ABSTRACT
The local well-posedness proved in the previous chapter holds for rather general situations, i.e. anisotropic media and for one or more space di mensions. As it is well known for evolution equations, the global in time existence of smooth solutions cannot be expected in general. Also for sys tems with some kind of dissipation as it is given here by heat conduction, the global existence might fail to hold; moreover in our case, there is the hyperbolic part that is even more likely to produce singularities in finite time. In general, the development of singularities depends on the space di mension and on the special type of nonlinearity considered. A general tool to obtain global solutions in some cases is to exploit the decay of solutions as was described earlier for the linearized systems, cf. [103] for a survey.
