On the Diffusive Profiles for the System of Compressible Adiabatic Flow through Porous Media

We study the Cauchy problem for the system of one dimensional compressible adiabatic flow through porous media and the related diffusive problem. We introduce a new approach which combines the usual energy methods with special L1 -estimates and use the weighted Sobolev norms to prove the global existence and large time behavior for the solutions of the problems. The asymptotic states for the solutions are given by either stationary solutions or similarity solutions depending on the behavior of the initial data when |x|→ ∞. Our estimates provide asymptotic time decay rates.