A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the <i>H</i>-theorem are essential elements of our analysis.

As a typical example of hyperbolic conservation laws, the system of Euler equations representing the conservation of mass, momentum and energy has three basic wave patterns. They are two nonlinear waves called shock and rarefaction wave, and one linearly degenerate wave called contact discontinuity. These basic wave patterns have their counterparts both in other physical models for gas motions in equilibrium involving viscosity and heat conductivity, and gas motion in non-equilibrium. The stability of these basic wave patterns in the system of Navier-Stokes equations and the Boltzmann equation has been an active research topic. Even though the stability of the two nonlinear wave patterns has been extensively studied, the stability of the linearly degenerate contact wave was not solved until recently. In this paper, we will briefly present our recent results in [32] on the stability of the contact wave patterns for both the Navier-Stokes equations and the Boltzmann equation.

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For a bipolar hydrodynamic model of semiconductors in the form of EulerPoisson equations with Dirichlet or Neumann boundary conditions, in this paper we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted L^1 space, a faster and optimal convergence rate is also given.