  Here we investigate global strong solutions for a class of partially
dissipative hyperbolic systems in the framework of critical homogeneous Besov
spaces. Our primary goal is to extend the analysis of our previous paper [10]
to a functional framework where the low frequencies of the solution are only
bounded in L p-type spaces with p larger than 2. This enables us to prescribe
weaker smallness conditions for global well-posedness and to get a more
accurate information on the qualitative properties of the constructed
solutions. Our existence theorem in particular applies to the multi-dimensional
isentropic compressible Euler system with relaxation, and provide us with
bounds that are independent of the relaxation parameter. As a consequence, we
justify rigorously the relaxation limit to the porous media equation and
exhibit explicit rates of convergence for suitable norms, a completely new
result to the best of our knowledge.
