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In the paper under consideration we study several model examples of degenerate linear hyperbolic operators - mainly of second order, which are not (micro) locally solvable and do not propagate singularities, i.e. possesses solutions with (micro) locally smooth right-hand sides and stationary singularities on characteristic rays of the principal symbols. These two effects contrast with the properties of the strictly hyperbolic operators local solvability (including of the Cauchy problem) and propagation of singularities. As our paper is a survey on some results of the general theory of ψdo with real values principal symbols we have illustrated it by examples, not by long and complicated theorems that can be found in the corresponding monographs and papers. In some cases of degenerate hyperbolic operators propagation of singularities including branching ones is possible and we discuss it at the end. [ABSTRACT FROM AUTHOR]

We study the existence of ground states for the Beris-Edward system. Under reasonable symmetry assumption we prove the existence, regularity and decay property of the ground states. Using a linearization around these ground state solutions we study the corresponding Cauchy problem with small initial data. Some dispersive estimates for the perturbed linearized problem are established also. [ABSTRACT FROM AUTHOR]