This article is intended to present a construction of structural representations of solutions to the Cauchy problem for wave equations with time-dependent dissipation above scaling. These representations are used to give estimates of the solution and its derivatives based on L q (R n ), q 2.The article represents the second part within a series. In [Jens Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations 222 (2) (2006) 487-514] weak dissipations below scaling were discussed. IntroductionThe purpose of this paper is to investigate asymptotic properties of solutions to the Cauchy problem for a wave equation with time-depending dissipationis assumed to be positive and satisfy a lower bound of the form tb(t) → ∞ as t → ∞. As usual we denote D = −i∂ and by = j ∂ 2 j the Laplacian. We refer to [13] for an exposition of results and for the classification of time-dependent dissipation terms. In [18] it was shown that, roughly speaking, under the non-effectivity assumption b(t) = O(t −1 )

The goal of this article is to construct structural representations of the solutions to Cauchy problems for weakly dissipative wave equations below scaling and to deduce estimates of the solution and its energy based on L q (R n ), q 2. Furthermore, the sharpness of the obtained estimates is discussed. IntroductionCauchy problems for wave equations with weak dissipationis assumed to be positive and tends to zero as t tends to infinity, provide an important model problem for the study of asymptotic behaviours and the influence of lower-order terms on them. As usual we denote D = −i* and = j * 2 x j . We refer to [RW05] for an exposition of results and for the classification of * Fax: +49373 139442.

SUMMARYWe consider the Cauchy problem for the weakly dissipative wave equationparameterized by ¿0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain Lp-Lq estimates for the solution and for the energy operator corresponding to this Cauchy problem.Especially for the L 2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of and that = 2 is critical.

In this paper we prove L p Fourier multiplier theorems for invariant and also noninvariant operators on compact Lie groups in the spirit of the well-known Hörmander-Mikhlin theorem on R n and its variants on tori T n . We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.

In this paper we give several global characterisations of the Hörmander class m (G) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.