple roots, the size of the sets of multiplicity and the order with which characteristics meet the real axis, yield different rates of decay. As an example, we show how the results obtained can be applied to the Fokker-Planck equation. In the second part, we derive Lp-Lq estimates for wave equations with a bounded time dependent coefficient. A classification of the oscillating behaviour of the coefficient is given and related to the estimate which can be obtained.Imperial Users onl
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